v = 2Io + rv. (2.2) where Vo is the ..... ronments such as EMTP [20], SPICE [21], and SABER [22] may provide rapid prototyping and time-domain modeling ...
STABILITY CRITERIA FOR AC POWER SYSTEMS WITH REGULATED LOADS
A Thesis Submitted to the Faculty of Purdue University by Mohamed Belkhayat
In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 1997
- ii -
To Jackie, Noah, and Nora
My mercy pervades everything Quran
- iii -
ACKNOWLEDGMENTS
I wish to thank a few key people who made this thesis possible. David Clayton of Naval Sea Systems Command has encouraged this research since its inception and has been unflinching in his support throughout the three-year development of this thesis. Dana Delisle and Chip Tucker, fellow engineers at the Naval Surface Warfare Center, the old David Taylor Laboratory, started the stability work many years before me and developed dc system stability testing devices which have lent great insight in the development of the present work in AC system stability criteria. Brain-storming sessions with Roger Cooley of NSWC have always sparked unforeseen and fruitful directions not only in the present work but also in the area of nonlinear systems stability. I thank Mr. J. P. Goodman of NSWC/Annapolis for the historical cases related to ac stability on US aircraft carriers. Finally, I thank the committee and especially the chair, Prof. Wasynczuk, for his guidance, his emphasis on rigor, consistency, and above all his patience. My thanks also go to all fellow graduate students in the Energy Sources and Systems area whose encouragements have made the task a little easier.
- iv -
TABLE OF CONTENTS
Page LIST OF TABLES ............................................................................................................ vi LIST OF FIGURES .......................................................................................................... .ix ABSTRACT .....................................................................................................................xiii INTRODUCTION .............................................................................................................. 1 Stability of a Stand-Alone Converter ..................................................................... 2 DC System Stability ............................................................................................... 7 STABILITY CONSIDERATIONS IN THREE-PHASE SYSTEMS .............................. 11 Time-Domain Model of Ideal Three-Phase CPL ................................................. 11 Time-Domain Model of CPL with Input Filter ..................................................... 14 Frequency-Domain Model of CPL........................................................................ 16 Generalized Nyquist Criterion............................................................................... 18 SYSTEM MODELING .................................................................................................... 23 Synchronous Machine Model ............................................................................... 27 Induction Motor Model ........................................................................................ 30 Converter and Inverter Models ............................................................................. 32 Field-Oriented Controlled Induction Motor ......................................................... 36 Component Model Interconnection and Interface Compatibility.......................... 38 Synchronous Machine and Converter System ...................................................... 39 Inverter and Converter System .............................................................................. 40 Source and Load Impedance Matrix Realizations................................................. 41 Example System ................................................................................................... 47 SMALL-GAIN STABILITY CRITERIA ......................................................................... 65 Small-Gain Stability Criteria ...................................................................................... 65 Ideal Three-Phase CPL with Input Filter..................................................................... 69
-vPage Stability Using Small-Gain Criteria ................................................................................. 69 Conservatism .............................................................................................................. 71 Stability of Non-Ideal System Using Small Gain Criteria .......................................... 73 THREE-PHASE FILTER DESIGN BY SINGULAR VALUES ..................................... 77 First-Order RL Filter ................................................................................................... 77 Second-Order RLC Filter ............................................................................................ 71 RLC Filter with Damping Network ............................................................................ 81 STABILITY ANALYSIS OF A HYBRID AC/DC SYSTEM ........................................ 87 System Description ..................................................................................................... 87 System Topology......................................................................................................... 87 System Model.............................................................................................................. 89 System Parameters ...................................................................................................... 89 Generalized Nyquist and Small Gain-Criteria Analysis.............................................. 92 Existence of Minimal Realizations At Arbitrary Interfaces ..................................... 102 Stabilizing controls.................................................................................................... 104 Effect of Phase-Lock Loop Control ......................................................................... 109 Effect of Prime Mover .............................................................................................. 114 CONCLUSION AND RECOMMENDATIONS ............................................................ 115 Conclusion ................................................................................................................ 115 Recommendations .................................................................................................... 115 LIST OF REFERENCES ............................................................................................... 117 VITA .............................................................................................................................. 121
- vi -
LIST OF TABLES
Table
Page
3.1
Eigenvalues of load subsystem (ZL) at Interface 1................................................ 56
3.2
Eigenvalues of source subsystem (Ys) at Interface 1............................................. 56
3.3
Overall system Eigenvalues for stable operating point ........................................ 56
3.4
Eigenvalues of load subsystem (ZL) at Interface 1 ............................................... 62
3.5
Eigenvalues of source subsystem (Ys) at Interface 1............................................. 62
3.6
Overall system Eigenvalues for unstable operating point .................................... 62
3.7
Eigenvalues of load subsystem (ZL) at Interface 2 ............................................... 64
3.8
Eigenvalues of source subsystem (Ys) at Interface 2............................................. 64
3.9
Overall system Eigenvalues for stable operating point ........................................ 64
6.1
Synchronous machine parameters ........................................................................ 89
6.2
Induction motor parameters .................................................................................. 89
6.3
System and subsystem eigenvalues at Interface 1 for stable operating point........ 98
6.4
System and subsystem eigenvalues at Interface 2 for stable operating point........ 99
6.5
System and subsystem eigenvalues at Interface 3 for stable operating point...... 100
6.6
System and subsystem eigenvalues at Interface 2 for unstable operating point..................................................................................................... 101
- vii -
Table 6.7
Page System and subsystem eigenvalues with nonlinear stabilizing controller present................................................................................................ 108
- viii -
LIST OF FIGURES
Figure
Page
1.1
DC-DC converter with input filter ......................................................................... 4
1.2
Equivalent Thevenin circuit for DC-DC converter with input filter ...................... 4
1.3
Allowed area by small gain criterion....................................................................... 7
1.4
DC system with self-regulated converter loads....................................................... 8
1.5
Allowed region by the opposing-argument criterion.............................................. 9
1.6
Allowed region in the 6-dB 30-degree criterion................................................... 10
2.1
High-efficiency three-phase converter with PWM controls................................. 12
2.2
Three-phase CPL with input filter......................................................................... 14
2.3
Large-displacement equivalent circuit of three-phase CPL with filter in the synchronous reference frame........................................................................... 15
2.4
Small-displacement equivalent circuit of three-phase CPL with input filter ........ 16
2.5
MIMO feedback system ........................................................................................ 18
2.6
Equivalent feedback control block diagram of three-phase CPL with input filter...................................................................................................... 19
- ix Figure
Page
2.7
Stable loci for P0 = 20 kW, L = 77 µH, C = 300 µF, r = 0.1Ω , V0q = 367V, Vod = 0V ............................................................................................................... 21
2.8
Unstable loci for P0 = 200 kW, L = 77 µH, C = 300 µF, r = 0.1Ω , V0q = 367V, Vod = 0V ............................................................................................................... 21
2.9
System response to step increases in real power of ideal three-phase CPL .......... 23
3.1
Block diagram of synchronous machine model ................................................... 30
3.2
Block diagram of induction motor model ............................................................. 32
3.3
Inverter and converter configuration ................................................................... .33
3.4
Hysteresis current control..................................................................................... 34
3.5
Synchronous PI regulator for converter and inverter control .............................. .35
3.6
Bounded proportional current control .................................................................. 35
3.7
Block diagram of converter-inverter model .......................................................... 36
3.8
Indirect field-oriented control of induction motor ................................................ 38
3.9
Model input-output configuration for synchronous machine and converter system................................................................................................................... 39
3.10
Model input-output configuration for system with phase-lock loop .................... 40
3.11
Model input-output configuration for inverter-converter system with dc interface ................................................................................................... 41
3.12
Thevenin partitioning of system at two different interfaces.................................. 43
3.13
System Thevenin equivalent at Interface 1........................................................... 43
3.14
One-line diagram of system studied ...................................................................... 48
3.15
Comparison of system responses calculated using detailed and averaged computer models ............................................................................ 50
3.16
System response with nominal controller parameters ........................................... 52
-x-
Figure
Page
3.17
Induction motor response with nominal controller parameters ............................ 53
3.18
Characteristic loci at Interface 1 for stable operating point................................... 54
3.19
Enlargements of characteristic loci for stable system around (-1+j0) ................... 55
3.20
System response with modified controller parameter ....................................... 58
3.21
Induction motor response with modified controller parameter ........................... 59
3.22
Characteristic loci at Interface 1 for modified controller parameter .................... 60
3.23
Enlargements of characteristic loci around (-1+j0) .............................................. 61
3.24
Characteristic loci at Interface 2 for stable system ............................................... 63
4.1
Small-gain measures for P0 = 20 kW, L = 77 µH, C = 300 µF, r = 0.1Ω , V0q = 367V, Vod = 0V .......................................................................................... 70
4.2
Small-gain measures for P0 = 200 kW, L = 77 µH, C = 300 µF, r = 0.1Ω , V0q = 367 V, Vod = 0V ......................................................................................... 70
4.3
Small-gain measures Stable loci for P0 = 25 kW, L = 77 µH, C = 300 µF, r = 0.1Ω , V0q = 367V, Vod = 0V.......................................................................... 71
4.4
Gain margins based on different measures for ideal three-phase CPL with input filter ............................................................................................ 72
4.5
Small-gain measures at Interface 2 for example system under stable conditions ............................................................................. 74
4.6
Zoom A for small-gain measures at Interface 2 for example system under stable conditions ............................................................................. 74
5.1
Plots of σ (Zqde) and |Zs| for a first-order RL filter. R = 1m Ω, and L = 100 µH ................................................................................... 79
5.2
Plots of σ (Zqde) and |Zs| for a first-order RL filter. R = 1m Ω, C = 1 mF, and L = 100 µH ............................................................... 82
5.3
Single-phase RLC circuit with damping branch .................................................. 83
- xi -
Figure
Page
5.4
Plots of σ (Zqde) (solid line) and |Zs| (dashed line) for a second order RLC with and without damping. R = 100m Ω, C = Cd = 300 µF, and L = 77 µH. ....... 84
5.5
Zoom around the resonance point for plots of σ (Zqde) (solid line) and |Zs| (dashed line) for a second order RLC with and without damping. R = 100m Ω, C = Cd = 300 µF, and L = 77 µH .......................................................................... 84
5.6
Small-gain measure for ideal three-phase CPL with damped input filter and P set to 200 kW ............................................................................ 85
5.7
Eigen-loci of example system return ratio at Interface 2 with damped filter capacitor.................................................................................. 86
5.8
Small-gain measures for example system return ratio at Interface 2 with damped filter capacitor.................................................................................. 86
6.1
High-frequency AC/DC System........................................................................... 88
6.2
Input-output block diagram of system model........................................................ 90
6.3
Excitor/regulator model........................................................................................ 91
6.4
Detailed and averaged model response to increased torque commands............... 96
6.5
Detailed and averaged ac voltages and currents.................................................... 97
6.6
Eigen-loci for stable operating point at Interface 1 .............................................. 98
6.7
Small-gain measures for stable operating point at Interface 1 ............................. 98
6.8
Eigen-loci for stable operating point at Interface 2 .............................................. 99
6.9
Small-gain measures for stable operating point at Interface 2 ............................. 99
6.10
Eigen-loci for stable operating point at Interface 3 ............................................ 100
6.11
Small-gain measures for stable operating point at Interface 3 .......................... 100
6.12
Eigen-loci for unstable operating point at Interface 2 ........................................ 101
6.13
Small-gain measures for unstable operating point at Interface 2 ....................... 101
- xii -
Figure
Page
6.14
Nonlinear stabilizing controller........................................................................... 105
6.15
System response with and without nonlinear stabilizing controller.................... 107
6.16
Eigen-loci at Interface 2 with nonlinear stabilizing controller present .............. 108
6.17
Small-gain measures at Interface 2 with nonlinear stabilizing controller present ............................................................................................... 108
6.19
Detailed model of phase-lock-loop ..................................................................... 109
6.20
Detailed and averaged system response with PLL included ............................... 111
6.21
Phase error response with sample and hold model included............................... 112
6.22
Phase error response with sample and hold bypassed ......................................... 112
6.23
Eigen-loci at Interface 2 with PLL included ....................................................... 113
6.24
Small-gain measures at Interface 2 with PLL included....................................... 113
6.23
Eigen-loci at Interface 2 without PLL ................................................................ 113
6.24
Small-gain measures at Interface 2 without PLL ............................................... 113
- xiii -
ABSTRACT
Belkhayat, Mohamed. Ph.D., Purdue University, December, 1997. Stability Criteria For AC Systems With Regulated Loads. Major Professor: Oleg Wasynczuk.
High-bandwidth regulated converter loads, or constant-power loads (CPLs) exhibit a negative incremental input resistance within the regulation bandwidth of the power converter. Distributed power systems which include a large percentage of CPLs and contain energy storage devices may be susceptible to potentially destabilizing interactions of these elements. For the stand-alone dc-dc converter, and in distributed dc systems, design requirements that guarantee stability have been previously advanced and are extensively used in the design and specification of these systems. However, prior to this research, design-oriented stability criteria have not been developed for distributed ac and hybrid ac/dc systems with CPLs. In this research, a mathematical framework based on the generalized Nyquist criterion, reference frame theory, and multivariable control is set forth for dynamic stability assessment of distributed ac systems. Numerical and analytical techniques are established for determining the frequency-dependent impedance characteristics of representative sources and loads in the appropriate frame of reference. Moreover, design-oriented stability criteria analogous to those used in dc systems are derived.
The purpose of these criteria is to ensure a safe stability margin without
imposing undue restrictions on component designers and system integrators. Finally, the advanced criteria are used to assess the stability of several example systems and to evaluate different stabilizing control strategies, including passive and active approaches.
1. INTRODUCTION Some of the earliest documented cases of instabilities caused by negative incremental input resistance are the 400-Hz electric power systems on board US aircraft carriers including USS America and USS Kitty Hawk. In 1964, the Navy installed a large number of Line Voltage Regulators (LVRs) at the Aircraft Electric Service Stations (AESS) to eliminate voltage losses due to cable length. The LVRs consisted of saturable core transformers with solid-state controls that were designed to provide voltages regulated within ± 0.5%. Upon energization of all the LVRs, the 400-Hz ship service power system, including the generator, went into a sustained oscillation, i.e., a limit cycle. The Navy decreased the number of LVRs that may operate simultaneously, and redesigned them to avoid such instabilities. The redesigned LVRs had lower regulation bandwidths which decreased the effect of the negative incremental input resistance. The system designers traded regulation bandwidth for a safer stability margin [1]. More recently, the designers of power systems on various platforms, such as NASAs Space Station Freedom [2], US Air Force Copper Bird [3], and the Navys Integrated Power System IPS [4] have all paid close attention to the negative incremental input resistance of self-regulated converter loads and its effect on system stability. In a self-regulated converter load, the regulator (e.g. load voltage) is included as part of the converter controls. For the IPS system, very promising nonlinear controls have been shown [5] to alleviate the trade off between stability margin and regulation bandwidth. The rapid advances in power IGBTs (Insulated Gate Bipolar Transistors) and MCTs (MOS-Controlled Thyristors) have allowed the development of solid-state power converters which possess the desirable qualities of high-bandwidth regulation and high power conversion efficiencies. Namely, 1200-V and 1600-V IGBTs are capable of switching rates that exceed 100 kHz which allows the regulation bandwidths of power converters to easily reach the 1-kHz range. Also, the presently available soft switching and control techniques allow
-2power conversion efficiencies that can exceed 95%. Although very desirable, these qualities may cause system instabilities and eventually system failure if not prevented. With a constant output current, high-bandwidth voltage regulation gives rise to constant output power. When combined with high conversion efficiency, this results in essentially constant input power which, in turn, translates into a negative incremental input resistance. Due to the constant input power characteristic, these converters are commonly called Constant Power Loads (CPLs). When the negative incremental resistance is combined with the filters that are usually used to block the effects of the high-frequency switching, negatively damped oscillations may result [6]. The stability of the power converter as well as the system to which it is connected depends on the proper damping of the negative incremental resistance due to the CPL characteristic. For the stand-alone DC-DC converter, the Nyquist criterion has been applied to develop an impedance ratio condition that ensures converter stability during small disturbances. For distributed dc systems with the majority of loads consisting of CPLs, similar but less restrictive criteria have been imposed on the source-to-load impedance ratio to ensure smalldisturbance system stability. Prior to this research, comparable stability criteria have not been developed for distributed ac systems with a high percentage of CPLs. This thesis develops stability criteria for ac systems that not only include CPLs but also rotating machines such as synchronous generators and induction motors. Due to the time-varying and periodic nature of these systems, reference frame theory [7] will be applied so that linear time-invariant analysis tools, such as the generalized Nyquist criterion can be applied to investigate small-disturbance stability. The criteria will be optimized to establish a safe stability margin and to allow maximum system design flexibility. A typical ac system with self-regulated loads will be modeled and simulated to evaluate the criteria. The results will be compared with an eigenvalue-based analysis and detailed time-domain simulations. 1.1
Stability of a Stand-Alone Converter
For the stand-alone converter, a small gain criterion was developed in [6] to ensure stability when an input filter is to be designed for the closed-loop converter. The self-regulated
-3converter gives rise to a negative incremental input resistance that, when combined with an input filter, may cause negatively damped oscillations. Since this criterion is fundamental to future developments, a brief derivation will be given here. The negative input resistance can be shown to be the result of two properties of power converters, namely the high regulation-bandwidth and the efficiency of conversion as shown in the following analysis. It is assumed that the output voltage vL of the converter shown in Figure 1.1 is maintained constant by the action of a regulator and a PWM controller. If the load current does not change, then Pout is constant whereupon Pin = µPout ≈ Pout = Constant
(1.1)
The efficiency µ is close to unity in modern converters. The differential change in the input power may be expressed dPin = 0 = idv + vdi
(1.2)
The input incremental resistance rin may be defined as rin =
dv v =− di i
(1.3)
Another expression for incremental resistance can be developed from the fact that the input current to the converter will be inversely proportional to the input voltage since Pin is constant. In particular i=
Pin v
(1.4)
Substituting (1.4) into (1.3)
v2 dv ri = =− di Pin
(1.5)
which is always negative for positive Pin . When combined with an input filter, this negative incremental resistance may cause negatively damped oscillations. Therefore, a stability analysis is necessary when a filter is to be added to the converter. The converter with input filter may be represented as shown in Figure 1.1.
-4-
Pin
Pout
i + -
voc
vs
=
+ v -
hf vs
Filter
Zs
vL = hc v Converter
+ R vL -
Zi
Figure 1.1 DC-DC converter with input filter. The filter may be characterized by its voltage transfer function hf when unloaded, and the converter is characterized by its voltage transfer function hc while loaded with some resistance R. Then the transfer function for the overall system can be derived by using the equivalent Thevenin model shown in Figure 1.2 as vL hf hc = vs 1 + Zs /Zi
(1.6)
where Zs is the output impedance of the filter and Zi is the input impedance of the converter under load.
