Robust sliding-mode control of dc/dc boost converter ...

IET Power Electronics Research Article

Robust sliding-mode control of dc/dc boost converter feeding a constant power load

ISSN 1755-4535 Received on 16th July 2014 Revised on 18th December 2014 Accepted on 19th January 2015 doi: 10.1049/iet-pel.2014.0534 www.ietdl.org

Suresh Singh, Deepak Fulwani ✉, Vinod Kumar Department of Electrical Engineering, Indian Institute of Technology Jodhpur, Jodhpur 342 011, India ✉ E-mail: [email protected]

Abstract: Tightly regulated power electronic converters show negative impedance characteristics and behave as a constant power load (CPL) which sink constant power from their input bus. This incremental negative impedance characteristics of tightly regulated point-of-load converters in multi-converter power systems have a destabilising effect on source converters and may destabilise the whole system. Similar phenomena also occur in many situations like dc microgrid, vehicular power system. Here, the authors present a robust pulse-width modulation-based sliding-mode controller for a dc/dc boost converter feeding the CPL in a typical dc microgrid scenario. A non-linear surface is proposed which ensures constant power to be delivered to the load. The existence of sliding mode and stability of the sliding surface are proved. The proposed controller is implemented using OPAL-RT real-time digital simulator on a laboratory prototype of dc/dc boost converter system. The effectiveness of the proposed sliding-mode controller is validated through simulation and experimental results under different operating conditions.

1

Introduction

Twenty first century is witnessing wider collaborative efforts in developing renewable energy technologies. Owing to developments in power electronics technology, there is an increasing trend in cascaded multi-converter power systems, including renewable energy systems and vehicular power systems (sea, land, air and space vehicles) [1, 2]. dc Microgrids, in particular, are getting much attention because of ease in integrating renewable energy sources producing dc power such as photovoltaic (PV), fuel cells etc. Furthermore, the availability of efficient dc loads, compatibility of storage systems with dc, absence of reactive power compensation are the main drivers for dc microgrids, among many others. However, cascaded multi-converter power systems, including microgrids pose severe stability challenges because of converter non-linearities and inter-converter dynamics, even though individual converters are stable [1, 3–8]. Generally loads in the cascaded multi-converter power systems are of two types, namely, constant power loads (CPLs) and constant voltage loads (CVLs). Tightly regulated point-of-load (POLs) converters in the cascaded multi-converter power systems behave as CPL, sinking constant power from input bus and show negative incremental impedance characteristics [9]. On the other hand, CVLs require constant voltage for their operation. Tightly regulated POL converters sinking constant power have a destabilising effect on poorly damped input LC filters and source converters operating in continuous conduction mode, inducing oscillations in input bus voltage and sometimes lead voltage collapse as well. Instabilities caused by such CPLs are termed as negative impedance instabilities [10–18]. Some solutions have been suggested in the literature to mitigate negative impedance instabilities such as load shedding, passive resistance damping, placement of capacitors and batteries at dc bus and controller-based methods [4, 6]. Some of the existing control solutions are based on the admittance space method [1, 17], feedback linearisation [13, 19], loop cancellation [15], pulse width adjustment [18], virtual capacitance emulation [11], sliding-mode control (SMC) [13, 20, 21] and other non-linear theory-based techniques such as Lyapunov and Bryaton–Mayer’s mixed potential functions [14]. Middlebook’s Nyquist impedance criterion-based admittance space solution requires reshaping of

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source or load impedance to compensate CPL in the small-signal sense [5, 16]. Controllers designed using linearising the plants about equilibrium point are generally insufficient to mitigate CPL effect. Therefore to mitigate CPL induced instabilities in highly non-linear dc distribution systems, there is a need of robust control design approaches which can ensure system stability in global or semi-global sense, see [22] and the references therein. In the presented paper, we use SMC approach to mitigate the CPL caused instabilities of multi-converter dc distribution systems. SMC is a well established and matured technique to design robust controllers for a variety of systems including power converters. SMC technique ensures robustness with respect to disturbances and parameter variations [23–28]. Preliminary results of the proposed paper are published in [21] wherein a discontinuous version of an SMC is presented for a boost converter. This paper presents non-linear switching function-based sliding-mode controller to mitigate CPL effect in a dc–dc boost converter. The presented scheme uses an actual non-linear model for control design and to prove the existence of sliding mode and stability of the switching function. To the best of the authors’ knowledge, this is the first work which addresses the mitigation of the CPL effect in dc–dc boost converter using non-linear switching function-based SMC. 1.1