Zs + -
hf vs
+ Zi v = vL / hc -
Figure 1.2 Equivalent Thevenin circuit for DC-DC converter with input filter. This transfer function is important because the overall system dynamics are described in terms of the individual subsystem characteristics. Hence, stability conditions on this transfer function may be directly converted into component design requirements. The previous transfer function may also be obtained from a linearized state-space averaged [8] model of the converter as follows vL = h(s) = C [sI − A]−1 B vs
(1.7)
Although stability conditions may be imposed on this transfer function in terms of the
-5eigenvalues of A, there is no direct relationship between the system eigenvalues and the component characteristics. Questions regarding observability and controllability which become important at the system level were not treated in [6]. In order to address these questions and to provide a more general framework for stability analysis, more specific definitions for stability such as Bounded Input Bounded Output (BIBO) and Bounded Input Bounded State (BIBS) stability will be stated here from [10]. Definition: A system is BIBO stable if, for each admissible bounded input u(t) (i.e., for ku(t)k∞ < ∞) the response y(t) is bounded (i.e., ky(t)k∞ < ∞). Here u(t) is an n × 1 vector-valued function, y(t) is an m×1 vector-valued function, ku(t)k∞ =max kui (t)k∞ , i
and kui (t)k∞ =sup |ui (t)| . t
The previous definition is valid for Multiple Input Multiple Output (MIMO) systems as well as Single Input Single Output (SISO) systems. BIBO stability is concerned with external behavior; if all bounded inputs cause bounded output responses, the system will be termed BIBO stable regardless of the internal states of the system. BIBO stability can exist even if some of the internal system states are unstable [11]. BIBO stability for the system in Figure 1.2 can be determined from a simple examination of the SISO transfer function h(s) in (1.7); the system is BIBO stable if and only if all the poles of h(s) are in the open Left Hand Plane (LHP). Hence, the converter will be BIBO stable if the poles of vL /vs are in the open LHP. Consequently, the numerator product hf hc must have all associated poles in the open LHP and the denominator zeros must also be in the open LHP. The condition on the numerator transfer function may be regarded as BIBO stability of the open-loop hf hc . This is easily satisfied since the filter transfer function hf is always BIBO stable and the closedloop converter transfer function hc is designed a-priori to be stable. Hence, the denominator zeros become the determining factor for BIBO stability of the filter-converter system. This condition may be verified by Nyquist stability theory, i.e. the denominator 1 + Zs /Zi will have all zeros in the open LHP if the number of counter-clockwise encirclements around (−1 + j0) of the Nyquist map of Zs /Zi is equal to the number of RHP poles of Zs /Zi . If Zs /Zi has no RHP poles, the number of encirclements must be zero for stability. A morethan-necessary but simpler condition, given that the filter and converter are BIBO stable, is to restrict the Nyquist contour of Zs /Zi within the unit circle; i.e.,
-6-
¯ ¯ ¯ Zs ¯ ¯ ¯ −1 Zck
for − ∞ < ω < ∞
(1.13)
Since the above condition avoids the (−1 + j0) encirclement, stability is guaranteed and flexibility in design is increased so that the source and load impedances are allowed to overlap.
Im Allowed region -1
Re
Figure 1.5 Allowed region by the opposing-argument criterion.
- 10 In order to provide more flexibility in the design of converter loads, other allowable regions on the Nyquist plane have been advanced. Namely, the 6-dB 30o and 3-dB 60o infinite cones have been used as shown in Figure 1.6. The 6-dB 30o forbidden region criterion has already been applied to such systems as Navy shipboard dc distribution. Im Allowed Region
-1
Re
Figure 1.6 Allowed region in the 6-dB 30-degree criterion. For ac systems, however, none of the previous criteria are directly applicable since ac systems with rotating machines and power converters are inherently time-varying systems. Historically, ac systems were largely studied for frequency stability especially in the utility industry [15]. A power grid may have hundreds of generators operating at 60 Hz and therefore synchronism is of utmost importance. In the systems studied here, the number of generators is usually one or two and synchronism issues arise only in certain isolated cases. Such systems exist on board most aircraft and marine vessels. Hence synchronism will not be considered as the main thrust of this thesis. Other areas such as Sub-Synchronous Resonance (SSR) caused by series capacitors in transmission lines were studied in [16]. Solutions based on current injection techniques were offered and the stability analysis was based on generalized Nyquist theory. Since there are no series capacitors in the systems to be studied, SSR will not be analyzed. Recently, voltage stability criteria for utility type networks were set forth based on a frequency-domain characterization of constituent subsystems in [17, 18] . However, these criteria require the impedance matrices to be skew symmetric and to have equal elements in the diagonals which is not always the case, especially in the systems at hand. Therefore, new criteria are needed; stability criteria will be developed for these systems in the following chapters.
2. STABILITY CONSIDERATIONS IN THREE-PHASE SYSTEMS For dc systems with a large number of self-regulated converter loads, it has been common to represent these CPLs as voltage-dependent current sources whose current magnitude is given by i=
P v
(2.1)
where P is the rated power of the converter and v the instantaneous input voltage. This nonlinear model is a good approximation at frequencies that are below the regulation bandwidth of the converter. It also provides a linear model through a Taylor series expansion of the previous expression. In particular i(v) ' 2
P P − 2 v = 2Io + rv Vo Vo
(2.2)
where Vo is the operating point voltage and Io is the operating point current. Note that this linear model accurately predicts the negative input resistance of the model as was shown in the introduction. Since ac systems are inherently time-varying systems, and the voltages cross zero periodically at steady state, the above model is not applicable. Reference frame theory [7] may be used to circumvent this problem. In the next section, a model is developed for the three-phase CPL with variables expressed in the synchronous reference frame which sets the stage for linearization using Taylor series. 2.1
Time-Domain Model of Ideal Three-Phase CPL
In the following development, it is assumed that the three-phase converter shown in Figure 2.1 has a high-bandwidth regulation and high-efficiency conversion. The instantaneous power in terms of the line-to-neutral voltages and currents may be expressed as T P (t) = va ia + vb ib + vc ic = vabc iabc
(2.3)
- 12 If the fundamental frequency of the voltages is ωe, a transformation to the synchronous reference frame [7] can be defined as
cos θ cos (θ − 2π/3) cos (θ + 2π/3) 2 Kes = sin θ sin (θ − 2π/3) sin (θ + 2π/3) 3 1/2 1/2 1/2
Pin
va vb vc
+
iabc
(2.4)
iout Pout
vout iabc
vout PWM CONT
vref
Figure 2.1 High-efficiency three-phase converter with PWM controls. where θ =
Rt 0
ωe (ξ)dξ + θ(0). Then P (t) can be written as ¡ ¢T ¡ e−1 e ¢ e P (t) = Ke−1 Ks iqdo s vqdo
(2.5)
e and ieqdo are the transformed voltages and currents. After some algebra, the where vqdo
previous expression becomes 3 eT e iqdo P (t) = vqdo 2
(2.6)
which represents an expression for the instantaneous power in terms of the transformed variables. Since there is no neutral in the converter considered, the sum of the currents is zero. Therefore the zero-sequence current io is zero whereupon P (t) =
¢ 3¡ e e vq iq + vde ied 2
(2.7)
- 13 The instantaneous reactive power Q(t) is defined such that P 2 (t) + Q2 (t) =
¢¡ ¢ 9 ¡ e2 e2 vq + vde2 ie2 q + id 4
(2.8)
It can be shown that Q(t) which satisfies (2.8) can be expressed as Q(t) =
¢ 3¡ e e vq id − vde ieq 2
(2.9)
The expressions for the instantaneous power in the synchronous reference frame, given in (2.7) and (2.9) are valid for balanced as well as unbalanced load currents. The only assumption is that the zero-sequence current is zero, i.e., the neutral current is zero. For balanced conditions, P (t) and Q(t) represent the conventional real and reactive powers which are constant at steady state. From (2.7) and (2.9), the q and d-axis currents may be expressed in terms of P and Q as 2 P vqe − Qvde ¡ ¢ 3 vqe2 + vde2 2 P vde + Qvqe ¡ ¢ = 3 vqe2 + vde2
ieq =
(2.10)
ied
(2.11)
This consists of a nonlinear model that has a form similar to the dc case expressed in (2.1) and the expressions for the currents are valid regardless of the form of the applied voltages. Since the voltages are no longer periodic with zero-crossings at steady state, the model can be linearized using the two-variable Taylor series up to the second term, e e e e e e + OF(v )|Ve v ee = Ieqd + YqdL eqd ) ' F(vqd )|Vqd v ieqd = F(vqd qd qd qd
e e e eqd where v = vqd − Vqd . If P and Q are constant, the small-displacement model is e eie = Y e v qd qdL eqd
e where YqdL is given by e YqdL
=
2
¯ e ¯2 ¯ 3 ¯Vqd
·
P − 3Vqe Iqe −Q − 3Vde Iqe Q − 3Vqe Ide P − 3Vde Ide
¸
=
·
yqq yqd ydq ydd
¸
(2.12)
(2.13)
(2.14)
¯ e ¯2 ¡ ¢ ¯V ¯ = Vqe2 + V e2 , and Vqe , V e , Iqe , I e are the operating-point variables. This gives a qd d d d linear model of the three-phase CPL in terms of the nominal values of the constant power
- 14 e and qd-components of the currents and voltages. The matrix YqdL is not skew-symmetric
nor are the diagonal entries equal as was assumed in [17]. Now, frequency analysis can be e is readily available. The following applied to the model since the input admittance YqdL
section presents a frequency analysis of an ideal self-regulated power converter with an input filter and a three phase source. 2.2
Time-Domain Model of CPL with Input Filter
In order to illustrate the use of generalized Nyquist methods, an ideal three-phase CPL with input filter is subsequently investigated in this chapter. In this section, a time-domain model is developed for the following system:
vas vbs vcs
-
+
r ia ib
+
ic -
+
L
va
C
vb vc
Three-Phase CPL
Figure 2.2 Three-phase CPL with input filter. The state equations for the inductor currents can be written in vector form as diabc = −L−1 riabc − L−1 vabc + L−1 vabcs dt
(2.15)
where r, L,and C are the resistance, inductance, and capacitance diagonal matrices respectively. Similarly, for the voltages across the capacitors dvabc = C−1 iabc − C−1 iabcL dt
(2.16)
Taking the source voltages and the load currents as inputs to the system, the previous equations can be expressed in a more compact form ¸ · ¸ · ¸ · −1 dxabc d iabc −L−1 r −L−1 L vabcs = = x+ C−1 0 −C−1 iabcL dt dt vabc
(2.17)
Even though the present system has a constant state matrix A, this is generally not the case for ac systems. For example, a synchronous machine will have time-varying inductances
- 15 [7] when modeled in physical variables as done in the previous equation, and the load will have time-varying switching functions. Also, since the CPL model is expressed in the synchronous reference frame, the input filter section has to be transformed accordingly. With the same transformation used in (2.4), the previous state equations can be transformed as follows xeqdo
=
·
Kes 0 0 Kes
¸
xabc = Kexxabc
(2.18)
Since there are no zero-sequence currents in the example network, the transformed system can be expressed as ¸ · · −1 e ¸ dxeqd L vqds −L−1 r − T −L−1 e xqd + = e C−1 −C−1 YqdL −T −2C−1 Ie0qdL dt which is a 4 × 4 system and T is defined as T=
dKe−1 s Kes dt
=
·
0 ωe −ωe 0
¸
(2.19)
(2.20)
with the zero-sequence contribution dropped from the matrices. The load current ieqdL was replaced by the linear expression derived in the previous section and Ie0qdL represents the nominal load current. The previous equations are linear with constant coefficients. This model can be expressed in equivalent-circuit form as shown in Figure 2.3. ω e Lide
L
r
+ + -
e vqs
e vds
e C ω e Cvd
L
r + -
-
iqe
ide
e iqL
+ vqe
C ω e Cvqe
3(vqe2 + vde2 )
-
ω e Liqe - +
2( Pvqe − Qvde )
e idL
+ vde
-
2( Pvde + Qvqe ) 3(vqe 2 + vde2 )
Three-Phase CPL Figure 2.3 Large-displacement equivalent circuit of three-phase CPL with input filter expressed in the synchronous reference frame.
- 16 The linearized version of the previous circuit is obtained from the derivation given in the previous section as shown in Figure 2.4. At this point, the stability of the system can be studied by conventional techniques such as eigenvalue analysis. However, in order to use Nyquist, Bode, and root locus techniques, a frequency-domain model is required. Since the systems at hand are inherently MIMO systems, a frequency-domain model will be developed so that generalized Nyquist techniques may be applied. r e v∼qs
L ∼e iq
+ -
e v∼ds
+ -
∼e id
+ -
+ v∼ e
C ω Cv~ e e d L
r
∼ iqLe
∼ ω e Lide
-
q
yqq
yqd v∼de
ydd
ydq v∼qe
∼e idL
∼ ω e Liqe
- +
+ ∼ v de
C ω e Cv~qe
-
Input Filter
Three-Phase CPL e Zqds
e YqdL
Figure 2.4 Small-displacement equivalent circuit of three-phase CPL with input filter.
2.3
Frequency-Domain Model of CPL
Before generalized Nyquist techniques can be applied, a dq Thevenin model of the above system will be developed so that linear system stability analysis techniques are applied to ac systems. For dc systems, it was shown that the source impedance and the load impedance, both complex scalar quantities, can be used to determine stability. In this section, the source dq impedance and the load dq impedance, which are both complex 2 × 2 matrices, will be used to determine system stability. To develop the Thevenin equivalent, first the open circuit voltage will be determined by removing the three-phase CPL in Figure 2.4; second the short circuit current will be determined by short-circuiting the CPL; finally the Thevenin impedance will be derived from step 1 and 2. For simplicity in notation, the superscript is neglected in the following derivations. When the constant power load is removed, the
- 17 e open-circuit voltage voc,qd can be established as follows e e voc,qd = Heqd vqds
(2.21)
£ ¤ where Heqd is defined as CH (sI − A)−1 BH where CH = 0 0 1 1 , BH = £ ¤T 1 1 0 0 and A is given in (2.19). After some matrix manipulation Heqd = [I + (sL + r + TL) (sC + TC)]−1
(2.22)
The short circuit current at the terminals of the load can be determined in a similar fashion e e vqds iesc,qd = Ysc,qd e By defining Ysc,qd = CY (sI − A)−1 BY , BY = BH , CY =
(2.23) £
¤ 1 1 0 0 , and allowing
the inverse of the capacitance C−1 Y to go to zero during the short-circuit e Ysc,qd = [sL + r + TL]−1
(2.24)
e e By substituting vqds in terms of iesc,qd in voc,qd the Thevenin impedance Zeth,qd at the output
of the filter can be determined as follows e e−1 e voc,qd = Heqd Ysc,qd isc,qd = Zeth,qd iesc,qd
(2.25)
When the CPL is connected, the voltage at the terminals of the load can be expressed in the terms of the source voltage as e e e e e = Heqd vqds − Zeth,qd ieqdL = Heqd vqds − Zeth,qd YqdL vqd vqd
(2.26)
e where YqdL is the load admittance developed in the previous section. Rearranging,
£ ¤−1 e e e e = I + Zeqds YqdL Hqd vqds vqd
(2.27)
where Zeqds = Zeth,qd . This is a frequency-domain model that expresses the output voltage in terms of the impedances at the interface between the CPL and the input filter. In the next section, it will be shown how this dq partitioning technique along with the generalized Nyquist criterion can be used for stability studies.
- 18 2.4
Generalized Nyquist Criterion
In the 1970s, MacFarlane and Postlethwaite [19] extended the theory of Nyquist stability, developed for scalar transfer functions, to a generalized Nyquist stability criterion which addresses matrix transfer functions. In this section, the generalized Nyquist criterion will be used to characterize the stability of the previous CPL system under different loading conditions, and the results will be compared with an eigenvalue analysis. The generalized Nyquist theorem [19] may be stated for the system depicted in Figure 2.5 as follows:
Theorem: Let the multivariable feedback system shown in Figure 2.5 have no open-loop uncontrollable and/or unobservable modes whose corresponding characteristic frequencies lie in the right-half plane. Then this configuration will be closed-loop stable if and only if the net sum of anti-clockwise encirclements of the critical point (−1 + j0) by the set of characteristic loci of the return ratio L(s) = G (s) K (s) is equal to the total number of right-half plane poles of G (s) and K (s).
u +
G(s)
y
K(s) Figure 2.5 MIMO feedback system. The output of the MIMO system in Figure 2.5 may be expressed as y(s) = [I + G(s)K(s)]−1 G(s)u(s)
(2.28)
This is very similar to the expression obtained in (2.27) for the CPL with input filter. If e−1 e G (s) = Heqd (s) and K (s) = Ysc,qd (s) YqdL (s), then the return ratio matrix is G (s) K (s) = e−1 e e Heqd (s) Ysc,qd (s) YqdL (s) = Zeqds (s) YqdL (s), which is very similar to the SISO case (DC
systems), and can be realized in a control block diagram format as shown in Figure 2.6. The characteristic loci of return ratio are defined as the graphs of λi (s), the eigenvalues of the return ratio, as s goes once around the Nyquist contour. This is different from SISO systems where the Nyquist locus is generated by plotting the magnitude and phase
- 19 -
e vqds
e vqd
e Hqd
+ -
e Ysce −1 ,qd YqdL
Figure 2.6 Equivalent feedback control block diagram of three-phase CPL with input filter. of the return ratio as s goes around the Nyquist contour. Since the poles of the return rae tio, Zeqds (s) YqdL (s) , are always on the left-hand s-plane for an RLC circuit, stability for
the present system can be determined from the net sum of counterclockwise encirclement of the critical point (−1 + j0) by the set of characteristic loci of the return ratio matrix only. The proof for the previous theorem is treated in great detail in [19] and only a cursory discussion will be given here for the purpose of future derivations. If the system shown in Figure 2.5 has a closed-loop realization, Ac Bc Cc Dc , and an open loop realization ABCD. Then using the Laplace transform and some linear Algebra, the Closed Loop Characteristic Polynomial CLCP may be related to the Open Loop Characteristic Polynomial OLCP as follows CLCP (s) det [sI − Ac ] = = det [I + L(s)] OLCP (s) det [sI − A]
(2.29)
where D and Dc were assumed zero for simplicity. Now, If λi (s) is an eigenvalue of L(s) = G (s) K (s) , then 1 + λi (s) is an eigenvalue of I + L(s) and since the determinant is the product of the eigenvalues, the right-hand side may be expressed as Y [1 + λi (s)] det [I + L(s)] =
(2.30)
i
By an extended version of the principle of the argument theorem [19] , it may be shown that 4 arg det [I + L(s)] =
X
4 arg [1 + λi (s)]
(2.31)
where 4 arg denotes the change in the argument (angle) as s traverses the Nyquist Contour. Hence, the closed-loop stability may be determined by counting the total number of encirclement around the origin made by the graphs of 1 + λi (s) or the graphs of λi (s) around (−1 + j0).
- 20 In some cases the generalized Nyquist will close through infinity and interpretation may become difficult. In these cases, the generalized inverse Nyquist may yield easier interpretation. The generalized inverse Nyquist theorem [19] may be stated as follows: Theorem: Let the multivariable feedback system shown in Figure 2.5 have no open-loop uncontrollable and/or unobservable modes whose corresponding characteristic frequencies lie in the right-half plane. Then this configuration will be closed-loop stable if and only if the net sum of anti-clockwise encirclements of the critical point (−1 + j0) by the set of inverse characteristic loci of the return ratio L(s) = G (s) K (s) is equal to the total number of right-half plane zeros of G (s) and K (s) . The generalized inverse Nyquist theorem necessitates knowledge of the zeros of the return ratio L(s), which may require extra effort. However, if the return ratio L(s) is square and nonsingular, then its zeros and poles are the poles and zeros, respectively, of its inverse L−1 (s). This fact will be used to apply the inverse generalized Nyquist in later chapters. For illustrative purposes, the feedback system in Figure 2.6 was simulated in MATLAB at nominal values of the load current for different constant power levels. The system parameters were designed for a 200-kW source. The initial test was performed for a 20-kW e (s) is a CPL and the characteristic loci are shown in Figure 2.7. Since L(s) = Zeqds (s)YqdL
2 × 2 matrix, only two characteristic loci are expected, and due to symmetry only the loci corresponding to positive frequencies were plotted (s was varied from 0 to +∞). The loci do not encircle the critical point (−1 + j0), and since there are no right-hand poles of the return ratio, the system is stable according to the generalized Nyquist theorem. The eigenvalues for the previous conditions were computed from the system state matrix in (2.19) as
−0.6573 + 6.8849i −0.6573 − 6.8849i λ =103 × −0.6414 + 6.2057i −0.6414 − 6.2057i
(2.32)
As expected, all eigenvalues have negative real parts which is in agreement with the Nyquist criterion. Next, the constant power load was increased by a factor of 10 to 200 kW and the results are shown in Figure 2.8. There exists one clockwise encirclement (two as ω varies from -∞
- 21 -
0.15
0.1
0.05
0
-0.05
-0.1
-1.5
-1
-0.5
0
0.5
Figure 2.7 Stable loci for P0 = 20 kW, L = 77 µH, C = 300 µF, r = 0.1 Ω, v0q = 367 V, v0d = 0 V.
1.5
1
0.5
0
-0.5
-1
-1.5 -3
-2
-1
0
1
2
3
Figure 2.8 Unstable loci for P0 = 200 kW, L = 77 µH, C = 300 µF, r = 0.1 Ω, v0q = 367 V, v0d = 0 V.
- 22 to +∞) which implies that the closed-loop system is unstable. The system eigenvalues for the new operating conditions are
+0.9546 + 6.1591i +0.9546 − 6.1591i λ =103 × −2.2533 + 6.5071i −2.2533 + 6.5071i
(2.33)
As shown, there are two eigenvalues with positive real parts which agrees with the Nyquist loci in Figure 2.8. To illustrate the nature of these instabilities, a time domain simulation of the nonlinear model in Figure 2.3 was done. Figure 2.9 shows the response of the system subject to step changes in real power P while the reactive power Q is kept null. The filter parameters are as given in Figure 2.8. The variables VQ and IQ denote the q-axis capacitor voltage and inductor current respectively whereas VA and IA denote the corresponding phase voltage and current. Initially, the system is allowed to reach steady state. At t = 0.02 sec, the real power is increased to 20 kW and the system is allowed to reach steady state again. At t = 0.05, the real power is increased to 82.1 kW which destabilizes the system. At high power levels, the negative damping due to the ideal three-phase CPL becomes more prominent and excites the resonance frequency of the filter. In the phase variables VA and IA the instability appears as a high-frequency oscillation that is modulated by the fundamental. In the q-axis variables VQ and IQ the fundamental is filtered out and only the high-frequency instability is apparent. This renders the interpretation more straightforward since the fundamental does not contribute to the instability. The high-frequency instability occurs at exactly the resonance frequency of the filter as expected. This study shows that the time domain simulation of the nonlinear model supports the previous results obtained from the eigenvalue analysis and the frequency-domain analysis which were both based on linear models. The previous analysis is based on two assumptions. First, the output voltage of the converter is regulated with a high bandwidth controller. Second, the load current is constant. This simplified model shows that CPLs can indeed destabilize three-phase ac systems as in dc systems. In the next chapter, the simplifying assumptions made thus far will be removed and non-ideal limited-bandwidth self-regulated converter loads will be considered.
- 23 -
Figure 2.9 System response to step increases in real power of ideal three-phase CPL.
- 25 -
3. SYSTEM MODELING In this chapter, appropriate mathematical models of various power system components are set forth so that linear multivariable analysis tools may be used to assess system stability. The interconnection of these models is considered in terms of input-output algebraic compatibility and reference frame compatibility. Methods of computing the impedance and admittance matrices at different interfaces are then described. Finally, the mathematical models and procedures set forth herein are used to analyze the stability characteristics of an example power system which includes regulated ac loads. The ac systems addressed in this research include rotating machines such as synchronous alternators and induction motors. They also include static electronic power converters such as solid-state switching converters and inverters. The associated controllers and regulators that control power flow and regulate voltage and current are also considered. The dynamics of these controllers and regulators may have a significant influence on system stability. For example, the synchronous alternator may have a voltage regulator that controls the output voltage by adjusting the field voltage. The dynamics of the regulator may be governed by a controller that is proportional, integral, or a combination thereof. The induction motors may employ speed or torque control using voltage, current, or field oriented control. Finally, the converters and inverters may have pulse-width modulation (PWM) to regulate current and/or voltage. Due to the complex dynamics that may emerge from the interconnection of these components, it is crucial that the appropriate mathematical models are developed for system stability assessment. It is also important that these models are expressed in the appropriate reference frame or frames so that stability techniques such as linearization, eigenvalue analysis, and frequency-domain analysis are possible. Also, in order to use these models, the choice of simulation environment has to be carefully made. Otherwise, a rigorous stability analysis may not be possible. For instance, although circuit-based simulation envi-
- 26 ronments such as EMTP [20], SPICE [21], and SABER [22] may provide rapid prototyping and time-domain modeling capabilities for electrical and electromechanical systems, they are not well suited for stability assessment.. In these environments, the system solutions are not obtained from a purely state space representation, i.e., controllers may be represented in state form but circuit elements are replaced by discrete algebraic models for discrete time steps. Hence, the system eigenvalues cannot be calculated directly from the system representation. Linearization and eigenvalue analysis techniques which are central to stability assessment are not yet available in these environments. On the other hand, in state-variable-based simulation environments, such as MATLABs Ordinary Differential Equations (ODE) utilities or the Advanced Continuous Simulation Language (ACSL), the state models can be linearized numerically which allows direct calculation of the eigenvalues, and plotting of Evans root and Nyquist loci. For this research, ACSL was selected as a simulation language because of its flexibility in modelling, its wide selection of integration algorithms, and its extensive library of analysis tools. Generally, ACSL requires the system model to be in a state space form which may involve extensive model development for large systems and systems that include switching devices. However, an automatic state model generation (ASMG) has recently been set forth in [24] which allows the user to specify the system in terms of nodes and branches that may include switching devices, whereupon the state space equations are generated automatically for all possible configurations. Using averaging and linearization techniques, the ASMG provides linear models from switching discontinuous models at steady state. In general, the component and system models to be used for stability assessment must have the following four necessary properties. (a) The models can be expressed in an explicit state space form such as dx = f(x(t), u(t)) dt y = g(x(t), u(t))
(3.1) (3.2)
where x(t) and u(t) are constant vectors for steady-state operation at a stable operating point.
- 27 (b) The models are differentiable, and in the case of switching networks, it is assumed that state-space averaging has been applied so that the resulting models are of the form described in (a). (c) The input-output variables of interconnecting models are reference-frame compatible. (d) The input-output variables of interconnecting models are algebraically compatible. Using these properties as a goal, mathematical models of several ac components are set forth herein. 3.1
Synchronous Machine Model
In general, a synchronous machine includes multiple windings on the stator and multiple windings on the rotor. The field winding on the rotor is excited by an external dc voltage source and as a generator, the rotor is driven by a prime mover. The changing flux linkages due to the rotor and stator winding currents induce voltages in all of the windings. These voltages are given by Faradays law in matrix form as dλ = v − ri dt
(3.3)
where λ is a vector of time-varying rotor and stator flux linkages, v represents the voltages at the terminals of the windings, i the winding currents which are positively directed into the machine, and r a diagonal matrix whose entries represent the resistances of the windings. If a three-phase machine with 120o displaced and sinusoidally distributed stator windings is considered, λ = [λabcs , λqdr ]T , v = [vabcs , vqdr ]T , i = [iabcs , iqdr ]T , r = diag[rabcs , rqdr ] where [f abcs , fqdr ] =[fas, fbs, fcs, fkq1, fkq2, ff d, fkd ]T . Here, λqdr , vqdr , and iqdr represent variables for a rotor with one field winding, one d-axis and two q-axis damper windings. The rotor variables are assumed to be referred to the stator by the appropriate turn ratios. If magnetic linearity is assumed, λ may be expressed as λ = Li
(3.4)
where L is a 7×7 inductance matrix relating the winding flux linkages to the winding cur-
- 28 rents. For a salient machine L may be written as [7] · ¸ Ls Lsr L= 2 T L Lr 3 sr
(3.5)
where all the inductances are referred to the stator by the appropriate turn ratios. Here, Ls relates the stator flux linkages to the stator winding currents, Lr the rotor flux linkages to the rotor winding currents, and Lsr the stator flux linkages to the rotor winding currents. Both Ls and Lsr are time-varying matrix-valued periodic functions which directly depend on the rotor position. If (3.4) is substituted into (3.3), the induced voltages become dλ = v − rL−1 λ dt
(3.6)
This is a state-space model whose inputs are the terminal voltages v and outputs are the winding currents given by (3.4). However, this model is not computationally efficient because the inductance matrix needs to be inverted at each time step. Furthermore, for stability analysis, this model is not useful because the system matrix rL−1 is time-varying even during steady state operation. To eliminate the time-varying coefficients, R. H. Park [23] introduced a transformation of the stator variables to the rotor reference frame. The transformation Krsr is defined as Krsr where
Here, θr =
=
·
Krs 03×4 04×3 I4×4
¸
Rt 0
cos θr cos (θr − 2π/3) cos (θr + 2π/3) 2 Krs = sin θr sin (θr − 2π/3) sin (θr + 2π/3) 3 1/2 1/2 1/2
(3.7)
(3.8)
ωr (ξ)dξ + θr (0) and ωr denotes the rotor speed. The variables in (3.4)
may be transformed to the rotor reference frame as λr = Krsr λ, vr = Krsr v, ir = Krsr i where r r T r r r r r r T f r = [fqdos, fqdr ] = [fqs, fds, fos, fkq1, fkq2, ffrd, fkd ] . In terms of the transformed variables,
(3.3) becomes £ ¤ r dλr = vr − rL−1 qd + T λ dt
(3.9)
- 29 where Lqd = Kr L [Kr ]−1 . More explicitly · r ¸ Ks Ls [Krs ]−1 Krs Lsr Lqd = 2 T L [Krs ]−1 Lr 3 sr
(3.10)
which is a constant matrix. It is interesting to observe that this matrix is not diagonal. However, Krs Ls [Krs ]−1 and Krs Lsr LTsr [Krs ]−1 are diagonal and constant hence the rows of Krs are the time-varying transposed left eigenvectors of the time-varying matrices Ls and Lsr LTsr . The same is true for Krs Lsr LTsr [Krs ]−1 and Lsr LTsr . In (3.9), T is given by 0 ωr 0 −ωr 0 0 03×4 T = 0 0 0 04×3 04×4
(3.11)
The previous state model yields a system whose variables are constant at steady state and £ ¤ whose system state matrix −rL−1 qd − T is constant if the rotor speed is constant which is true at steady state. Also, the matrix Lqd is sparse and constant, hence it needs to be
inverted only once and this is usually done by substitution. In terms of flux linkage per second Ψr = ωb λr , the state-space equations for the synchronous machine may be written as £ ¤ r dΨr = ωb vr − ωb rX−1 qd + T Ψ dt r ir = X−1 qd Ψ
(3.12)
(3.13)
where and Xqd = ωb Lqd . Finally, for a P -pole machine, the electromechanical torque may be to shown to be [7] Te =
¤ 3P 1 £ r r Ψds iqs − Ψrqs irds 2 2 ωb
(3.14)
The previous three equations constitute a state-space model of the synchronous machine in the rotor reference frame which can be readily linearized about an operating point setting the stage for eigenvalue and frequency-domain analysis. For interconnection purposes, the only required currents are the stator currents irqs , irds , assuming that the zero-sequence current iros is zero. Sometimes, the rotor field current irf d is an output variable. Also, all rotor windings are usually short-circuited except for the field
- 30 -
v rqds
State Equations (3.12-3.14)
irqds
r e xfd
r ifd
Te
ωr
Figure 3.1 Block diagram of synchronous machine model. winding which is connected to a dc voltage source. This voltage is conventionally expressed in terms of the d-axis magnetizing reactance Xmd and the field winding resistance rf d as erxf d = vfr d Xmd /rf d . The input-output variables for the synchronous machine model are shown in Figure 3.1. 3.2
Induction Motor Model
In this section, a three-phase induction motor with sinusoidally distributed windings and 120o displacement is considered. The rotor is assumed to be round and to have three-phase windings. When the stator terminal voltages are applied, the induced voltages due to the winding currents are given by dλ = v − rL−1 λ dt
(3.15)
where λ = [λabcs , λabcr ]T represent the flux linkages of the stator and rotor windings, and r = diag[rabcs , rabcr ]. The inductance matrix L is given by · ¸ Ls Lsr L= LTsr Lr
(3.16)
where Ls relates stator fluxes to stator currents, Lr relates rotor fluxes to rotor currents, and Lsr relates the stator fluxes to rotor currents. Since the rotor is assumed to be round, Ls and Lr are 3 × 3 constant matrices but Lsr is a time-varying 3 × 3 matrix that depends on the rotor position. Although this model is in state space form, its state matrix is timevarying and requires the inverse of the inductance matrix at each time step. To eliminate the time-varying inductances, the rotor and stator variables may be transformed to a common reference frame. If an arbitrary reference frame is chosen, we may define the following
- 31 transformation Kasr
=
·
Kas 03×3 03×3 Kar
¸
(3.17)
where Kas transforms the stator variables to the arbitrary reference frame and Kar transforms the rotor variables likewise. Normally, the superscript a is not used for the arbitrary reference frame transformation. However, it is used here to differentiate arbitrary reference frame variables from abc variables which do not carry any superscript. Both Kas and Kar are similar to the transformation given in (3.6) with θr replaced by θ and θ − θr respectively. Rt Namely, for Kas , θ = 0 ωa (ξ)dξ + θa (0) and ωa is the speed of the arbitrary reference Rt frame. For Kar , θ = 0 ωa (ξ) − ωr (ξ)dξ + θ(0) − θr (0) and ωr is the speed of the rotor. a a fqdor ]= The new variables may be expressed as λa = Kasr λ, va = Kasr v where f a = [fqdos,
a a a a a a T [fqs, fds, fos, fqr, fdr, for ] . In this reference frame, the induced voltages become
£ ¤ a dλa + T λ = va − rL−1 qd dt
(3.18)
where Lqd = Ka L [Ka ]−1 is constant but not diagonal. Again, it is interesting to note here that Kas Ls [Kas ]−1 , Kas Lsr LTsr [Kas ]−1 are constant and diagonal. Therefore, the rows of Kas must be the time-varying transposed left eigenvectors of the time-varying matrices Ls and Lsr LTsr . The same may be stated for the matrices Kar , Lr and LTsr Lsr . In the previous equation, T is given by
T =
0 −ωa 0 0 0 0
ωa 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 − (ωa − ωr ) 0
0 0 0 (ωa − ωr ) 0 0
0 0 0 0 0 0
(3.19)
In terms of flux linkage per second, the state space equations for the induction machine may be written as £ ¤ a dΨa = ωb va − ωb rX−1 qd + T Ψ dt a ia = X−1 qd Ψ
(3.20)
(3.21)
where Ψa = ωb λa and Xqd = ωb Lqd . Finally, for a P -pole motor, the electromechanical
- 32 -
a v qds
i
State Equations (3.20-3.23)
TL Te
a qds
ωr
ωe
Figure 3.2 Block diagram of induction motor model. torque may be expressed Te =
¤ 3P 1 £ a a Ψds iqs − Ψaqs iads 2 2 ωb
(3.22)
If a mechanical load is assumed to exert a torque TL on the rotor, then the rotor speed will be given by dωr Te− TL = dt J
(3.23)
where J denotes the inertia of the motor and connected load. If the speed of the arbitrary reference frame is set equal to the frequency of the applied voltages, the model is said to be transformed to the synchronous reference frame. In this case, the previous four equations constitute a state-space model of the induction motor which is suitable for linearization, eigenvalue analysis, and frequency-domain analysis. For interconnection purposes, the only required currents are the stator currents iaqs , iads , assuming that the zero-sequence current iaos is zero. Also, all the rotor windings are usually short-circuited. The input-output variables for the induction motor model are shown in Figure 3.2. 3.3
Converter and Inverter Models
The converter topology considered here is the conventional six-switch bridge with antiparalleling diodes as shown in Figure 3.3. This allows the bridge to operate as an inverter when supplied by a dc source and as a rectifier when supplied by an ac source. A current control strategy [25, 26] was considered here since it offers fast response times and it also decouples the interactions between the output filter capacitance and inductance, thus eliminating the potentially destabilizing effects of the high bandwidth voltage control within the
- 33 idc
+
DC/AC Inverter/Converter
sa
sc
sb ea
eb
vdc
iabcI
Lac
iabcL
ec
iaC
g
LC Filter
iab
ibc
va vb vab + - Load vc
ica Cac Switching Signals
---
iabcI
i*abc
Current Controller
vabc
iabcL
Voltage Controller
vdc
v *dc
* vabc
Figure 3.3 Inverter and converter configuration. stand-alone converter. Since the converter model is identical to the inverter model, except for some minor modifications of the controls, only the inverter model is described here. For each leg in the inverter, only one switch is on at any given time. This ensures that there is never a shoot-through or a short-circuit of the dc source. The output voltage eabc = [ea , eb , ec ]T with respect to the ground node g can be written as eabc = vdc sabc
(3.24)
where sabc = [sa , sb , sc ]T are the switching functions of the top switches and assume the value of 1 for a closed switch and the value of 0 for an open switch. The input dc voltage is denoted by vdc . Similarly the dc current idc is given by idc = sTabc iabcI
(3.25)
where iabcI represent the inductors currents [iaI , ibI , icI ]T and are governed by the state equa-
- 34 -
1
+ -
i*abcI i abcI
-h
sabc
h
Figure 3.4 Hysteresis current control. tion diabcI 1 (eabc − vabc ) = dt Lac
(3.26)
where the output voltages vabc are measured with respect to node g and Lac is the filter inductance. In the synchronous reference frame, this may be written as ¢ dieqdI 1 ¡ e e = eqd − vqd − Lac TieqdI dt Lac
(3.27)
e Here, fqdo = Kes fabc where Kes is defined in (2.4). The zero-sequence current has been
neglected since there is no neutral and T is given by · ¸ 0 ωe T= −ωe 0
(3.28)
Similarly, for the capacitors it may be shown that e ¢ dvqd 1 ¡e = iqdC − Cy Tveqd dt Cy
(3.29)
where Cy = 3Cac is the per phase Y equivalent capacitance of the filter and ieqdoC = Kes iabcC . The value of sabc is determined from a hysteresis current control as shown in Figure 3.4. The reference current i∗abcI = [i∗aI , i∗bI , i∗cI ]T is a obtained from a synchronous PI regulator e∗ ∗ [25] whose inputs are the desired output voltages vqdo and vdc , the measured output voltages e vqdo and vdc , and the measured load current ieqdoL , as shown the Figure 3.5. e∗ with the zero-sequence voltage As an inverter, the output ac voltage is regulated to vqd
voe∗ = 0. The control law is given by ie∗ qdI
=
ieqdL
+
Cy Tve∗ qd
+
e Kp ∆vqd
+ Ki
Z
t e ∆vqd dt
(3.30)
0
where the first term on the right-hand side denotes the measured load currents, the second term estimates the required capacitor currents for a given targeted output voltage and the
- 35 -
θe i
* abcI
[ K es ]−1
i
e* qdoI
e vqdo e iqdoL
Synchronous PI Regulator e* * v qdo vdc
θe K es
v abc i abcL
vdc
Figure 3.5 Synchronous PI regulator for converter and inverter control. last two terms constitute a PI (proportional-integral) compensator with the voltage error e e∗ e ∆vqd = vqd − vqd as an input. For rectifier operation, the previous control law is replaced
with ie∗ qI
= Kp ∆vdc + Ki
Z
t
∆vdc dt
(3.31)
0
∗ with ∆vdc = vdc −vdc . The d-axis current is usually set to zero for unity power factor control.
Finally, the reference angle θe , for the transformations in Figure 3.5, is usually obtained from a voltage-controlled-oscillator (VCO) which is phase-locked to the zero crossings of the ac voltage [27] . In this case, the synchronous reference frame speed corresponds to the output frequency of the VCO. These transformations are used in the previous controller so that no phase delays result and compensation is independent of the input signal frequency. The dynamics of the previous control strategies are treated in great detail in [25, 26]. Due to the presence of switching which is modeled by the discontinuous functions sabc , the inverter and converter models are not linearizable. In order to obtain a linearizable model, the hysteresis controller is replaced by a bounded proportional gain controller whose gain magnitude, within the hysteresis band, is equal the inverse of the hysteresis band h as shown in Figure 3.6. 1
i*abcI iabcI
+ -
-h
h
sabc
Figure 3.6 Bounded proportional current control.
- 36 For small disturbances, the switching functions sabc are approximated as continuous functions proportional to the current error. Since h is small, the response of the bounded proportional controller closely approximates the actual system response for both large and small disturbances. For small disturbances, sabc will deviate from a nominal value but will remain in the linear region, hence the bounded proportional controller is linearizable. The input-output variables for the converter and inverter model are shown in Figure 3.7.
vdc
State Equations (3.24-3.31)
e vqd
e iqdL
idc * vdc
e* vqd
ωe
Figure 3.7 Block-diagram of converter-inverter model.