Major contributions

This paper proposes a pulse-width modulation (PWM)-based SMC using non-linear sliding surface to mitigate instabilities caused by the CPLs in the multi-converter dc distribution systems with the following important features: (a) Considers total load on the system of CPL nature which represents the worst-case scenario from stability point of view. (b) The controller ensures supply of constant power demanded by CPL. The performance and the robustness of the proposed controller is validated through numerical simulations and experiments under different operating conditions. The brief outline of this paper is as follows. Section 2 contains system modelling and introduces instabilities caused by CPL in a

1

dc microgrid scenario. Section 3 proposes a non-linear sliding surface. The PWM-based SMC design and proof of the existence of sliding mode are given in Section 4. The stability of the sliding surface is proved in Section 5. Simulation results, experimental setup and experimental results are discussed in Section 6. Finally, the conclusion is given in Section 7.

2 2.1

Simplified equivalent circuit diagram of the system of Fig. 1a is shown in Fig. 1b. Source dc/dc boost converter has input voltage E. The total load is assumed to be CPL, which represents a worst-case situation from a stability point-of-view. The state space averaging model of systems of Fig. 1b becomes 1 x˙ 1 = [E − (1 − u(t))x2 ] L 1 P (1 − u(t))x1 − x˙ 2 = C x2

System modelling and small-signal stability System modelling

A typical dc microgrid in island mode consists of a source dc/dc boost converter, tightly regulated POL converter and a CVL as shown in Fig. 1a. The source dc/dc boost converter may obtain its supply from solar PV, fuel cells or another dc/dc converter. The function of source converter is to regulate the dc bus voltage within prescribed limits. The combination of CVL RL and load with front end POL converter R is connected to the dc bus. The POL converter feeding the resistive load R is tightly regulated to maintain VR constant at load terminals. Therefore if the load connected to POL converter has a one-to-one voltage–current characteristics, the POL converter paired with fixed resistive load R behaves as CPL drawing constant power from the dc bus. Tightly regulated POL converter behaving as a CPL can be modelled as follows P ; icpl (t) = vB (t)

x2 . 1,

(1)

where P is the rated power of the CPL, icpl is the current drawn by the CPL, vB is the dc bus voltage and ε is an arbitrary small positive value. CPLs in contrast to the conventional CVLs have negative values of incremental impedance, although the magnitude of impedance is positive. Fig. 2 shows voltage–current characteristics of a CPL. It can be seen from Fig. 2 that value of incremental impedance is negative. Therefore the current drawn by the load, having a negative incremental impedance increases/decreases with decrease/ increase in the dc bus voltage, which adversely affects the system stability [29].

(2b)

x1 ≥ 0

where x1 and x2 are the moving averages of inductor current iL and capacitor voltage vc, respectively. Model in (2b) is non-linear in nature. P is rated power of CPL and E is source converter’s input voltage. The switch resistance Rs, diode ON resistance Rd, inductor resistance rL and capacitor ESR (equivalent series resistance) are assumed to be zero; this keeps the natural system damping at its minimum. It implies that the effects of the CPL induced instability would be more pronounced and designed controller has to deal with the most serious situation. The u ∈ {0, 1} is the control input or switching function of the converter. The ideal model of the boost converter is justifiable as source converters usually have very high efficiency [3]. 2.2

vB (t) . 1

(2a)

Small-signal stability of uncontrolled system

In what follows, we investigate the small-signal stability of the dynamic model of the boost converter system given in (2b). The source dc/dc boost converter is operated in open loop with fixed duty cycle, that is, switching function of insulated gate bipolar transistor (IGBT) switch u is replaced by its fast average (instantaneous duty cycle) U. The resistance of connecting cable is neglected assuming source converter and load in close proximity [3]. With this the dynamic model of (2b) becomes 1 x˙ 1 = [E − (1 − U )x2 ] L 1 P (1 − U )x1 − x˙ 2 = C x2 x2 . 1,

(3a) (3b)

x1 ≥ 0

A dynamical system is stable in the small-signal sense at a given equilibrium point, if the system trajectory asymptotically converges to an equilibrium point after a small disturbance about the equilibrium point. In other words, the system is stable in the