3.4
Field-Oriented Controlled Induction Motor
Field oriented control is used to provide rapid control of electromagnetic torque for an induction motor [27]. There are two general methods of field oriented control, direct and indirect. The direct method requires measurement of rotor flux whereas the indirect method does not depend on these measurements hence is easier to implement. In the indirect method, the q-axis rotor flux is targeted to be zero and the d-axis rotor flux is set to a constant. The electromagnetic torque is directly proportional to the q-axis current which can be rapidly controlled. In this section, a state-space model of the indirect method will be set forth. From the induction motor state equations presented in Section 3.2, the rotor flux state equations may be written in the scalar form as dΨeqr = −ωb rr ieqr − (ωe − ωr ) Ψedr dt
(3.32)
dΨedr = −ωb rr iedr + (ωe − ωr ) Ψeqr dt
(3.33)
- 37 where Ψedr and Ψeqr are the rotor flux linkages per second in the synchronous reference frame, iedr and ieqr are the rotor currents, and ωe, ωr , and ωb are the synchronous frequency, the rotor speed and the base frequency. Finally, rr denotes the rotor winding resistance. For simplicity of notation, all the rotor quantities are assumed to be referred to the stator by the appropriate turns ratio. The rotor flux linkages are related to the stator and rotor currents by Ψeqr = Xm ieqs + Xrr ieqr
(3.34)
Ψedr = Xm ieds + Xrr iedr
(3.35)
where Xm is the magnetizing reactance, Xrr = Xm +Xlr , and Xlr denotes the rotor leakage reactance. For indirect control, the aim is to have Ψeqr =
dΨeqr =0 dt
(3.36)
If this is substituted in the previous equations, the d-axis flux linkage and the machine slip may be obtained from dΨedr rr ωb = (Xm ieds − Ψedr ) dt Xrr ωslip = ωe − ωr =
rr ωb Xm ieqs Xrr Ψedr
(3.37)
(3.38)
The angle of the required currents may then be obtained from the measured rotor speed ωr and the estimated slip ωslip as θe(t) =
Z
t
(ωslip + ωr ) dt + θe (0)
(3.39)
0
If Ψeqr = 0 the torque expression from Section 3.2 becomes Te =
3 P 1 Xm e e Ψ i 2 2 ωb Xrr dr qs
(3.40)
If the inverter is current controlled as described in the previous section, the output currents will track the commanded currents with a fast response time and, for constant Ψedr , the torque response will be essentially instantaneous. This explains the similarity of field
- 38 controlled induction motors to armature-controlled dc motors. FOC induction motors offer fast response times and very tight over-current control. On the other hand, they may exhibit negative input resistance within the regulation bandwidth [5]. A simplified block diagram of the field oriented controller is given in Figure 3.8. e* i qd
e −1 s
[K ]
i e* abc
θe 1/s Equations (3.37-3.38) ω + slip
ωe
+
DC-AC 3-Phase Inverter
e v qdos
i
e qdos
3-Phase Induction Motor
ωr
Figure 3.8 Indirect field-oriented control of induction motor.
3.5
Component Model Interconnection and Interface Compatibility
As shown in previous sections, a component model may include variables that are represented in different reference frames. For example, the converter model described includes two reference frames. The filter and the regulator were developed in the synchronous reference frame while the switching functions were left as abc variables. Therefore, a system model may include numerous reference frames and the interconnection of the various component models deserves special attention. Specifically, there are two forms of compatibility that the input-output variables of the interconnecting models have to satisfy: algebraic compatibility and reference frame compatibility. Algebraic compatibility is satisfied when the input-output variables at the interface are consistent. For example, in a synchronous machine model, voltages are inputs and currents are outputs. If the load is inductive, i.e., the load model inputs must be voltages and outputs must be currents, there will be an algebraic incompatibility at the interface and the models cannot be connected without modification. It is interesting to observe here that the automatic state space model generator [24] discussed earlier is useful in these situations and may represent an alternative. A less computationally efficient method is to insert a small capacitor or a large resistor at the interface, hence
- 39 -
e v rqds = v qds
r e xfd
Synchronous Machine
Three-Phase Converter e irqds = iqds
r ifd
ωr
Te
Figure 3.9 Model input-output configuration for synchronous machine and converter system. making the interconnections algebraically compatible. Reference frame compatibility is satisfied when all the input-output variables at each interface are represented in the same reference frame. This allows for different reference frames at different interfaces. In hybrid ac/dc systems, numerous reference frames may be needed to satisfy all of the modeling conditions set forth at the beginning of this chapter. 3.5.1
Synchronous machine and converter system
When a synchronous machine is modeled as described in Section 3.1, the input variables include the stator voltages which are assumed to be expressed in the rotor reference frame. In turn, the model provides stator currents as output variables represented in the same reference frame. The converter model described previously expects the ac source currents as inputs in the synchronous reference frame and the ac capacitor voltages as outputs in the same reference frame. Hence, the input-output variables of the two models are algebraically compatible. However, the models are not reference frame compatible at the interconnecting interface if no further assumptions are made. If synchronism between the converter and the synchronous machine is assumed at all time, then the synchronous and rotor reference frames are the same. Hence, the interface variables will be reference frame compatible. A block diagram depicting the resulting input-output relationship is shown in Figure 3.9. On the other hand, if a phase-lock-loop (PLL) is used to synchronize the converter switching to the ac frequency, then the speed of the synchronous and rotor reference frames will differ somewhat during transients. In this case, it is convenient to define synchronous frequency as the frequency of the PLL whereupon the transformation from the synchronous
- 40 to the rotor reference frame may be expressed cos δ − sin δ e r K = sin δ cos δ 0 0 Rt where δ = θr − θe = 0 ωr (ξ) − ωe (ξ)dξ + θr (0) −
0 0 1
(3.41)
θe (0). The rotor referred voltages
r and currents may be expressed in terms of the synchronous voltages and currents as vqdo =
e
e Kr vqdo and irqdo = e Kr ieqdo . With this transformation, the dynamics of synchronization
between the converter and the synchronous machine may be accurately represented. The reference angle for the converter is usually generated by a phase-lock loop which senses ac voltage and determines the frequency and reference angle for the converter. A block diagram depicting the input-output configuration is shown in Figure 3.10
ωe
PLL
Three-Phase Converter
v eqds
e
r Ks
r
i
e qds
v rqds
e
Ks
r e xfd
Synchronous Machine
i rqds
ifdr
ωr
Te
Figure 3.10 Input-ouput configuration for system with PLL.
3.5.2
Inverter and converter system
There are two possible interconnection configurations for an inverter and a converter system. They may be connected either at the dc terminals in which case they will act as an ac-ac power converter or they may be connected at the ac terminals in which case they will act as a dc-dc power converter. If a dc interconnection is considered first, both converter and inverter models as set forth in Section 3.3 expect voltage as an input variable and current as an output variable. Therefore, the two models are algebraically incompatible at the interface. However, if a capacitor is connected at the interface, which is usually done to filter the voltage ripple due to converter harmonics, the interface variables become
- 41 -
Three-Phase Converter
vdc icnv
Link Capacitor
vdc
Three-Phase Inverter
iinv
Figure 3.11 Model input-output configuration for inverter-converter system with dc interface. algebraically compatible. The input-output configuration for the dc interconnection is illustrated in Figure 3.11. With the capacitor Clink in place, and assuming that the converter is supplying positive current, the dc link voltage is given by dvdc 1 (icnv − iinv ) = dt Clink
(3.42)
Note that for dc interconnection, the models are obviously compatible in terms of reference frame. If an ac interconnection is considered, a similar situation arises. The models are algebraically incompatible because both models expect voltages as inputs and provide currents as outputs. However, if the filter capacitors are combined into one equivalent capacitor, the algebraic incompatibility is eliminated. The voltage at the interface is given by an equation similar to (3.42) and the equivalent capacitance is equal to the sum of the individual capacitances. 3.6
Source and Load Impedance Matrix Realizations
In this section, the numerical techniques used to generate the transfer functions necessary for the application of the generalized Nyquist criterion to ac systems are discussed. It is assumed in this section that the system is first modeled in detail so that the switching of the converters is accurately portrayed. Then, the switching functions are averaged to yield linearizable models. The detailed model may be represented in a nonlinear state from as dx = f (x (t) , s (t) , u (t)) dt y = g (x (t) , s (t) , u (t))
(3.43) (3.44)
where si (t), an element of s (t), is a discontinuous switching function representing the
- 42 two states of a switch in the converters as discussed in Section 3.3. A value of one in si (t) indicates the switch is on, a value of zero indicates the switch is off. The state vector x (t) may include the stator and rotor fluxes of the synchronous generator and induction motor, the voltages across the filter capacitors, the currents in the inductors, and the states associated with the PI controllers of the converters. The input vector u (t) may include the fixed field voltage of the generator, the fixed speed of the generator and motor, and voltage control references for the converters. This yields a coupled nonlinear and discontinuous set of differential equations. When the system variables, including switching, are averaged, the system may be described by a new set of equations: dx = f (x (t) , s (t) , u (t)) dt y = g (x (t) , s (t) , u (t))
(3.45) (3.46)
where v (t) is a continuous function representing the average of the variable v(t). In the present research, the switching functions were averaged by replacing the hysteresis controller with a linear bounded controller as discussed in Section 3.3. It is beyond the scope of this thesis to discuss these averaging techniques in detail. However, it is important to note that, in general, such averaging [8] is possible only for systems where the switching ripple is small. These techniques yield nonlinear averaged models which are differentiable and hence linearizable. The following discussion assumes that the system model is linearizable. To obtain the transfer functions needed for the generalized Nyquist criterion, the system must be partitioned into a source subsystem and a load subsystem as shown in Figure 3.12. At the interface between the generator and filter capacitor, the transfer function between generator output voltage and the field input voltage may be expressed as r vqds = [I + Zqds1 YqdL1 ]−1 Hqd1 vfr d
(3.47)
r where vqds is the generator output voltage, vfr d is the field voltage, Hqd1 is the no-load
generator voltage transfer function, Zqd1 is the generator output qd impedance matrix at interface 1 and YqdL1 is the load subsystem impedance matrix at the same interface. The
- 43 -
Interface 1
Interface 2
v rqds
Synchronous Generator
irqds
Capacitor Bank
v rqd i
r qd
v rqdc
Three-Phase Inductor r Bank iqdc Converter
v rfd Yqds1
ZqdL1
Zqds2
YqdL2
Figure 3.12 Thevenin partitioning of system at two different interfaces. overall system may be represented by the equivalent circuit shown in Figure 3.13.
Zqds1 v rfd
+ -
r r voc ,qd = H qd1v fd
Generator
YqdL1
Load
Figure 3.13 System Thevenin equivalent at Interface 1. r r The voltage voc,qd is the generator no-load voltage, i.e., it is the voltage vqds with the
load current being zero. Since Hqd1 describes the synchronous machine only, its poles should be in the OLHP. Hence, BIBO stability of the selected interface depends only on the placement of the zeros of I + Zqds1 YqdL1 . Given the number of poles of Zqds1 YqdL1 , stability is determined by the number of encirclement of the characteristic loci of Zqds1 YqdL1 around (−1 + j0). In order to obtain the transfer functions for Zqds1 and YqdL1 , the appropriate ABCD realizations must be computed from the non-linear state model. In general, a state model of
- 44 the form dx = f (x (t) , u (t)) dt y = g (x (t) , u (t))
(3.48) (3.49)
may be linearized about an operating point as dx ' Ax + Bu dt y ' Cx + Du
(3.50) (3.51)
where A = 5x f
(3.52)
B = 5u f
(3.53)
C = 5x g
(3.54)
D = 5u g
(3.55)
where, for example, 5x f denotes the Jacobian matrix of f with respect to x. Any transfer function matrix T with inputs u and outputs y may be obtained then as T = C [sI − A]−1 B + D
(3.56)
This, however, is limited to proper matrix transfer functions, i.e., the degree of the numerator polynomials cannot be greater than the degree of the denominator polynomial as can be shown from the following observation [sI − A]−1 =
Adj [sI − A] det [sI − A]
(3.57)
Since the degree of the determinant in (3.57) will always be greater than the degree of the cofactors in Adj[sI − A], the ABCD realization in (3.56) will always yield a proper transfer function. Hence there are no ABCD realizations for nonproper transfer functions such as inductive impedances or capacitive admittances since these are nonproper. For example, Zqds1 YqdL1 from Interface 1 in Figure 3.12 has no ABCD realization, i.e., these transfer function matrices cannot be obtained from the system model numerically. There are two possible solutions. The first is to find the ABCD realization for T−1
- 45 = [Zqds1 YqdL1 ]−1 = ZqdL1 Yqds1 , as shown in Figure 3.12 and to invert in the frequency domain to obtain T. The second is to realize T−1 and use the inverse generalized Nyquist criterion to determine stability. Even though both options are feasible, the second one will be used here since the first option requires calculation of T and its poles. Since T has no ABCD realization, its poles are difficult to find. In the second option, however, the zeros of T are needed for the generalized inverse Nyquist, but these are exactly the poles of the realizable T−1 . Of course, T is assumed to be square and nonsingular. At Interface 2 in Figure 3.12, the impedances are realizable and the generalized Nyquist criterion can be applied directly. Consideration is also required for transfer functions which have system states as inputs and outputs. For example, in order to compute Zqds , the impedance looking towards the generator, the inputs to the transfer function are currents which are state variables in the generator model and the outputs are voltages which are state variables in the capacitor model. If the transfer function involves inputs and outputs that are states, a slightly different formulation is required. For a transfer function matrix with j state inputs and k state outputs, a system with n states and m inputs may written as dx = f (xn , um ) = f (xn−j , xj , um ) dt y = g (xn , um ) = g (xn−j−k , xk , xj , um )
(3.58) (3.59)
The ABCD realization for the transfer function with j state inputs and k state outputs may be obtained as: A = 5xn−j f
(3.60)
B = 5xj f
(3.61)
C = 5xk g
(3.62)
D = 5xj g
(3.63)
The order of the Jacobian is reduced by the number of state inputs j. It may be further reduced by freezing , i.e., maintaining constant all the states that do not effect the transfer function required. For example, if Zqds is desired, the load states may be frozen. In ACSL, the Jacobian calculation is automatic once the lists of inputs and outputs are specified. Any
- 46 mixture of states and inputs may be used to generate a transfer function from the non-linear state model. If states are specified as inputs, they must be set to a constant nominal value, which serves as the operating point. Once this is done, the Jacobian calculations are done automatically according to user specified error tolerance [28]. For Zqds , the inputs should be currents and the outputs should voltages. However, since this is an inductive impedance, i.e., a strictly nonproper transfer function, Z−1 qds must be realized first. Note also that the generator stator current cannot change discontinuously because of the stator inductance, voltages instead of currents must be used as inputs. This is resolved by first calculating Yqds then inverting the matrix to obtain Zqds . The ABCD realization for Yqds is then: A = 5xn−2 f
(3.64)
B = 5Vqds r f
(3.65)
C = 5irqds g
(3.66)
D = 5Vqds r g
(3.67)
Yqds = C (sI − A)−1 B + D = Z−1 qds
(3.68)
and
The load subsystem admittance YqdL , is calculated in a similar fashion. Once the ABCD realizations are obtained, they can be transferred to MATLAB to calculate the transfer matrices as functions of frequency. Zqds Yqds is then calculated and the eigenvalues at each frequency are determined and sorted based on continuity. The following algorithm illustrates the overall steps involved in determining the transfer functions needed for the generalized Nyquist test. (a) Model system in state form according to conditions set forth in the beginning of this chapter. (b) Simulate and run the model to establish an operating point. (c) Compute the source and load susbsystems ABCD realizations for impedance Zqds and YqdL . (d) Determine the poles of the return ratio Zqds and YqdL .
- 47 (e) Transfer state matrices to MATLAB and compute return ratio Zqds YqdL as a function of frequency. (f) Compute the eigenvalues of return ratio at each frequency. (g) Sort the eigenvalues obtained such that the characteristic loci are continuous. (h) Plot the characteristic loci on the complex plane. (i) Count the number of encirclements to determine stability. (j) If Zqds YqdL are not realizable then calculate ZqdL Yqds and use the inverse Nyquist criterion.
3.7
Example System
In this Section, an example ac system is considered which is similar in structure to an ac system considered by NASA for commercial aircraft. The stability of the system under different conditions is predicted using the approach described heretofore. To determine the validity of the results, an eigenvalue analysis and time domain simulations will be performed for comparisons. The system consists of a synchronous machine that supplies power to two current-controlled three-phase converters, each of which is connected to a currentcontrolled inverter and its respective load as shown in Figure 3.14. Inverter 1 supplies a field-oriented controlled (FOC) induction motor while Inverter 2 supplies an RL load. The RL load is representative of lighting, heating, and other services while the FOC induction motor is representative of an electric actuator. The system shown in Figure 3.14 was simulated in ACSL. The parameters for the synchronous machine in pu are given in [7] (Table 5.10-1). The 50-hp induction motor parameters are also given in [7] (Table 4.10-1). To minimize the voltage ripple on the link bus, a 36 mF capacitor was placed at each converter-inverter interface. The inductor at the input of each converter was set to 190 µH whereas the ac capacitors were combined, as shown in Figure 3.14, and set to 180 µF line-to-neutral, thus resulting in a resonance frequency of 860 Hz. Hence, the ac filter provides 80 dB attenuation for an assumed switching frequency of 100 kHz. The regulators gains were set to Kp = 3 and Ki = 30 for the inverter, and Kp = 10 and Ki = 300 for the converter. The approximate regulator bandwidths are:
- 48 -
3-Phase Bus
Interface 1
3-Phase Generator
+ v rfd
DC-AC 3-Phase Inverter 2
AC-DC 3-Phase Converter 2
3-Phase RL Load
Interface 2
3-Phase Capacitor v rqds
3-Phase Bus
DC Link
v rqds
Converter Regulator
Inverter Regulator 3-Phase Bus
DC Link
AC-DC 3-Phase Converter 1
DC-AC 3-Phase Inverter 1
Induction Motor
FO Control
Yqds ZqdL
Zqds YqdL
Converter Regulator
Figure 3.14 One-line diagram of system studied. Kp /(2πCac ) = 2.6 kHz for the inverter, and Kp /(2πClink ) = 13 Hz for the converter. The generator is represented in the rotor reference frame as described in Section 3.1. The speed of the synchronous reference frame for both converters is assumed to be equal to the speed of the rotor thus making the interface variables compatible. The controls for both converters are implemented as described in Section 3.3. The reference angle for the converter regulator is assumed equal to the synchronous machine rotor angular position. The commanded rotor reference frame q-axis current is calculated as in (3.30) while the d-axis current is set to zero. This strategy does not guarantee unity power factor control but minimizes the stator currents for round-rotor machines. If unity power factor is desired, the reference angle may be obtained from a PLL that is synchronized to the a-phase voltage. Unity power factor control will be implemented and discussed in Chapter 6. Inverter 2 is implemented in a synchronous reference frame with angular speed fixed at 60-Hz which represents the commanded load frequency. The commanded synchronous reference frame currents are calculated as in (3.29). Inverter 1 and induction motor are implemented in a synchronous reference frame whose speed is equal to the sum of the induction motor speed
- 49 and controlled slip as described in Section 3.4. Thus, the overall system model incorporates four different reference frames including the switching functions which were left as abc variables. This system model satisfies all the criteria set forth in the beginning of his chapter. To compare the detailed system model which uses hysteresis current control in each of the converters and inverters to the averaged model which uses proportional bounded controllers, the following scenario was simulated. The synchronous machine was scaled to 400 kW 450V by multiplying the pu stator currents and voltages by the desired base current and voltage respectively. The inverters and converters were rated at 200 kW with an output dc voltage of 750-V and ac line-to-line voltages of 450-V. The induction motor was rated at 50 hp 450-V AC. The synchronous machine speed is assumed constant at 377 rad/sec and the induction motor speed at 370 rad/sec. First, the system is allowed to reach steady state with no load connected to the converters. Then, at 200 ms, a 90-kW three-phase RL load is connected to inverter 2. The results from the detailed simulation are shown on the left side in Figure 3.15, while the results of the averaged model simulation are shown on the right side. VQSR and IQSR denote the q-axis voltage and current of the generator, VLINK1 and ILINK1 denote the voltage and current at the interface between Converter 1 and Inverter 1, VLINK2 and ILINKDC2 denote the voltage and current at the interface between Converter 2 and Inverter 2. All currents are assumed to be positive from left to right. When the ac load is connected to Inverter 2, VLINK2 drops momentarily and ILINKDC2 increases to a nominal value. The source current IQSR increases to 25 % of rated and the source voltage VQSR remains steady. At 400 ms, a 110-kW dc load is applied to converter 1 at the link bus whereupon the bus voltage, VLINK1, drops momentarily while the link current, ILINKDC1, increases to a nominal value. The source current IQSR increases to 50 % while the source voltage VQSR remains steady. Note that since the second load is dc, there is no switching ripple in the link current ILINKDC1. The two representations are in close agreement in an average-value sense. This study shows that the bounded linear current controller is a good approximation for the hysteresis band control. The bounded linear controller allows linearization for smalldisturbance stability analysis.
- 50 -
AC load on DC load on
Figure 3.15 Comparison of system responses calculated using detailed and averaged computer models.
- 51 In the next study, the system is energized with a 55-kW load connected to the link bus of Converter 1. Then, at t = 100 ms, the load on the link bus is doubled and a 200-kW load is connected to the output of Inverter 2. At t = 200 ms, the induction motor d-axis commanded current is set to 0.5 per unit. Then, at t = 300 ms, the commanded q-axis current of the induction motor is set to 1.0 per unit. The system was then allowed to reach steady state and the simulation is stopped at 0.75 sec. The system voltages and currents are shown in Figures 3.16 and 3.17 where VQSR, VLINK1 and VLINK2 are as described previously. The variables VQE1 and VQE2 denote the q-axis voltages at the output of Inverters 1 and 2, respectively. The currents shown in Figure 3.17 are the input currents to the induction motor. These currents track the commanded currents very closely. The electromagnetic torque TE tracks the q-axis current once SIDRE reaches steady state as expected. The q-axis rotor flux SIQRE is very small as predicted in Section 3.4. The system is stable at steady state. Interface 1, between the synchronous machine and the input capacitor to the converter, was chosen for Nyquist calculations. Since this is an LC interface, the synchronous machine is an inductive subsystem and the load with input filters is a capacitive subsystem, the ABCD realizations for Yqds and ZqdL were calculated using ACSL as described in Section 3.6. These matrices were then transferred to MATLAB, and the inverse Nyquist criterion was used. The characteristic loci of ZqdL Yqds are shown in Figure 3.18. Since, the impedances ZqdL and Yqds consist of 2 × 2 matrices, two eigenvalue loci are plotted. The enlargements of the top Figure in 3.18 around (−1 + j0) are shown in Figures 3.18 and 3.19. Figure 3.18 illustrates the existence of one counter-clockwise encirclement even though the system is stable. However, the eigenvalues obtained from the realization of ZqdL , shown in Table 3.1, indicate the presence of an unstable open-loop pole. Together with the counter-clockwise encirclement, the number of unstable zeros for the closed-loop system is zero and the system is stable as predicted by the Nyquist criterion. This is in agreement with the time-domain simulation and the system eigenvalues shown in Table 3.3.
- 52 -
Figure 3.16 System response with nominal controller parameters.
- 53 -
Figure 3.17 Induction motor response with nominal controller parameters.
- 54 -
6000 4000
Zoom A
2000 0 -2000 -4000 -6000 -1000
0
1000
2000
3000
4000
5000
6000
30
40
Zoom A 15
Zoom B
10 5 0 -5 -10 -15 -20 -30
-20
-10
0
10
20
Figure 3.18 Characteristic loci at Interface 1 for stable operating point.
- 55 Zoom B 5
Zoom C
4 3 2 1 0 -1 -2 -3 -4 -5 -12
-10
-8
-6
-4
-2
0
Zoom C 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -1.5
-1
-0.5
0
0.5
Figure 3.19 Enlargement of characteristic loci around (-1+j0).
- 56 -
Table 3.1 Eigenvalues of load (ZL) subsystem at Interface 1. REAL 1 2 3 4 6 7 8 9 10 11 13 15 17 18 19 20
+800.508000 -6.42263000 -9.69469000 - 6.43677000 - 29.5449000 - 35.0443000 - 179.532000 - 655.056000 - 1557.39000 - 21009.3000 - 71794.5000 - 79934.9000 - 95309.7000 - 96702.1000 -98012.7000 -98621.2000
IMAGINARY
+/-13.9426000
+/-132.560000 +/-383.185000 +/-269.837000
Table 3.2 Eigenvalues of source (Ys ) subsystem at Interface 1. REAL 1 2 3 4 5
Table 3.3 Overall system eigenvalues for stable operating point.
IMAGINARY
-0.50027100 -18.1308000 -23.3359000 -49.8278000 -3.95740000 +/-376.934000
REAL 1 2 3 4 5 7 8 9 10 11 12 13 15 17 19 21 23 24 25 26
-0.15864900 -6.22318000 -6.42263000 -9.69599000 -6.43790000 -16.8681000 -33.5115000 -36.0842000 -49.8278000 -175.856000 -196.125000 -304.322000 -314.353000 -21009.3000 -71794.5000 -79934.9000 -95311.3000 -96702.7000 -98014.3000 -98621.2000
IMAGINARY
+/-13.9435000
+/-4852.04000 +/-5706.24000 +/-132.559000 +/-383.185000 +/-269.837000
- 57 In the next computer study, it is assumed that the same sequence of events occurs as in the previous study. However, the proportional gain of Converter 1 is set to a negative value (Kp = −1.0) which causes the system to become unstable. The time-domain responses are shown in Figure 3.20 and 3.21. The induction motor stator currents in Figure 3.21 track the commanded currents closely and the electromagnetic torque tracks the q-axis current. It can be observed that although the output voltage of the synchronous machine as well as the output voltages of the converters are unstable, the induction motor is stable. This is due to the field oriented controller which keeps the induction motor input currents, IQSE1 and IDSE1, constant even though the inverter input voltage varies. Similar observations [5] were made in hardware tests. Again the computer study was stopped at t = 0.75 sec and the ABCD realizations were computed and transferred to Matlab. The resulting characteristic loci are shown in Figures 3.22 and 3.23. The enlargements show no encirclements of the point (−1+j0). However, the eigenvalues in Table 3.4 indicate the presence of two unstable poles for ZqdL . The Nyquist criterion predicts two unstable poles for the overall system which is in agreement with the system eigenvalues in Table 3.6. In the previous two studies, information regarding the eigen-loci of the return ratio was insufficient in determining the stability of the system because of the presence of positive open-loop poles at the interface studied. In the final test, Interface 2 in Figure 3.14 is chosen for the stable system case. Since this is a CL interface, Zqds YqdL is realizable and the generalized Nyquist test may be applied directly. As shown in Tables 3.7-9, the positive open-loop poles are no-longer present. Thus, the return ratio is sufficient for the determination of stability at this interface. Indeed, the eigen-loci in Figure 3.24 exhibit no encirclement of the critical point (-1+j0) for the stable system considered. In this section, it has been shown that the generalized Nyquist criterion is very well suited to the analysis of ac systems with regulated converter loads. Stability can be predicted based on the characteristic loci of subsystem qd impedances under stated nonideal conditions. It was also shown that, by a suitable choice of the interface, the return ratio is sufficient for the determination of stability. Time-domain simulation and eigenvalue analysis were used to support this claim. In the next chapter, potential small-gain design-oriented criteria are discussed.
- 58 -
Figure 3.20 System response with modified controller parameter.
- 59 -
Figure 3.21 Induction motor response with modified controller parameters.
- 60 -
6000 4000
Zoom A
2000 0 -2000 -4000 -6000 -1000
0
1000
2000
3000
4000
5000
6000
Zoom A 40 30 20 10
Zoom B
0 -10 -20 -30 -40 -50 -30
-25
-20
-15
-10
-5
0
5
10
Figure 3.22 Characteristic loci at Interface 1 for modified controller parameters.
- 61 Zoom B 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.5
-1
-0.5
0
Figure 3.23 Enlargement of characteristic loci around (-1+j0).
- 62 -
Table 3.4 Eigenvalues of load (ZL) subsystem at Interface 1.
1 3 4 5 7 9 10 11 13 15 17 19 20
REAL
IMAGINARY
+276.287000 -6.42263000 -9.70455000 -6.44793000 -52.4335000 -831.407000 -1195.16000 -21223.7000 -53480.7000 -72888.4000 -79657.0000 -96501.4000 -97942.3000
+/-11.8639000
+/-14.1844000 +/-5.32566000
+/-501.895000 +/-383.050000 +/-1131.26000 +/-174.302000
Table 3.5 Eigenvalues of source (Ys) subsystem at Interface 1. REAL 1 2 3 4 5
IMAGINARY
-0.50015800 -18.1303000 -23.3369000 -49.8283000 -3.95745000 +/-376.896000
Table 3.6 Overall system eigenvalues for unstable operating point.
REAL 1 3 4 5 6 7 9 10 11 12 13 15 17 19 21 23 25 26
+22.3952000 -0.16110500 -6.18479000 -6.42263000 -9.70316000 -6.45234000 -16.8810000 -36.2472000 -49.8283000 -173.859000 -356.018000 -360.936000 -21223.7000 -53480.7000 -72890.6000 -79657.0000 -96502.4000 -97943.1000
IMAGINARY +/-87.5137000
+/-14.1848000
+/-4877.97000 +/-5752.71000 +/-501.895000 +/-383.050000 +/-1131.29000 +/-174.302000
- 63 -
20
Zoom A 15 10 5 0 -5 -10 -15 -20 -5
0
5
10
-0.5
0
15
20
25
Zoom A 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5
-1
0.5
1
1.5
Figure 3.24 Characteristic loci at Interface 2 for stable system
- 64 -
Table 3.7 Eigenvalues of load (YL) subsystem at Interface 2.
REAL 1 2 3 5 6 7 8 9 11 13 15 16 17 18
-6.42263000 -9.69597000 -6.43789000 -33.5238000 -36.1049000 -173.505000 -189.243000 -21009.3000 -71794.4000 -79934.9000 -95702.9000 -96936.4000 -98559.5000 -98670.5000
1 2 3 4 5 7
-0.15977300 -6.17793000 -16.8719000 -49.8278000 -6.11221000 -7.22393000
REAL
IMAGINARY
+/-13.9435000
+/-132.558000 +/-383.183000 +/-269.831000
Table 3.8 Eigenvalues of source (Zs) subsystem at Interface 2. REAL
Table 3.9 Overall system eigenvalues for stable operating point.
IMAGINARY
+/-4882.40000 +/-5794.99000
1 2 3 4 5 7 8 9 10 11 12 13 15 17 19 21 23 24 25 26
-0.15864900 -6.22318000 -6.42263000 -9.69599000 -6.43790000 -16.8681000 -33.5115000 -36.0842000 -49.8278000 -175.856000 -196.125000 -304.322000 -314.353000 -21009.3000 -71794.5000 -79934.9000 -95311.3000 -96702.7000 -98014.3000 -98621.2000
IMAGINARY
+/-13.9435000
+/-4852.04000 +/-5706.24000 +/-132.559000 +/-383.185000 +/-269.837000
- 65 -
4. SMALL-GAIN STABILITY CRITERIA 4.1
Small-Gain Stability Criteria
For the stand-alone converter, it was shown in Chapter 1 that for dc SISO systems, limiting the magnitude of the impedance ratio Zs YL inside the unit circle ensures stability of the converter given that the filter is stable and the converter by itself is stable. In this section, analogous criteria will be discussed for ac systems. Since the return ratio, Zs YL in three-phase systems with no zero-sequence current is a 2×2 matrix, it is not clear which norm will yield the desired stability condition on the characteristic loci. Although Zs YL is used as the return ratio in the following development, similar criteria may be stated for Ys ZL since the two ratios are related by the inverse Nyquist criterion. If the induced norm is chosen, at a given frequency kZs YL k = sup
kxk6=0
If kxk =
kZs YL xk kxk
(4.1)
√ xH x, the previous norm is referred to as the 2-norm and kZs YL k = σ (Zs YL , ω)
(4.2)
where σ is the largest singular value of Zs YL and is frequency dependent. σ (ω) is referred to as a principal gain in MIMO systems. If the induced norm of Zs YL is restricted to be less than unity for all frequencies, then σ (ω) < 1 for − ∞ < ω < +∞
(4.3)
Now, if R = Zs YL is normal, i.e. R commutes with its Hermitian, then from linear algebra, each singular value of R, denoted σi , may be written in terms of the corresponding eigenvalue of R, denoted λi , as σi (ω) = |λi (ω)|
(4.4)
- 66 Since the largest singular value σ is less than one, Y i
λi (ω) = det(R) = det(Zs YL ) < 1 for − ∞ < ω < +∞
(4.5)
Hence the characteristic loci will lie within the unit circle. Given that Zs and YL are stable subsystems, the system interface will be BIBO stable. It is possible then to use the norm of the product Zs YL or its principal gain σ and impose stability conditions on the system. Furthermore, by Cauchys inequality kZs k kYL k > kZs YL k
(4.6)
and if kZs k kYL k < 1, then the characteristic loci will not encircle (−1+j0). Equivalently, σ (Zs , ω) σ (YL , ω) < 1
(4.7)
This is a design-oriented criterion that allows for the design of the source subsystem impedance given the load characteristic impedance and vice versa. It is important to note that even if the return ratio Zs YL is not normal stability will still be ensured given that (4.7) holds. This is due to the fact that for any eigenvalue λi of Zs YL |λi | ≤ σ (Zs YL )
(4.8)
Another design-oriented stability criterion for ac systems may be derived in terms of the magnitude of the entries of the return ratio matrix Zs YL and the use of Gershgorans theorem. For convenience, this theorem is stated here from [9]. Theorem: If Z is a complex matrix of dimensions m × m, then the eigenvalues of Z lie in the union of the m circles, each with center z ii and radius ri =
m X
j=1, j6=i
|zij | , i = 1, ..., m
(4.9)
Since the return ratio matrix Zs YL is 2 × 2, restricting the largest entry in magnitude to less than 1/2 for all frequencies will ensure that the eigenvalues of Zs YL will be within the unit circle. Hence, the characteristic loci will not encircle the critical point (−1 + j0).
- 67 A new norm, k.kG , may be defined as kRkG = kZs YL kG = max (|r11 | , |r12 | , |r21 | , |r22 |)
(4.10)
kRkG < 1/2 for − ∞ < ω < +∞
(4.11)
If
the interface under consideration is BIBO stable provided Zs and YL have no poles in the RHP. This criterion does not require the return ratio to be normal. Hence, it is applicable to all forms of Zs YL . The advantage of this criterion lies in the fact that the eigenvalues of the return ratio matrix are not required for a stability test. For example, given YL , Zs may be designed according to the previous condition. Since the previous condition is in a coupled form, it is not clear how changes in the individual entries of Zs or YL will affect system stability. With some matrix algebra, it is possible to decouple this return ratio norm condition into a condition that is based on the individual norms of Zs and YL . Specifically, if Z s YL =
·
a 1 b1 c1 d1
¸·
a2 b2 c2 d2
¸
(4.12)
then a sufficient and necessary condition for (4.11) to hold is that |a1 a2 + b1 c2 | < 1/2
(4.13)
|c1 a2 + d1 c2 | < 1/2
(4.14)
|a1 b2 + b1 d2 | < 1/2
(4.15)
|c1 b2 + d1 d2 | < 1/2
(4.16)
The previous conditions will be satisfied if |a1 | |a2 | + |b1 | |c2 | < 1/2
(4.17)
|c1 | |a2 | + |d1 | |c2 | < 1/2
(4.18)
|a1 | |b2 | + |b1 | |d2 | < 1/2
(4.19)
|c1 | |b2 | + |d1 | |d2 | < 1/2
(4.20)
- 68 In turn, the previous conditions are satisfied if kZs kG kYL kG < 1/4
(4.21)
The interface under study will be stable if the previous condition is true. This criterion is in a form similar to the small gain criterion presented in Chapter 1 for SISO dc systems. Although, this is much more conservative than the characteristic loci criterion, this criterion may be very useful since the source impedance is usually much smaller than the load impedance. In cases where the previous criteria are too conservative, other norms may be considered. For instance, if the infinity norm is defined as # " X |zij | kZs k∞ =max i
(4.22)
j
which denotes the maximum row-sum of the largest magnitudes of the entries of Zs over all possible frequencies. The unity norm is defined as " # X kYL k1 =max |yij | j
(4.23)
i
which denotes the maximum column-sum of the largest magnitudes of the entries of YL over all possible frequencies. Then, based on Gershgorans theorem and the fact that each entry in R = Zs YL consists of a row-column inner product, BIBO stability of the interface in question is satisfied if kZs k∞ kYL k1 < 1/2
(4.24)
This appears to be a less conservative criterion than the previous one and extends the range of possible values for the characteristic matrices. It is important to note at this point that if the small-gain criteria proposed so far are satisfied, the system will be stable provided there exist no unstable open-loop poles. However, if these criteria are not met, the system may still be stable but cannot be guaranteed to be stable unless a complete generalized Nyquist test is performed. These criteria will be tested in the next section using the ideal three-phase CPL with input filter example.
- 69 4.2
Ideal Three-Phase CPL with Input Filter
In Chapter 2, it was shown that the generalized Nyquist criterion is applicable to the ideal three-phase CPL with input filter, and that stability may be predicted from the dq impedance characterization done at the interface of the CPL and the input filter. In this section, the same example will be used to: (1) discuss stability in light of the new criteria; (2) discuss the conservatism involved. 4.2.1
Stability using small-gain criteria
The initial test of the aforementioned criteria was performed for the 20-kW CPL with e input filter that was presented in Section 2.4. The small gain measures of Zeqds YqdL are
shown in Figure 4.1. All traces are below their established threshold for frequencies plotted. According to the previous analysis, the system must be stable. This is in agreement with the eigenvalues found for this case in (2.32). It is interesting to note that the singular value measure exhibits two sidebands with peak frequencies at f1,2 = fo ±
ωe 2π
(4.25)
where fo is the resonant frequency of the filter and ωe is the speed of the synchronous reference frame in rad/s. This result might be expected since it is known that the eigenvalues of an inductive circuit are shifted by ±jωe in the synchronous reference frame. However, this result is significant because the resonant frequency, which is usually used as a design parameter, is easily identified in the singular value measure plot. This fact will serve to show that the traditional techniques of filter design are also applicable using the proposed small-gain measures. Next, the CPL was increased to 200 kW and the results are shown in Figure 4.2. All small-gain traces are above the established threshold indicating that stability cannot be guaranteed in this case. In fact, the eigenvalue analysis in section 2.4 shows that the system is unstable. Hence, there is a good correspondence between the small-gain criteria and the eigenvalue analysis in this case.
- 70 -
0.35
Zs
∞
YL
G
YL
< 1/ 2
1
σ (Z s ) σ( YL ) < 1
0.3
Zs
0.25
G
< 1/ 4
0.2 0.15 0.1 0.05 0
0
500
1000
1500 2000 2500 Frequency in Hz
3000
3500
Figure 4.1 Small-gain measures for P0 = 20 kW, L = 77 µH, C = 300 µF, r = 0.1 Ω, v0q = 367 V, v0d = 0 V.
3.5
Zs
∞
YL
1
> 1/ 2
σ (Z s )σ ( YL ) > 1
3
Zs
2.5
G
YL
G
> 1/ 4
2 1.5 1 0.5 0
0
500
1000
1500 2000 2500 Frequency in Hz
3000
3500
Figure 4.2 Small-gain measures for P0 = 200 kW, L = 77 µH, C = 300 µF, r = 0.1 Ω, v0q = 367 V, v0d = 0 V.