Fig. 1 Typical dc microgrid system and its simplified equivalent circuit diagram considering boost converter supplying a CPL a A typical dc microgrid system b Simplified equivalent circuit diagram of (a), considering boost converter supplying a CPL

Fig. 2 Voltage–current characteristic of a CPL

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2

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small-signal sense if all eigenvalues have negative real part. Therefore to investigate small-signal/linearised stability of the dynamic model of the boost converter system of (3b), the eigenvalues of the system Jacobian matrix at the equilibrium point are analysed. The equilibrium point of the dynamic system of (3b) is given as (4) x2 ] := [ x1 ,

P , E

E (1 − U )

(4)

where x1 and x2 are the equilibrium values of x1 and x2, respectively. Jacobian matrix at the equilibrium point becomes ⎡ 0

⎢ J =⎢ ⎣ (1 − U ) C

⎤ (1 − U ) L ⎥ ⎥ P(1 − U )2 ⎦ CE 2

−

(5)

Here, trace (P(1 − U )2/CE 2) > 0 and determinant ((1 − U )2/LC) > 0 of the Jacobian given in (5) are positive quantities. This indicates that the equilibrium point of (4) is unstable. Furthermore, all the parameters in the expression of the trace and the determinant are positive and there is no control parameter which can be used to stabilise the system. Therefore the boost converter system of (3b) is unstable in open loop and can be operated only in closed loop. Furthermore, the system is non-linear in nature which requires a special attention.

3 3.1

The expression of the instantaneous duty cycle of converter u(t) consists of three terms. The first two terms represent equivalent control ueq and the third term represents discontinuous control uN, which ensures robustness. The control law (8) can be rewritten as u(t) = ueq + uN

Non-linear sliding surface

s := x1 x2 − x1r x2r

(6)

where x1r and x2r are the reference values of x1 and x2, respectively. In the second step, a controller is designed to bring sliding mode in finite time and force the system trajectory to stay on the sliding surface then on. It is intuitive from the selected switching function that when sliding mode is established the controller would ensure supply of constant power to the load. In the next section, we shall propose a PWM-based sliding-mode controller.

PWM-based sliding-mode controller

In case of conventional discontinuous SMC, the switching frequency may become very high and may cause excessive losses. In this section, we propose a PWM version of SMC to ensure a fixed switching frequency. To compute control law u(t) for the PWM-based sliding controller, the following reaching dynamics is chosen [30] s˙ = −ls − K sgn(s)

(7)

where l > 0 and K > 0 are the tuning parameters which are used to control convergence speed of switching function s. Using (3b), (6) and (7) and solving for u(t) results in the equation of instantaneous converter duty cycle in terms of the state variables and the converter parameters

(9)

The value of constant K in (8) should be chosen such that the term K (x21 /C) − (x22 /L)

Proposed PWM-based sliding-mode controller

This section proposes a sliding-mode controller for the non-linear system model defined in (2b). The first step in the design of a sliding-mode controller for a given system is to design a stable sliding surface which meets the system requirements. The proposed non-linear switching function is defined as follows

4

Fig. 3 Implementation scheme of the proposed PWM-based SMC

is some small percentage of ueq to ensure u(t) ∈ (0, 1). The implementation scheme of the proposed PWM-based SMC realisation is shown in Fig. 3. The controller requires information of inductor current and capacitor voltage references (x1r, x2r), sensed variables (x1, x2, iload), converter parameters (L, C ) and parameters (K, l) to compute desired control law. The inductor current reference x1r is computed online using relation x1r = (P/E) = (iloadx2/E). The computed control is compared with the triangular carrier signal of desired switching frequency to produce PWM pulses. 4.1

Existence of sliding mode

It is essential that trajectory starting from any initial condition reaches the sliding surface in finite time and constrained to the surface then on. The control law should be designed to ensure reachability condition. The existence of the sliding mode for the proposed PWM-based sliding-mode controller of (8) is proved based on the reaching dynamics given in [30] as follows. The reaching dynamics s˙ = −ls − K sgn(s)

(10)

for l and K > 0 ensures that reachability condition sT s˙ , −h|s|

(11)

is satisfied for some η > 0. Now, considering the reaching dynamics of (10), the left-hand side of the reachability condition sT s˙ becomes sT s˙ = sT [ − ls − K sgn(s)]

(12)

This implies (Px1 /Cx2 ) − (Ex2 /L) − ls K sgn(s) + 2 u(t) = 1 − (x21 /C) − (x22 /L) (x1 /C) − (x22 /L)

(8)

sT s˙ = [ − ls2 − K|s|]

(because sT s = |s|2 )

(13)



3

where K > 0 and l > 0. Therefore sT s˙ = −|s|[l|s| + K]

Table 1 Converter and controller parameters Sl. no.