- 71 -
0.4
Zs
∞
YL
1
< 1/ 2
σ (Zs )σ( YL ) < 1
0.35 0.3
Zs
G
YL
G
< 1/ 4
0.25 0.2 0.15 0.1 0.05 0
0
500
1000
1500 2000 2500 Frequency in Hz
3000
3500
Figure 4.3 Small-gain measures for P0 = 25 kW, L = 77 µH, C = 300 µF, r = 0.1 Ω, v0q = 367 V, v0d = 0 V. 4.2.2
Conservatism
For comparison purposes, the norms associated with the small gain criteria set forth in (4.7), (4.21), and (4.24) are plotted as a function of frequency in Figure 4.3. Therein, it is assumed P = 25 kW and Q = 0 which represents a stable case. The three plots in Figure 4.3 indicate that the singular value criterion set forth in (4.7) is the least conservative since its measure, σ (Zs ) σ (YL ) , is far below unity for all frequencies. The most conservative appears to be the G-norm criterion set forth in 4.24 whose measure approaches the threshold (1/4) closely. In order to quantitatively assess the conservatism assumed by the small-gain measures, gain margins are defined and compared with the gain margin obtained from the eigen-loci of the return ratio Zs YL . Based on the eigen-loci, the gain-margin, Gm , is defined as 1/ |G| where G is the vector starting at the origin and ending where the eigen-loci cross the negative real axis. For the small-gain measures, the gain margins were defined as
- 72 -
Gm−σ =
1 sup [σ (Zs ) σ (YL )]
(4.26)
ω
Gm−∞1 =
1/2 sup [kZs k∞ kYL k1 ]
(4.27)
1/4 sup [kZs kG kYL kG ]
(4.28)
ω
Gm−G =
ω
The previous gain margins were calculated for the ideal three-phase CPL with input filter while the real power was increased from 10 to 100 kW. The results are shown in Figure 4.4. All the gain margins are plotted in dB vs real power. At low power levels, all the gain margins are positive indicating a stable system. At approximately 90 kW, Gm turns negative indicating an unstable system whereas Gm−σ turns negative slightly below 80 kW. Throughout the power range studied the difference between Gm and Gm−σ is no more than 2 or 3 dB. Thus Gm−σ is not a very conservative measure as a gain margin for the example studied. The other two norms are obviously more conservative since the corresponding gain margins turn negative at much lower power levels. 20 15
Gm
dB 10
Gm −σ
5 0 -5
Gm −∞1 -10 -15
Gm − G 1
2
3
4
5
6
7
8
9
10
Real Power in 10 kW
Figure 4.4 Gain margins based on different measures for the ideal three-phase CPL with input filter.
- 73 4.3
Stability of Non-Ideal System Using Small-Gain Criteria
In order to illustrate how the small-gain criteria apply to non-ideal systems, the example studied in Chapter 3 is considered in the following section. In Chapter 3, the example finitebandwidth system shown in Figure 3.14 was tested at two interfaces using the generalized Nyquist criterion. At the first interface, between the generator an the three-phase filter, the load subsystem acts as a capacitive constant-power load which causes an unstable openloop pole. This renders the generalized Nyquist more difficult to use as a design tool at the interface and the small-gain criteria become inapplicable. To apply the small-gain criteria, it is recommended that Interface 2, between the filter capacitor and the converters, be used. At this interface, no unstable open-loop poles may result if the converters are stable. In dc systems, the interface under test is usually taken as close to the stable closed-loop converter as possible [6]. Thus, the load subsystem is ensured to have no positive open-loop poles when the closed-loop converter is designed to be stable. Interface 2 in Figure 3.14 was chosen for the small-gain criteria test. The system was stable under conditions as in study 2 in Section 3.7. The three small-gain measures for the return ratio Zqds YqdL are plotted in Figure 4.5. The three measures are very close to each other for all frequencies plotted. Both the singular value norm and the infinity-one norm based criteria, are satisfied except at high frequencies. The G-norm is slightly greater than 1/4, thus the G-norm based criterion is not satisfied at low frequencies either. However, this criterion was shown to be the most conservative. The most prominent infraction of the small-gain criteria is at a frequency of approximately 850 Hz. The double spike or side-band behavior will be shown in Chapter 5 to be the result of resonance peaks due to undamped LC filters. In this example, the resonance frequency of the LC filter due to the three-phase capacitor and the generator inductance may be approximated by fo ' 00
1 p 00 2π Ld C
(4.29)
where Ld is the subtransient inductance of the generator [7] and C is the line to neutral 00
capacitance of the filter. For the system in Figure 3.14, C = 180 µF, Ld = 215 µH, and fo ' 810 Hz which is roughly the same as the resonant frequency in the small gain plots.
- 74 -
35 30 25 20
Zoom A 15 10 5 0 -4 10
-2
10
0
10
2
4
10
10
Frequency in Hz
Figure 4.5 Small-gain measures at Interface 2 for example system under stable conditions.
Zoom A 0.3 0.25 0.2 0.15 0.1 0.05 0 -4 10
-2
10
0
10
2
10
4
10
Frequency in Hz
Figure 4.6 Zoom A for small-gain measures at Interface 2 for example system under stable conditions.
- 75 Although the system is stable with the presence of these resonance peaks, the stability gain or phase margin is usually degraded due to their presence [6]. In Chapter 5, it will be shown for simple three-phase filters that the singular value frequency plots of the qdimpedance matrix is akin the Bode plot of the dc filter impedance magnitude. It will also be shown that traditional filter damping techniques may be used to tailor the singular value measure of filter impedances, thus increasing the stability gain/phase margin of the system and satisfying the small-gain criteria.
- 77 -
5. THREE-PHASE FILTER DESIGN BY SINGULAR VALUES As was shown in the previous chapter, the singular value measure of the return ratio, ¡ e ¢ ¡ e ¢ σ Zqds σ YqdL , may possess very high peaks at specific frequencies. The peaks were
shown to be due to resonance points of undamped filters. If left undamped, the filters
degrade the system gain/phase margin and may cause instabilities. The magnitude of the peaks may be reduced by appropriate filter design. In this chapter, three-phase filters will be analyzed in the synchronous reference frame using singular value analysis in order to improve filter characteristics by using small-gain criteria and hence improve the system phase/gain margin. It was shown in the previous chapter that the least conservative small-gain stability critee rion is based on the induced 2-norm or equivalently the largest singular value of Zeqds YqdL .
In the following treatment, the singular value measure will be used to establish exact relationships between the characteristics of dc filters and three-phase ac filters represented in a synchronous reference frame. The aim will be to show that traditional filter design tools can be used to satisfy the proposed small-gain stability criteria and thereby ensure ac system stability. The filter characteristics to be investigated will include resonance frequency, peak magnitude, bandwidth, and roll-off slope. In order to accomplish this goal, a few conventional filter topologies will be investigated. First, a first-order RL filter will be investigated, then a second-order RLC filter will be studied, and finally, a damping branch will be added to the RLC filter and its effects will be analyzed using the following treatment. 5.1
First-Order RL Filter
For a simple three-phase RL filter, the output impedance may be expressed in the synchronous reference frame as Zeqd
=
·
R + sL ωe L −ωe L R + sL
¸
(5.1)
- 78 ° ° where ωe is the speed of the synchronous reference frame in rad/s. In order to find °Zeqd ° , ¡ ¢ which is also σ Zeqd , the following expression from linear algebra may be used r ³£ ¤ £ ¤´ ¡ e ¢ H σ Zqd = λmax Zeqd Zeqd (5.2) Substituting jω for s
whereupon
£ e ¤H £ e ¤ Zqd Zqd = σ
· ¡
R2 + (ω 2 + ωe2 ) L2 −2jωωe L2 2jωωe L2 R2 + (ω 2 + ωe2 ) L2
Zeqd
¢
=
¸
q
R2 + (ω + ωe )2 L2
(5.3)
(5.4)
p ¡ ¢ At low frequencies, where ω is close to zero, σ Zeqd = σ dc ≈ R2 + ωe2 L2 which shows
a dependence not only on R but also on the inductance and the speed of the synchronous ¡ ¢ reference frame. At very high frequencies, σ ∞ Zeqd = ωL where the effect of R and ωe become negligible compared to ωL
For a single-phase RL filter, the magnitude of the output impedance is simply |Zs | = |sL + R| =
√ ω 2 L2 + R2
(5.5)
where at low frequencies, |Zs |dc ≈ R, and at very high frequencies, |Zs |∞ = ωL. In ¡ ¢ comparison, σ 2 Zeqd is positively shifted from |Zs |2 by ωe2 L2 at low frequencies, but at ¡ ¢ high frequencies, σ Zeqd and |Zs | are essentially the same, and their slope is exactly the same. That is, the roll-up slope is 20 dB/decade as expected from first-order filters. The bandwidth ωb of the dc RL filter may be defined as the frequency at which |Zs | =
√ √ 2 |Zs |dc = 2R
(5.6)
In this case, ωb = R/L. For the case of the three-phase filter in the synchronous reference frame, the previous definition for the filter bandwidth translates to √ p ¡ ¢ √ σ Zeqd = 2σ dc = 2 R2 + ωe2 L2
(5.7)
The positive bandwidth frequency ωσb may be obtained ωσb =
p 2ωe2 + R2 /L2 − ωe
(5.8)
- 79 which shows that the filter bandwidth expressed in the synchronous reference frame is dependent on R/L as in the dc case but there is also a dependence on the speed of the reference frame ωe as expected. These results are illustrated in Figure 5.1 for a simple RL filter. This ¡ ¢ example shows that σ Zeqd may indeed be shaped by the same parameters as in singlephase filters. Namely, the ratio R/L affects the bandwidth, both R and L affect the dc gain, and the roll-up has a constant 20 dB/decade as in first-order filters. 40
20
0
( ( ))
e 20 log10 σ Zqd
20 log10 Z s
-20
-40
-60 -1 10
0
10
1
10
2
10
3
10
4
10
5
10
Frequency in rad/s ¡ ¢ Figure 5.1 Plots of σ Zeqd and |Zs | for a first-order RL filter. R = 1 mΩ, and L = 100 µH. 5.2
Second-Order RLC Filter
The second example in this section is similar to the input filter used in Section 2.4. Figure 2.2 shows the topology and Figure 2.4 shows the equivalent circuit in the synchronous reference frame. The output impedance of this filter in the synchronous reference frame is given by ¤−1 £ Zeqd = sC + TC + (R + sL + TL)−1
(5.9)
- 80 where T is defined by T=
·
0 ωe −ωe 0
¸
(5.10)
£ ¤−1 e = Zeqd is and R, L, C are 2 × 2 diagonal matrices with entries: R, L, and C. Since Yqd ¡ e¢ ¡ ¢ relatively easier to manipulate than Zeqd , σ Yqd will be derived first, then σ Zeqd may be obtained from
¡ e¢ ¡ ¢ σ Zeqd = σ −1 Yqd
(5.11)
As presented previously for the first-order filter, r ³£ ¤ £ e ¤´ ¡ e¢ e H Yqd σ Yqd = λmax Yqd
(5.12)
2 2 (ωp −1) +(R/Ro )2 ωp2 ¡ ¢ 2 2 2 e σ 2 Yqd = 2 Ro2ωp +R 2 2 ω −1 +(R/R ( m ) o ) ωm
(5.13)
After extensive algebraic manipulation
2 +R2 R2o ωm
where ωp =
√ ¡ ¢ = 1/ LC, and Ro = L/C. σ Zeqd is then r Ro2 ωp2 +R2 µ· ¸¶ 2 ¡ ¢ 2 2 2 σ1 σ Zeqd = max = max q (ωp −1) +(R/Ro ) ωp σ 2 2 2 ω+ωe , ωm ωo
=
ω−ωe , ωo ωo
ω
2
ω
(5.14)
Ro ωm +R 2 −1)2 +(R/R )2 ω 2 (ωm o m
There are several features to note in the previous expression. At low frequencies, where 2 , both singular values are equal and may be expressed as ω ' 0, ωp2 = ωm µ· ¸¶ s ¡ e ¢ L2 ωe2 + R2 σ1 σdc Zqd = max = (5.15) σ2 ω=0 (ωe2 LC − 1)2 + R2 C 2 ωe2 ¡ ¢ As expected, the dc magnitude of σ Zeqd depends not only on R but also on L, C and ωe .
2 At very high frequencies, ωp2 ' ωm , and σ1 and σ2 assymtotically fall down with a roll-off
slope of 20 dB/decade due to the presence of the capacitance. At frequencies close to the resonant frequency, ωo , σ1 and σ2 reach the same peak σ peak but at different frequencies. For σ1 , the peak is reached at ω1 = ωo −ωe ,and for σ2 , the peak is reached at ω2 = ωo +ωe . Substituting ω1,2 for ω in the previous equation yields
- 81 -
σ peak =
Ro2 /R
q
1 + (R/Ro )2
(5.16)
This is exactly the same value as the impedance magnitude-peak of the dc RLC filter at resonance which may be obtained from ¯ ¯ ¯ s/C + R/ωo2 ¯ ω 2 /C 2 + R2 /ωo4 ¯ ¯= |Zs | = ¯ 2 s + sR/L + ωo2 ¯ (ω 2 − ωo2 )2 + ω 2 R2 /L2
(5.17)
and substituting ωo for ω. High impedance values at resonance are undesirable and may be the source of instabilities. The fact that the peak singular value is reference frame speed independent is of high importance. This shows that the singular value stability criterion proposed in Section 4.1 may be satisfied by very simple and well established techniques of filter design. For example, by appropriate selection of Ro = L/C, or by adding damp¡ ¢ ing branches, the peak value of σ Zeqd may be reduced while maintaining the same cut-off p frequency 1/LC so that the small-gain stability criteria are met. An example case illus¡ ¢ trating how this can be done will be presented. Figure 5.2 shows a plot of σ Zeqd and a ¡ ¢ plot of the impedance magnitude of a dc RLC filter. σ Zeqd exhibits two peaks due to the two singular values which assume slightly different magnitudes near the resonance point. For frequencies below ωo , σ1 dominates and for high frequencies σ2 dominates, which ex¡ ¢ plains the side-bands behavior of σ Zeqd . Both peaks are equal to |Zs |max which is 60 dB
in this case. The plots in Figure 5.2 are slightly shy of the 60 dB mark because of plotting resolution. 5.3
RLC Filter with Damping Network
To illustrate the significance of the previous results, an unstable ideal three-phase CPL with input filter is considered. The filter is redesigned so that the small-gain stability criterion in terms of the singular value is met. For the RLC filter configuration presented in Section 2.4, a 200 kW CPL drives the system unstable as was shown in Chapter 2. In section 4.2, it was shown that the singular value stability criterion was also not met. According to the previous analysis, traditional filter design techniques may be used to reduce the singular value peak. For example, for a dc RLC filter, selecting a damping branch as shown
- 82 -
60 40
( ( ))
e 20 log10 σ Zqd
20
20 log10 Z s 0 -20 -40 -60 -2 10
0
10
2
10 Frequency in rad/s
4
10
6
10
¡ ¢ Figure 5.2 Plots of σ Zeqd and |Zs | for a second-order RLC filter. R = 1 mΩ, C = 1 mF, and L = 1 mH. in Figure 5.3 will reduce the impedance magnitude by a factor of 1/(1 + Ro2 /RRd ) for a large damping capacitor Cd , and a damping resistor Rd . The output impedance of this single-phase filter may be obtained by summing the admittances of the three branches then inverting " #−1 µ ¶−1 1 −1 + Rd + (sL + R) Zs = sC + sCd
(5.18)
In a similar fashion, the output impedance in the synchronous reference frame of the equivalent three-phase filter may be obtained o−1 n ¤−1 £ Zeqd = sC + TC + (sCd +TCd )−1 + Rd + (sL + R)−1
(5.19)
For this example, Rd was selected to be Rd = 1 Ω, and Cd = C. The previous equations were implemented in MATLAB and the singular values were computed. The result is shown in Figure 5.4.
¡ ¢ A dramatic reduction in the peak values of both σ Zeqd and |Zs | results due to the ad-
dition of the damping branch. A zoom of the area around the resonance point is shown in
- 83 -
R + _
L
Damping Branch
Rd Cd
C |Z s|
Figure 5.3 Single-phase RLC circuit with damping branch. Figure 5.5. Next, this filter was connected to the ideal three-phase CPL with P set to 200 kW. Figure 5.6 shows the three small-gain stability measures reflecting the effect of the damping branch. Compared to Figure 4.2, σ (Zs ) σ (YL ) is less than unity which indicate a stable £ ¤T e e system. Indeed an eigenvalue analysis readily supports this. If xeqd = ieqd , vqd , vqdD ,is e taken as the new state vector with ieqd denoting the inductor currents, vqd the capacitor volt-
e ages, and vqdD the damping capacitor voltages. The state matrix in (2.19), for the ideal
three-phase CPL with input filter, becomes −L−1 r − T −L−1 0 −1 −1 e −C−1 YqdL − T − R−1 R−1 A = C−1 D CD D CD −1 −1 −1 −1 0 RD CD −T − RD CD
(5.20)
where RD is the damping resistance matrix and CD is the damping capacitor matrix. With the damping branch included the eigenvalues of the previous state matrix are λ1,2 = −444.80 ± 5791.8i λ3,4 = −3467.2 ± 4997.6i λ5,6 = −4054.0 ± 0339.5i These are all negative indicating a stable system at P = 200 kW. The other two measures which are based on the infinity-one norm and the G-norm defined in Chapter 4 are not met in this case which implies their conservatism. This filter damping technique was also implemented in the finite-bandwidth 60-Hz system studied in Chapter 3 with Rd = 1 Ω and Cd = C = 180 µF. The results are shown in Figures 5.7 and 5.8. The eigen-loci are contained within the unit circle, thus dramatically
- 84 -
10
Zoom Area No Damping
0
Damping Present -10
-20
-30 -2 10
0
2
10
10
4
10
6
10
Frequency in rad/s
¡ ¢ Figure 5.4 Plots of σ Zeqd (solid line) and |Zs | (dashed line) for a second-order RLC filter with and without damping. R = 100 mΩ, Rd = 1. C = Cd = 300 µF, L = 77 µH.
No Damping 5
Damping Present 0
-5
4
10
Frequency in rad/s
¡ ¢ Figure 5.5 Zoom around the resonance point for plots of σ Zeqd (solid line) and |Zs | (dashed line) for a second-order RLC filter with and without damping. R = 100 mΩ, Rd = 1. C = Cd = 300 µF, and L = 77 µH.
- 85 -
Zs
1
∞
YL 1 > 1 / 2 σ (Z s )σ ( YL ) < 1
0.9 0.8
Zs
0.7
G
YL
G
> 1/ 4
0.6 0.5 0.4 0.3 0.2 0.1 0
0
500
1000
1500
2000
2500
3000
3500
Frequency in Hz Figure 5.6 Small-gain measures for ideal three-phase CPL with damped input filter and P set to 200 kW. improving the phase margin. When compared with Figure 4.5, the small-gain measures in Figure 5.8 indicate a dramatic reduction in the magnitude of the resonance peak, thus satisfying the small-gain criteria at all frequencies. It is interesting to note that all criteria are satisfied in this case. The selection of the damping capacitor and resistor is subject to weight, volume and power loss constraints. The latter is especially important to keep in mind since the systems studied here are ac. Other filter topologies, not presented here, may provide equivalent or even better filtering characteristics with satisfactory damping and less losses. In this chapter, design-oriented small-gain stability criteria were developed and it was shown that traditional filter design techniques can be used to satisfy these criteria. Expressions for filter characteristics, such as resonance frequency resonance peak, dc gain, and roll-off were derived in the synchronous reference frame in terms of the filter parameters and the speed of the synchronous reference frame.
0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.08 -0.06 -0.04 -0.02
0
0.02 0.04 0.06 0.08
Figure 5.7 Eigen-loci of example system return ratio at Interface 2 with damped filter capacitor.
0.3
Zs
∞
YL
1
< 1/ 2
σ (Z s )σ ( YL ) < 1
0.25
0.2
0.15 0.1
0.05
Zs 0 -4 10
G
YL -2
10
G
< 1/ 4 0
10
2
10
4
10
6
10
Frequency in Hz
Figure 5.8 Small-gain measures for example system at Interface 2 with damped filter capacitor.