This implies sT s˙ ≤ −h|s|

Parameter

Value

L C l K E Vref P

1 mH 1000 μF 16 × 104 24 × 106 50 V 200 V 1000 W

(14)

(15)

1 2 3 4 5 6 7

For all η ≥ l|s| + K. This completes the proof.

5

leads to

Stability of the sliding surface

The control law proposed in the previous section ensures that s = 0 is achieved in finite time. When s = 0 and the system is in the transient state x1x2 = x1rx2r, constant power is always delivered to the load. As t → ∞, x1 → (P/E) = x1r and this ensures x2 → x2r under the assumption of constant input voltage E. We prove although following lemma that at steady-state equivalent value of control equals to the duty cycle. This ensures that x2 approaches to x2r as t → ∞ and the demanded constant power to the CPL is maintained and thus system stability is ensured.

x2 − E =d x2

(16)

where d is the steady-state duty cycle. This ensures that x1 and x2 approach their respective reference values and constant power is supplied to the CPL. Proof: Equating s˙ = 0 in order to obtain an equivalent control law ueq implies x1 x˙ 2 + x2 x˙ 1 = 0

(17)

(1 − ueq ) =

(1 − ueq )x1 P E (1 − u)x2 + x2 − x1 − =0 Cx2 L C L

(18)

Solving (18) for (1 − ueq) gives (1 − ueq ) =

(Px1 /Cx2 ) − (Ex2 /L) (x21 /C) − (x22 /L)

(19)

At steady state x1 approaches its reference value x1r = (P/E). This

2 2 E (P /CE x2 ) − (x2 /L)

x2 (P2 /CE2 x2 ) − (x2 /L)

(21)

As t → ∞, (21) leads to ueq =

x2 − E x2

(22)

which is duty cycle d of the boost converter. This completes the proof of Lemma 1. □

6

Results and discussion

In this section, we present numerical simulation results using MATLAB SIMULINK and experimental results to validate the performance of the proposed controller. 6.1

Substituting x˙ 1 , x˙ 2 from the model given (2b) implies

(20)

This implies

Lemma 1: At the steady state, when s = 0, the equivalent control law becomes ueq =

(P2 /CEx2 ) − (Ex2 /L) (P2 /CE 2 ) − (x22 /L)

(1 − ueq ) =

Simulation results

The simulation has been conducted using the system of Fig. 1b and the proposed controller using the parameters given in Table 1. The transient response of the system has been simulated under two operating conditions: (1) the CPL power is increased by 50% at t = 0.6 s; restored back to its previous value at t = 0.65 s and the input voltage is increased by 30% at t = 1.1 s; restored back to its previous value at t = 1.15 s and (2) the CPL power is reduced by 50% at t = 0.6 s; restored back to its previous value at t = 0.65 s and the input voltage is reduced by 30% at t = 1.1 s; restored back to its previous value at t = 1.15 s. The results of the simulation are given in Figs. 4–6.

Fig. 4 Simulated start up response corresponding to the parameters given in Table 1 a Start up response of the output voltage b Start up response of the inductor current


4


Fig. 5 Simulated transient response corresponding to the operating condition (1) a Start up response of the output voltage b Transient response of the output voltage c Switching function d Start up response of the inductor current e Transient response of the inductor current f Control input

Fig. 6 Simulated transient response corresponding to the operating condition (2) a Transient response of the output voltage b Switching function c Transient response of the inductor current d Control input


5

Fig. 7 Image of the experimental setup

Table 2 Scaling parameters Sl. no. 1 2 3 4

Variable

Scale

output voltage and its reference inductor current and its reference input voltage load current