- 87 -
6. STABILITY ANALYSIS OF A HYBRID AC/DC SYSTEM In this chapter, the generalized Nyquist criterion will be applied at various interfaces of the selected system including dc interfaces to assess overall system stability. The smallgain criteria proposed in Chapter 4 will then be applied at the same interfaces and the results compared with the generalized Nyquist tests. The existence of minimal realizations at different interfaces will be investigated in order to relate BIBO stability to BIBS stability. Conditions guaranteeing observability and controllability at an arbitrary interface will be set forth. Also, conditions relating the stability of one interface to another will be established so that the number of tests required for a given system may be reduced. Stabilizing control techniques that tailor the qd characteristic impedances and hence alleviate or altogether eliminate the conservatism involved in the proposed criteria will be presented. Finally, effects, such as prime mover and generator excitation control and phase-lock loop dynamics on the source impedance characteristics will be discussed. 6.1
System Description
In Chapter 3, a system with two converters and two inverters was presented and the generalized Nyquist criterion was applied at one interface. The results agreed with the eigenvalue analysis as well as the time domain simulation. Since stability at one interface does not automatically imply stability of the overall system, stability tests at all subsystem interfaces are required. In order to limit the number of studies, a smaller system is considered in this chapter. 6.1.1
System topology
As shown in Figure 6.1, the ac source consists of a 1200-Hz, 100-kW regulated synchronous machine that supplies power to an ac/dc converter. The converter regulates the dc link bus to 300 V for an inverter-fed induction motor. The inverter supplies a 750-Hz
- 88 ilinkC ilinkI
+ Cac
Lac
Clink
vlink
SM
IM
exfd
Excitor & Regulator
_ sabcC
sabcI
iabcC
Field Oriented Controller
Synchronous PI Regulator
iabcI
Figure 6.1 High-frequency ac/dc system. 70-hp field-oriented controlled (FOC) induction motor as described in Section 3.4. The dc link bus may also be distributed to provide dc power to other loads. The converter and inverter controls as well as the topology were described in Section 3.3. In order to obtain unity power factor control, the converter regulator was slightly modified as follows. The converter ac voltages are referred to a synchronous reference frame whose speed and angle are such that the d-axis synchronous voltage at the capacitor is zero at all times. The e synchronous dq -voltages vqd are given in terms of the rotor reference frame variables by
e vqd
=
r
r Ke vqd
=
·
cos δ sin δ − sin δ cos δ
¸
r vqd
(6.1)
If δ is taken as −1
δ = tan
µ
vqr vdr
¶
(6.2)
the d-axis voltage vde will always be zero even during load variations. In this reference frame, the converter input real power P is expressed as P = 32 vqe ieq and for a fixed voltage vqe the real power is proportional to the q -axis current ieq . Likewise, the converter input
reactive power Q is expressed as Q = 32 vqe ied and for a fixed voltage vqe the reactive power is proportional to the d-axis current ied . In the studies conducted herein, the reference d-axis ∗
current ied is set to zero thus ensuring unity power factor operation at all time.
- 89 6.1.2
System model
The generator and ac capacitor dynamics are expressed in the a reference frame coincident with the rotor of the synchronous generator [2]. The converter is modeled in a synchronous reference frame whose speed is such that the d-axis synchronous voltage at the capacitor is zero as discussed previously. The generator and converter models are voltage-input current-output while the filter (capacitor) model is current-input voltage-output. These models are reference frame and inputoutput compatible. The electrical dynamics of the inverter was expressed in a synchronous reference frame whose speed is equal to the commanded output frequency which is determined by the FOC as described in Chapter 3. Figure 6.2 shows the input-output blockdiagram of the overall system model and the interfaces to be studied. The excitor/regulator for the synchronous machine was modeled as shown in Figure 6.3. 6.1.3
System parameters
In this section, the parameters of the different components of the system will be presented and the control gains of the converter will be set forth based on a linearized model. The inverter FOC was described in Chapter 3 and the parameter calculations are straightforward, hence no further analysis of the FOC will be presented here. The synchronous machine and induction motor parameters were obtained from their respective manufacturers and are shown in Tables 6.1 and 6.2. Table 6.1 Synchronous machine parameters rs = 0.00416 Ω Xmq = 0.375 Ω Xlkd = 0.0512 Ω rs = 0.00688 Ω
Xls = 0.025 Ω Xlf d = 0.0152 Ω rkd = 0.0114 Ω fb = 1200 Hz
Xmd = 0.755 Ω rf d = 0.00127 Ω Xlkq = 0.0216 Ω P = 12
Table 6.2 Induction motor parameters rs = 0.051 Ω rr = 0.0308 Ω
Xls = 0.122 Ω fb = 750Hz
Xm = 1.82 Ω P =6
The proportional gain Kp = 8.7 and the integral gain Ki = 438.0 were selected to provide a regulator bandwidth of 10 Hz assuming that the generator open-loop time constant 0
is τdo [7]. For the converter, Clink = 1 mF was selected to maintain a 2% ripple on the link
- 90 -
Interface 1
ωr
Interface 2
Te vrqds
Synchronous Machine r e xfd
irqds
Excitor & Regulator
Yqds1
Te
Induction Motor ωr
v
e qdI
i
e qdI
vrqds
3-Phase Capacitor
e Ks
e vqds
e
irqdC
r Ks
e i qdC
Zqds1
ZqdL1 Interface 3 vlink 3-Phase Inverter
r
ilinkI
3-Phase Inductor
e eqdC
e iqdC
YqdL2 vlink
Link Capacitor ilinkC
e* i qdI
FOC
3-Phase Converter e* iqd Synch-PI Controller
YqdL3
Zqds3
Figure 6.2 Input-ouput block diagram of system model. bus voltage with a 2% capacitor current at 100% loading conditions. Cac = 32 µF, and Lac = 8 µH were selected to provide a 10-kHz cut-off frequency, which is about 10 times
the fundamental (1.2 kHz), thus providing a 40-dB attenuation at a switching frequency of 100 kHz. Kp = 1.3 and Ki = 2000 were selected from the following more formal analysis. Assuming a resistive load R on the right side of the link capacitor, the voltage at the link capacitor may expressed as dvlink 1 ³ vlink ´ ilinkC − = dt Clink R
(6.3)
where ilinkC is the converter current supplying the link capacitor as shown in Figure 6.1. This dc current may be expressed in terms of the input power, or the input q-axis current
- 91 -
exfd K K p + i s
+
vqe
K es
v abc
-vref Figure 6.3 Excitor/regulator model. and voltage in the synchronous reference frame as ilinkC = µ
Pin 3 vqeieq = µ vlink 2 vlink
(6.4)
Where µ is the efficiency of the converter, Pin is the real input power and vqe and ieq are the q -axis capacitor voltage and inductor current, respectively. Assuming ied is regulated to zero
for unity power factor control and vqe = Vqe is constant, ilinkC may be linearized in terms of the two-variable Taylor series as 3 Vqe Iqe 3 Vqe Iqe 3 Vqe e ilinkC = f (vlink , iq ) ' µ − µ 2 vlink + µ i 2 Vlink 2 Vlink 2 Vlink q
(6.5)
Where the first term on the right may be considered as the nominal dc converter current Iodc , the second term as the load current, and the third term as the dependence on the q-axis
current. Thus ilinkC may be expressed as ilinkC ' Io,linkC −
vlink + Gv ieq R
(6.6)
Where Gv may be interpreted as the steady-state voltage gain which is approximately 1/2 for the present system. Assuming that the q -axis current ieq tracks the commanded current ie∗ q , and using the control law presented in Chapter 3, ieq
=
ie∗ q
= Kp
=
∗ Kp (vlink
dζ + Ki ζ dt
− vlink ) + Ki
Z
t 0
∗ (vlink − vlink ) dt
(6.7) (6.8)
Where ζ denotes the voltage error integral Z t ∗ ζ= (vlink − vlink ) dt 0
(6.9)
- 92 Substituting the previous expression for ieq in (6.5) yields ilinkC = Io,linkC −
dζ vlink + Gv Kp + Gv Ki ζ R dt
(6.10)
Finally, the state representation of the system may be obtained from (6.3), (6.9) and (6-10) as d dt
·
vlink ζ
¸
=
·
−2 RClink
−1
−
Gv Kp Clink
Gv Ki Clink
0
¸·
vlink ζ
¸
+
·
Gv Kp Clink
1 Clink
1
0
¸·
∗ vlink Io,linkC
¸
(6.11)
This is a second-order representation of the converter based on three assumptions: (1) the input ac voltage deviations from the nominal value are negligible; (2) the inductor currents track the commanded currents; (3) the d-axis current is regulated to zero for unity power factor operation. Using this state space representation, the gains Kp and Ki were selected such that the converter regulation bandwidth is 200 Hz and the damping factor is 0.8. In the next section, results of the generalized Nyquist test and the small-gain criteria will be presented for three interfaces in the given system. 6.2
Generalized Nyquist and Small Gain-Criteria Analysis
In this section, the stability of several interfaces in the system will be investigated. First, a time-domain simulation showing both stable and unstable system behavior will be presented; the detailed and averaged models responses will be compared. Second, the averaged model will be linearized at stable and unstable operating points, and the generalized Nyquist eigen-loci of the return ratio at the given interfaces will be computed. Third, the small-gain stability measures at the same interfaces will be computed and presented. Finally, the frequency-domain results will be compared with an eigenvalue analysis wherein all the pertinent eigenvalues will be presented. To compare the detailed system model to the averaged model under stable and unstable operating conditions, the following scenario was simulated. Initially, it is assumed that all loads are disconnected from the system. The computer simulation is executed and the system is allowed to reach its steady-state equilibrium point. Then, the d-axis current of the induction motor is set to 50 A at 25 ms and the commanded q-axis current is set to 100 A at 80 ms. Once steady-state conditions are achieved, t = 120 ms, the commanded q-axis
- 93 current is increased to 400 A at 120 ms then to 500 A at 150 ms. The simulation is stopped at 190 ms. The results from the detailed simulation are shown on the left side in Figure 6.4, while the results of the averaged model simulation are shown on the right side. VQSE1 and IQCE1 denote the q -axis voltage and current of the converter. VLINK1 and ILINKC1 denote the voltage and current at the interface between the converter and inverter. TE1 is the induction motor electromagnetic torque. The ac variables are shown in Figure 6.5. VAC1 denotes the line-to-neutral a-phase voltage at the capacitor, IAC1 the a-phase converter input inductor current, VAI1 the inverter output voltage, and IAI1 the induction motor input current. It is interesting to note that the negative incremental resistance of the converter is very pronounced at high torque commands. As the generator voltage VQSE1 drops due to the increase in load (t > 120 ms), the converter current IQCE1 increases in order to maintain the output link voltage close to the reference. The two representations in Figure 6.5 are in close agreement in an average-value sense. This shows that the bounded linear controller used in the averaged model is a good approximation for the hysteresis band control for both stable and unstable operating conditions. The bounded linear controller allows linearization for small-disturbance stability analysis. Next, the generalized Nyquist eigen-loci were computed for three interfaces at a stable operating point with t = 120 ms. Then, Interface 2 was studied at an unstable operating point with t = 190 ms. For each interface, the small-displacement ABCD realization for Zs YL was established using the ACSL command analyze/jacob as described in Chapter 3.
The ABCD models were exported to MATLAB for subsequent calculations. The interface between the generator and the three-phase ac capacitor, Interface 1 in Figure 6.2, was studied first at the stable operating point t =120 ms. Since Zs YL is improper at this interface, Ys ZL was computed instead, and the inverse generalized Nyquist was applied as discussed in Chapter 3. The eigen-loci of Ys ZL are shown in Figure 6.6. There is one CCW encirclement of the critical point (-1+j0) even though the system is stable. Inspection of the eigenvalues of ZL in Table 6.3 however, reveals that there is an unstable pole. The net number of unstable closed-loop poles is then zero and the generalized Nyquist test is satisfied for a stable system as predicted by the time domain simulation. It is important to note, however, that the generalized Nyquist criterion requires extra effort at this interface
- 94 since knowledge of the open-loop poles is a must. For hardware implementation, this will require more tests which may be time consuming. It will be shown later that testing adjacent interfaces will be sufficient to determine stability at the current interface. Next, the small gain measures are plotted in Figure 6.7. As expected from the previous eigen-loci, all of the small-gain measures exceed unity. Although the criteria proposed in Chapter 4 are not met, instability of this interface cannot be implied from these plots. It is important to keep in mind that the small-gain criteria were developed to ensure stability assuming that there are no open-loop unstable poles. Hence, they are not applicable in the current interface. Furthermore, if the criteria are not met given a stable open-loop, the closed-loop system is not necessarily unstable. In other words, the small-gain criteria guarantee stability given a stable open-loop, but cannot be used to indicate instability if they are not met. Next, the interface between the converter and the ac filter capacitor is considered for Nyquist calculations. Since Zs YL is proper at this interface, the generalized Nyquist criterion is applied directly. The eigen-loci of Zs YL are shown in Figure 6.8. Note that the return ratio is 1 × 2 at this interface since the converter d-axis current is regulated to zero and the d-axis voltage is also zero in the synchronous reference frame chosen for the controller. Hence, there is only one eigen-locus even though the interface is ac. The second eigen-locus is always zero and does not contribute to the number of encirclements around the critical point (-1+j0). Figure 6.8 shows that the single eigen-locus is completely inside the unit circle. Since there are no unstable open-loop poles, the system is closed loop stable. Note also that the small-gain measures in Figure 6.9 are completely satisfied at this interface and, since the open loop is stable, closed-loop stability is guaranteed from the small gain criteria. This is in agreement with the time-domain response and the eigenvalues of the ABCD model of the composite systems shown in Table 6.4. The third interface studied is between the link capacitor and the inverter. Zs YL is also proper at this interface, therefore the generalized Nyquist criterion was applied directly. Since the interface is dc, Zs YL is 1 × 1 whereupon the single eigen-locus consists of the magnitude of Zs YL . Hence, the generalized Nyquist reduces to the SISO Nyquist criterion. Figure 6.10 shows that the single eigen-locus is completely inside the unit circle. Since there are no unstable open-loop poles, the system is closed-loop stable. Figure 6.11 shows
- 95 the small-gain measures which are also satisfied, hence interface stability is also implied from this test. This is in agreement with the time-domain response and the eigenvalues of the ABCD model of the composite system shown in Table 6.5. A final study was performed at Interface 2 with the inverter q -axis current set to 500 A. Due to the high negative damping caused by the inverter and converter at these loading conditions, the system becomes unstable. The eigen-loci of Zs YL are shown in Figure 6.12. Since there is one clockwise encirclements of the critical point (-1+j0), the system is unstable. Figure 6.13 shows the small-gain measures which are clearly beyond the respective thresholds for stability. System instability may not be implied from the small-gain measures. As discussed earlier, system stability may not be inferred in this case either. For this last test, the generalized Nyquist criterion agreed with the eigenvalue analysis and time domain results. The previous results show that the generalized Nyquist criterion is satisfied in terms of the return ratio, Zs YL , at various interfaces. The results also show an agreement between the small-gain and the generalized Nyquist criteria when the system is stable and the openloop system is stable. In the case of unstable interfaces, the small-gain criteria as well as the Nyquist stability test were not satisfied. However, in the case of the small-gain criteria, failure to meet the proposed thresholds does not always imply instability as was shown in Interface 1. Hence, the small-gain criteria cannot be used as an absolute instability indicator. It is important to note that the main objective of the small-gain criteria is to always guarantee stability of the given interface assuming the criteria are satisfied and the open-loop is stable. In the next section, techniques to reduce the number of interfaces that need to be analyzed for system stability assessment will be presented and conditions that guarantee BIBS stability from a single test will be set forth.
- 96 -
Figure 6.4 Detailed and averaged model response to increased torque commands.
- 97 -
Figure 6.5 Detailed and averaged ac voltages and currents
- 98 -
600
1200
400
1000
200
800
0
600
-2 0 0
400
-4 0 0
200
-6 0 0 -2 0 0
0
200
400
600
800
1000
Figure 6.6 Eigen-loci for stable operating point at Interface 1.
σ ( Z s ) σ ( YL )
0 -2 10
Zs
G
YL
G
Zs
∞
YL
1
0
10
2
10
4
10
Frequency in Hz Figure 6.7 Small-gain measures for stable operating point at Interface 1.
Table 6.3 System and subsystem eigenvalues at Interface 1 for stable operating point. Overall System Eigenvalues λ1 = -0.94181500 λ2 = -74.7382000 λ3 = -91.5579000 λ4 = -143.112000 λ5,6 = -74.7510000 +/-153.769000 λ7 = -901.151000 λ8,9 = -434.685000 +/-1069.93000 λ10,11 = -5700.89000 +/-66731.8000 λ12,13 = -4097.00000 +/-80899.6000 λ14,15 = -599817.000 +/-4820.15000 λ16,17 = -3.7427E+06 +/-849.127000
6
10
Eigenvalues of Source Subsystem (Zs) λ1 = -133.410000 λ2,3 = -1299.39000 +/-62.2176000 λ4,5 = -743.161000 +/-7456.98000 Eigenvalues of Source Subsystem (YL) λ1 = +2262.47000 λ2 = -74.7382000 λ3,4 = -74.6841000 +/-153.748000 λ5 = -1099.13000 λ6,7 = -9273.65000 +/-3496.97000 λ8,9 = -599817.000 +/-4820.15000 λ10,11 = -3.7427E+06 +/-847.259000
- 99 -
σ(Zs ) σ( YL ) Zs 0.06
G
YL
G
Zs
∞
YL 1
0.07
0.04
0.06
0.02
0.05
0
0.04
-0.02
0.03
-0.04
0.02 0.01
-0.06 -0.04 -0.02
0
0.02 0.04 0.06 0.08 0.1 0.12
Figure 6.8 Eigen-loci for stable operating point at Interface 2.
0 -4 10
-2
10
0
10
2
10
4
10
Frequency in Hz Figure 6.9 Small-gain measures for stable operating point at Interface 2.
Table 6.4 System and subsystem eigenvalues at Interface 2 for stable operating point. Overall System Eigenvalues λ1 = -0.94181500 λ2 = -74.7382000 λ3 = -91.5579000 λ4 = -143.112000 λ5,6 = -74.7510000 +/-153.769000 λ7 = -901.151000 λ8,9 = -434.685000 +/-1069.93000 λ10,11 = -5700.89000 +/-66731.8000 λ12,13 = -4097.00000 +/-80899.6000 λ14,15 = -599817.000 +/-4820.15000 λ16,17 = -3.7427E+06 +/-849.127000
Eigenvalues of Source Subsystem (Zs) λ1 = -0.92879800 λ2 = -106.997000 λ3 = -123.260000 λ4 = -901.904000 λ5,6 = -5539.54000 +/-66883.5000 λ7,8 = -3729.93000 +/-80828.6000 Eigenvalues of Source Subsystem (YL) λ1 = -74.7382000 λ2,3 = -74.7517000 +/-153.770000 λ4,5 = -440.319000 +/-1070.36000 λ6,7 = -599817.000 +/-4820.14000 λ8,9 = -3.7432E+06 +/-2880.31000
- 100 -
0.08
0.15
σ ( Z s ) σ ( YL )
0.06 0.04
0.1
0.02 0
Zs
G
YL
G
Zs
∞
YL
1
-0.02
0.05
-0.04 -0.06 -0.08 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
Figure 6.10 Eigen-loci for stable operating point at Interface 3.
0 -4 10
-2
10
0
10
2
10
4
10
Frequency in Hz Figure 6.11 Small-gain measures for stable operating point at Interface 3.
Table 6.5 System and subsystem eigenvalues at Interface 3 for stable operating point. Overall System Eigenvalues λ1 = -0.94181500 λ2 = -74.7382000 λ3 = -91.5579000 λ4 = -143.112000 λ5,6 = -74.7510000 +/-153.769000 λ7 = -901.151000 λ8,9 = -434.685000 +/-1069.93000 λ10,11 = -5700.89000 +/-66731.8000 λ12,13 = -4097.00000 +/-80899.6000 λ14,15 = -599817.000 +/-4820.15000 λ16,17 = -3.7427E+06 +/-849.127000
Eigenvalues of Source Subsystem (Zs) λ1 = -0.94181700 λ2 = -91.4554000 λ3 = -143.355000 λ4 = -901.058000 λ5,6 = -506.196000 +/-1038.13000 λ7,8 = -5700.79000 +/-66731.8000 λ9,10 = -4096.91000 +/-80899.5000 λ11,12 = -3.7427E+06 +/-849.568000 Eigenvalues of Source Subsystem (YL) λ1 = -74.7382000 λ2,3 = -74.7029000 +/-153.754000 λ4,5 = -599745.000 +/-4763.59000
- 101 -
400
1000
σ ( Z s ) σ ( YL )
300 800
200 100
600
Zs
G
YL
G
Zs
∞
YL
1
0 400
-100 -200
200
-300 -400 -800
-600
-400
-200
0
200
Figure 6.12 Eigen-loci for unstable operating point at Interface 2.
0 -4 10
-2
10
0
10
2
10
4
10
Frequency in Hz Figure 6.13 Small-gain measures for unstable operating point at Interface 2.
Table 6.6 System and subsystem eigenvalues at Interface 2 for unstable operating point. Overall System Eigenvalues λ1 = +76.1260000 λ2 = -74.7382000 λ3,4 = -287.384000 +/-61.5958000 λ5 = -921.087000 λ6,7 = -225.587000 +/-1107.22000 λ8 = -1468.75000 λ9,10 = -1168.94000 +/-6488.30000 λ11 = -36723.3000 λ12,13 = -1945.75000 +/-72149.3000 λ14 = -158622.000 λ15 = -3.7151E+06 λ16 = -3.8268E+06
Eigenvalues of Source Subsystem (Zs) λ1 = -0.12218000 λ2 = -201.906000 λ3 = -916.445000 λ4 = -36702.4000 λ5,6 = -1373.86000 +/-72937.5000 λ7 = -158624.000 Eigenvalues of Load Subsystem (YL) λ1 = -74.7382000 λ2 = -344.755000 λ3,4 = -232.500000 +/-1121.16000 λ5 = -1468.48000 λ6,7 = -1180.48000 +/-6466.07000 λ8 = -3.7162E+06 λ9 = -3.8268E+06
- 102 6.3
Existence of Minimal Realizations At Arbitrary Interfaces
The advanced small-gain and Nyquist criteria guarantee the boundedness of the dq voltage vqd at an interface with respect to a selected input u such as the reference voltage of a regulated generator. In order to guarantee system stability, all subsystem interfaces have to satisfy the criteria. If the transfer function at a given interface has a minimal realization, however, all of the eigenvalues of the system are preserved and appear as the system poles. In this case, system (BIBS) stability is satisfied by a single test (BIBO). In order to ensure the minimalilty of the transfer function realization at a given interface, the observability and controllability matrices must be full rank. These matrices may be obtained from the individual realizations of each transfer function constituting the interface transfer function F. From Chapter 2, the interface voltage vqd in terms of a given input u is
vqd = Fu = [I + Zqds YqdL ]−1 Hqd u
(6.12)
If Zqds is assumed to have realization As Bs Cs Ds and states xs , YqdL is assumed to have realization AL BL CL DL and states xL , and Hqd is assumed to have realization AH BH CH DH and states xH , then F may be obtained by using simple matrix manipulation as follows. Since the output variables are usually states, such as voltages across capacitors or currents through inductors, the feedforward matrix D for all realizations is usually zero. In cases where the interface is at a purely resistive load, or a load such as the ideal CPL, D will be the only nonzero matrix in the realization of YqdL . For generality, D will be considered non-zero. Also, it is important to note that AH = As and xH = xs since these subsystems are the same and constitute the subsystem looking from the interface towards the source. For Zqds YqdL , it may be shown that AZY =
·
BZY =
CZY =
£
As Bs CL 0 AL ·
Bs DL BL
¸
¸
Cs Ds CL
(6.13)
(6.14) ¤
(6.15)
- 103 DZY = Ds DL
with states xZY = [xs , xL ]T which are the overall system states. Now for [I + Zqds YqdL ]−1 · ¸ As − Bs DL MCs Bs CL − Bs DL MDs CL A I+Z Y −1 = (6.16) [ qds qdL ] −BL MCs AL − BL MDs CL B I+Z Y −1 = [ qds qdL ]
·
Bs DL M BL M
¸
(6.17)
£ ¤ C I+Z Y −1 = − MCs MDs CL [ qds qdL ]
(6.18)
D I+Z Y −1 = M = [I + Ds DL ]−1 [ qds qdL ]
(6.19)
And finally for F AF =
·
As 0 BL M(CH − CL ) AL − BL MDs CL BF =
·
BH BL MDH
¸
£ ¤ CF = M CH − Cs −Ds CL
¸
(6.20)
(6.21)
(6.22)
DF = MDH
The controllability and observability matrices are CF = C(AF , BF )
(6.23)
OF = O(AF , CF )
(6.24)
If the rank of the previous matrices is full, then the interface realization is minimal and only one test is necessary for overall system stability. It is important to note that a necessary but insufficient condition is the minimaliy of the load subsystem realization since any associated pole-zero cancellation will automatically translate into non-minimality of the overall realization for F. Therefore, a simpler test for non-minimality of the overall realization is
- 104 to check for the rank of CL = C(AL , BL )
(6.25)
OL = O(AL , CL )
(6.26)
If the rank is lower than the rank of AL , (AF , BF , CF ) will be automatically non-minimal. As an example, Interface 2 was tested for minimilaty using the previous condition and the result indicated that both CL and OL were of lower rank than AL . Therefore, in general, minimality at an interface is not satisfied and more tests will be required. This is intuitively expected since the example studied in Chapter 3 shows an instability at one interface (generator/converter) but yet the system is stable at another interface (inverter/induction motor). The number of tests may still be reduced if the above conditions are not satisfied. For example, if two interfaces are separated by a capacitor, Zs YL will be different at the two interfaces. However, since the voltage is the same for both interfaces, both interfaces will have the same transfer functions, F1 = F2 and hence the same stability characteristics. This is useful, since interfaces with capacitive CPLs, as in Interface 1 in the previous system, have unstable open-loop poles which render the Nyquist criterion more difficult to use. By the previous reasoning, Interface 1 may be inferred to have the same stability characteristics as Interface 2 since they are separated only by a capacitor. Since the Nyquist test is easily applied at Interface 2, there is no need to study Interface 1. In this section, conditions for minimality at an arbitrary interface were set forth and the possibility of reducing the number of tests required for system-wide stability was presented. 6.4
Stabilizing controls
In Chapter 5, it was shown that the small-gain measures for stability can be met by appropriately designing the three-phase filters present in the system. For example, adding an RC -damping circuit may provide the stability margin required. These options however, require more weight and volume, which is not always possible on board mobile systems. Other options include modifying the existing controls such that the system response is stable, or implementing additional controls whose sole function is to stabilize the system dur-
- 105 -
Tedes e vqC
Te*
N N 1 D sτ + 1 D
n
Figure 6.14 Nonlinear stabilizing controller. ing disturbances. In this section, the latter solution will be implemented in the hybrid ac/dc system presented in Section 6.1. The stabilizing controller shown in Figure 6.14 was chosen for this application. This controller was initially proposed as a dc-link stabilizer for a field-oriented induction motor drive [5]. In the present application, the non-linear controller will be shown to stabilize an ac interface as well. In a field-oriented induction motor, the torque response is instantaneous given that an appropriate rotor field is established and the ac currents track the commanded ac currents. This is usually the case and the output torque is insensitive to variations in the input voltage. Hence, at constant speeds, the inverter supplying the induction motor will exhibit a negative incremental resistance at the input. In the controller shown in Figure 6.14, the commanded torque is made to depend on the input voltage disturbances; a decrease in the input voltage will result in a momentary decrease in the commanded torque; an increase in the output voltage will result in a momentary increase in the commanded torque. Hence, the effect of negative incremental resistance is reduced. Another property of this controller is the ability to tailor the input impedance characteristic by proper choices of n and τ [5]. e By making the commanded torque depend on the input converter voltage vqC as shown
in Figure 6.14, the converter incremental q -axis input admittance may be obtained as e yqq,conv (s) =
sτ (n − 1) − 1 Pdes e2 sτ + 1 VqC
(6.27)
where a zero-d-axis synchronous reference frame has been assumed. Pdes denotes the desired torque-speed product of the induction motor. If n is set to zero, the controller is bye e passed and yqq,conv is negative. If n is taken greater or equal to one, yqq,conv can be made
positive which will provide positive damping and hence improve system stability.
- 106 If the commanded reactive power of the converter is a non-zero constant, there will also e be an incremental d-axis input admittance yqd,conv due to the non-zero commanded d-axis
current. However, if the converter reactive power is made to depend on the input converter e e as shown in Figure 6.14 with T replaced by Q, yqd,conv may be tailored to a voltage vqC
positive value by an appropriate choice of n as follows e yqd,conv (s) =
sτ (n − 1) − 1 Qdes e2 sτ + 1 VqC
(6.28)
Thus, all elements of the converter impedance matrix may be designed to provide positive damping with negligeable effect on the system response. The design trade-off in this technique lies in choosing values for n and τ that provide positive damping and yet minimize the output sensitivity, such as motor torque or converter reactive power, to input voltage disturbances. This is easily accomplished by setting τ larger than the converter time constants and n close to unity. For the high-frequency system presented in Section 6.1, the controller was implemented with the input voltage signal as the q-axis voltage at the input of the converter. The d-axis voltage is zero for the synchronous reference frame chosen. For the controller, τ was selected to be 100 ms and n = 2. The results are shown in Figures 6.15, 6.16, and 6.17; an obvious improvement results from the implementation of the nonlinear stabilizing controller. The time domain response of the system, on the right side of Figure 6.15, is stable, the generalized Nyquist is satisfied in Figure 16, and the small-gain measure in terms of the largest singular value is also satisfied. Hence, stability may be inferred from the frequency domain as well. This is confirmed by the eigenvalues of the system shown in table 6.7. In this section, it was shown that a non-linear stabilizing controller may be used to tailor the dq -input impedances and thus stabilize the system and satisfy the stability criteria proposed thus far.
- 107 -
Figure 6.15 System response with and without nonlinear stabilizing controller.
- 108 -
1.2
0.3 1
σ ( Z s ) σ ( YL )
0.2 0.8
0.1 0
Zs
G
YL
G
Zs
∞
YL
1
0.6
-0.1
0.4
-0.2 0.2
-0.3 -0.4
-0.2
0
0.2
0.4
Fig. 6.16 Eigen-loci at Interface 2 with nonlinear stabilizing controller present.
0 -4 10
-2
10
0
10
2
10
Frequency in Hz Fig. 6.17 Small-gain measures at Interface 2 with nonlinear stabilizing controller present.
Table 6.7 System and subsystem eigenvalues with nonlinear stabilizing controller present. Overall System Eigenvalues λ1 = -1.50619000 λ2 = -4.66543000 λ3 = -74.7382000 λ4,5 = -106.773000 +/-137.221000 λ6 = -200.000000 λ7,8 = -76.2576000 +/-611.281000 λ9 = -910.767000 λ10,11 = -429.341000 +/-996.086000 λ12,13 = -55484.1000 +/-49384.6000 λ14,15 = -16737.0000 +/-78701.9000 λ16 = -182010.000 λ17 = -586796.000 λ18 = -3.6671E+06 λ19 = -3.7438E+06
4
10
Eigenvalues of Source Subsystem (Zs) λ1 = -1.00005000 λ2,3 = -125.212000 +/-77.8909000 λ4 = -900.791000 λ5,6 = -1966.14000 +/-73821.3000 λ7,8 = -55575.5000 +/-49720.5000 Eigenvalues of Source Subsystem (YL) λ1 = -10.0000000 λ2 = -74.7382000 λ3 = -200.000000 λ4,5 = -76.9679000 +/-611.934000 λ6,7 = -371.252000 +/-991.666000 λ8 = -201552.000 λ9 = -598699.000 λ10 = -3.6651E+06 λ11 = -3.7439E+06
- 109 -
ts = nT
θvas
φ err K Kp + i s
ω
1 s
θvco
Phase Detector
Loop Filter
Figure 6.19 Detailed model of phase-lock-loop. 6.5
Effect of Phase-Lock Loop Control
In order to parallel a PWM-controlled converter with an ac source or other converters, synchronization of the switching signals with the ac voltages is required. If the ac source is used as a reference, the converter firing angles are obtained from the measured reference angle of the source. Traditionally, the reference angle is generated using a phase-lock loop (PLL). The a-phase voltage is sensed and scaled to a low level whereupon the zerocrossings are monitored to determine the frequency. A phase difference is generated, filtered, then used to drive a Voltage-Controlled-Oscillator (VCO) which in turn generates the output frequency. By locking onto the phase, a PLL determines both phase and frequency of the input signal [27]. A model of this process is shown in Figure 6.19. Since the phase difference is updated only once every cycle of the fundamental frequency (1200 Hz) for the system considered, a sample and hold model was used. This was implemented in ACSL by a discrete section [28]. The closed-loop time constant of a PLL is usually on the order of 5-10 cycles. For the 1200-Hz system studied, this translates to 4-8 ms. Time-domain simulations of this type of reference angle generator have shown no noticeable difference from an assumed perfectly synchronized system as shown in the following study. This result may be attributed to the small variations in frequency and power angle during load changes and the fast response time of the PLL. The same study presented in Section 6.2 is repeated here with the previous PLL model included. The system is driven to a stable operating point then the torque command on
- 110 the induction motor is increased until the system goes unstable. The results are shown in Figure 6.20. The detailed simulation results, on the left of Figure 6.20, compare very well with those in Figure 6.4. All the variables are described in Section 6.2. For the averaged system model, the previous PLL model was modified by bypassing the sample and hold section; i.e., the phase error was updated continuously.. The results are shown on the right hand side of Figure 6.20. Again, there is no noticeable difference between the results in Figure 6.4 and those in Figure 6.20. The detailed and averaged phase error are shown in Figures 6.21 and 6.22. The phase error reaches a maximum of approximately 0.2 rad or 11 deg which is small compared to the maximum 2π reached every cycle. Finally, the eigenloci and the small-gain measures were generated for Interface 2 shown in Figure 6.2. The operating point at t = 120 ms was chosen for this study. The eigen-loci and the small-gains, shown in Figures 6.23 and 6.24, were generated with the PLL included in the averaged model. The eigen-loci and the small-gains, shown in Figures 6.25 and 6.26, were generated with the PLL bypassed in the averaged model; i.e., the synchronous machine and converter were assumed to be in perfect synchronism. The plots show very little difference with the PLL model present. A direct method of generating the reference angle from a three-phase ac source is to transform the abc voltages to the stationary reference frame [7] then take the arc-tangent of the d and q-axis voltages to obtain the reference angle. The transformation to the stationary reference frame Kss is obtained by setting the angle in (2.4) to zero. Thus 1 −1/2 −1/2 √ √ 2 Kss = 0 − 3/2 3/2 3 1/2 1/2 1/2
(6.29)
The advantage in this technique lies in the elimination of the zero-crossing detection-delay which is inherent to PLLs. Hence, reference angle acquisition-time is significantly reduced. With this technique, the system may be considered to be perfectly synchronized at all time and the assumption made in the previous sections is valid.
- 111 -
Figure 6.20 Detailed and averaged system response with PLL included.
- 112 -
Figure 6.21 Phase error response with sample and hold model included.
Figure 6.22 Phase error response with sample and hold bypassed.
- 113 -
0.3
0.015 0.25
0.01 0.005
0.2
0
0.15
-0.005
σ ( Z s ) σ ( YL )
0.1
-0.01
Zs
G
YL
G
Zs
∞
YL
1
0.05
-0.015 -0.06 -0.04 -0.02
0
0.02 0.04
0.06
0.08
0 -4 10
-2
0
10
10
2
10
4
10
Frequency in Hz Figure 6.24 Small-gain measures at Interface 2 with phase-lock loop.
Figure 6.23 Eigen-loci at Interface 2 with phase-lock loop.
0.3
0.015 0.25
0.01 0.005
0.2
0
0.15
σ ( Z s ) σ ( YL ) Zs
-0.005 0.1
Zs
-0.01 0.05
-0.015 -0.02 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
Figure 6.25 Small-gain measures at Interface 2 without phase-lock loop.
0.08
0 -4 10
G
YL
G
∞
YL
1
10
-2
10
0
10
2
Frequency in Hz Figure 6.26 Small-gain measures at Interface 2 without phase-lock loop.
10
4
- 114 It is important to note that the small-gain measures in Figure 6.26 are higher than those shown in Figure 6.9 for the same study in Section 6.2. The impedances in Figure 6.9 were expressed in the synchronous reference frame with the d-axis voltage being zero while the impedances in Figure 6.26 were expressed in the rotor reference frame. Based on these observations, it appears that the synchronous reference frame wherein the d-axis voltage is zero is the least conservative reference frame for the small-gain measures. 6.6
Effect of Prime Mover
The time constants associated with prime movers, such as hydraulic, steam, or gas turbines, is usually much larger than the time constants considered in the electrical distribution system. The mechanical time constants are usually on the order of a few seconds, while the present system has a fundamental electrical frequency of 1200 Hz. Although the generator regulator bandwidth was chosen to be 10 Hz in order to represent a soft ac source, the regulator time constant is still significantly smaller than the mechanical time constants. Therefore, for the dynamics studied in the present system, the prime mover effect may be ignored.
7. CONCLUSIONS AND RECOMMENDATIONS 7.1
Conclusions
In this thesis, it was demonstrated that ac systems that include regulated solid-state converters are susceptible to instabilities due to the negative incremental input resistance (NIIR) characteristic of constant-power loads or CPLs. In order to evaluate system stability, a nonconservative (necessary and sufficient) ac stability criterion was developed based on the generalized Nyquist criterion and reference frame theory. This criterion is expressed in terms of the dq -impedance matrices of the constituent subsystems at selected system interfaces. A mathematical framework suitable for small-disturbance stability assessment of ac systems and for the computation of the dq -impedance matrices has also been set forth. Several example systems were studied under various operating conditions using time-domain simulations, eigenvalue analysis, and frequency-domain analysis. The results are in agreement with the generalized Nyquist criterion. Other stability criteria based on small-gain measures were also derived. These smallgain criteria represent sufficient but not necessary stability conditions that introduce conservatism compared with the generalized Nyquist criterion. However, the small-gain measures provide insight into the system and subsystem design. Comparisons in terms of the gain margins indicate that the least conservative of the small-gain criteria is the 2-norm based criterion. This criterion may also be expressed in terms of the product of the maximum singular value of the source dq -impedance and the maximum singular value of the load dq -admittance. It was also shown that simple filter design techniques, analogous to those used in dc systems, may be used to alleviate the conservatism that may arise due to the small-gain criteria. Also, as a more cost-effective technique, active nonlinear controls were shown to be capable of tailoring the input impedance of ac-dc converters thus stabilizing ac systems and meeting the required criteria.
- 116 The existence of minimal system realizations at arbitrary interfaces was studied and conditions were established in terms of the subsystem realizations linking BIBO stability with BIBS stability. Capacitive CPL interfaces; i.e., interfaces with CPLs and shunt capacitors, were shown to have open-loop-unstable poles which complicates the generalized Nyquist interpretation and renders the small-gain criteria unsuitable. However, if the interface is chosen as close to the CPL as possible, the generalized Nyquist test becomes straightforward and the small-gain criteria become applicable. Finally, it was shown that the small-gain criteria are affected by the selection of the reference frame in which the interface variables are expressed. For example, if the small-gain measures are expressed in a synchronous reference frame wherein the d-axis voltage is zero, the criteria are less conservative than if the measures are expressed in the rotor reference frame. 7.2
Recommendations
Although ac system stability criteria have been developed, the optimum approach of satisfying these criteria is not yet apparent. Optimization of the system design may be accomplished using the analytical methods set forth herein to evaluate passive versus active means of stabilization. Parameters to consider in design optimization should include efficiency, weight, volume, losses, and cost. It is also recommended that the relationship between the reference frame an the conservatism of the small gain measures be studied and thoroughly understood. Other possible reference frames, not considered in this thesis, might be considered in order to establish the least conservative measure. Finally, in this thesis, only small-disturbance stability of ac systems was addressed. In order to guarantee global stability for large disturbances, other criteria must be developed. Lyapunov techniques in combination with optimization tools such as Linear Matrix Inequalities (LMI) may offer a viable option in the development of large-disturbance stability criteria. The models used in this thesis are capable of portraying large disturbance dynamics very accurately, and can be used to formulate suitable energy functions for a Lyapunov-based analysis. Other techniques such as describing functions may extend the frequency-domain criteria for large-disturbance stability evaluation.
- 117 -
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- 121 -
- 122 VITA
Mohamed Belkhayat received his M. S. degree in electrical engineering from the University of South Carolina in 1989. He then taught fundamentals of electromechanical energy conversion and system controls at the Naval Post Graduate School in Monterey California for two years. In 1991, he joined David Taylor Research Center (NSWC/A) in Annapolis Maryland as a research engineer. Since then, he has been involved in modeling, simulation, and analysis of finite-inertia dc/ac distribution systems that include rotating machines and solid state power converters. His areas of interest also include linear and nonlinear stability assessment techniques for dc/ac systems.