30 2 30 1

error. Fig. 4b shows that the inductor current tracks its reference value closely. The transient response corresponding to the operating condition (1) is shown in Fig. 5. In response to the changes in the CPL power and the input voltage corresponding to the operating condition (1), small magnitude (within ±1.5 V of reference value) spikes can be observed in the output voltage at the instants of step changes, as shown in Fig. 5a. The transient response of the inductor current to the changes in the CPL power and the input voltage is given in Fig. 5b, from which it is evident that it tracks changed reference values instantly. Fig. 5c shows that the average value of the switching function remains within acceptable limits and Fig. 5d gives plot of the generated PWM pulses which act as a control input to the converter switch. Simulated transient response corresponding to the operating condition (2) is shown in Fig. 6. In response to the 50% decrease in the CPL power at t = 0.6 s and 30% decrease in the input voltage at t = 1.1 s, the output voltage is maintained constant except transients within ±1.5 V at the instants of step changes (Fig. 6a). As shown in Fig. 6b, in response to the changes in the CPL power and the input voltage, the inductor current tracks changed references accurately. From Figs. 6c and d, shows switching function and generated PWM pulses, respectively. It can be seen from Fig. 6 that the average value of the switching function remains zero except at the instants of step changes in the CPL power and input voltage.

6.2 Fig. 4 shows start up response of the output voltage and the inductor current corresponding to the parameters are given in Table 1. It can be seen from Fig. 4a that the output voltage reaches its steady state in about 0.05 s with negligible steady-state

Experimental setup

To validate the performance of the proposed PWM-based SMC, an experimental setup was prepared in the laboratory as shown in Fig. 7. The experimental setup consists of a dc/dc boost converter, dc power supplies, dc programmable load, voltage/current sensors

Fig. 8 Experimental results corresponding to case 1 a Inductor current and its reference (scale: x-axis: 400 μs/div, y-axis: 100 mV/div) b Output voltage and its reference (scale: x-axis: 4 μs/div, y-axis: 500 mV/div) c Surface (scale: x-axis: 10 ms/div, y-axis: 500 mV/div) d Switching pulses (scale: x-axis: 10 μs/div, y-axis: 500 mV/div)


6


Fig. 9 Experimental results corresponding to case 2 a Inductor current and its reference (scale: x-axis: 400 μs/div, y-axis: 100 mV/div) b Output voltage and its reference (scale: x-axis: 4 μs/div, y-axis: CH2: 1 V/div, CH3: 200 mV/div) c Surface (scale: x-axis: 20 ms/div, y-axis: 500 mV/div) d Switching pulses (scale: x-axis: 20 ms/div, y-axis: 500 mV/div)

and OPAL-RT real-time digital simulator to implement the controller. The parameters of the dc/dc boost converter are L = 433 μH, C = 1000 μF and uses an IGBT switch (FGA25N120) and a fast recovery diode (MUR1520). First, dc power supply (30 V, 10 A) is used as an input power source for the dc/dc boost converter and the second dc power supply (30 V, 1 A) is used to power the voltage and current sensors. A dc programmable load is used as CPL to test the performance of the proposed controllers with the dc/dc converters feeding a CPL. The voltage and current signals required for the implementation of the proposed controller are sensed using Hall effect voltage (LEM LV 25-1000) and current (ACS 709) sensors.

A discrete fixed step size of 10 μs is used to compute the proposed controller. In all the experiments, the measured variables available in controller block running on real-time digital simulator were scaled down as given in Table 2, before sending them to real-time digital simulator output ports for their display on a digital phosphorous oscilloscope (DPO). Furthermore, measured variables are scaled down by (1/10)th when they reach at the monitoring ports of the real-time simulator. Measured variables from monitoring ports of the real-time digital simulator then directly displaced on the DPO. Therefore in order to determine actual value of a variable, DPO reading has to be multiplied by scaling parameter (from Table 2) × 10.

6.3

Experimental results

Experiments have been conducted on a dc–dc boost converter laboratory setup with proposed controller, implemented on OPAL-RT digital simulator. The values of converter and controller parameters are same as given in Table 1. The switching frequency is chosen to be 50 kHz. Experimental results are shown in Figs. 8– 10, corresponding to following operating conditions: (a) E = 50 V, Vref = 200 V and CPL power P = 100 W. (b) E = 50 V, Vref = 250 V and CPL power P = 150 W.

Fig. 10 Transient performance


Fig. 8 shows captured waveforms of the inductor current and its reference, the output voltage and its reference, switching function and switching pulses with E = 50 V, Vref = 200 V and CPL power P = 100 W. It can be verified that the steady-state error in the output voltage is