energies Article
Active Stabilization Control of Multi-Terminal AC/DC Hybrid System Based on Flexible Low-Voltage DC Power Distribution Wei Deng 1,2 , Wei Pei 1,2, * and Luyang Li 1,2 1 2
*
Institute of Electrical Engineering, Chinese Academy of Sciences, No. 6 Beiertiao, Zhongguancun, Beijing 100190, China;
[email protected] (W.D.);
[email protected] (L.L.) The School of Electronic, Electrical and Communication Engineering (EECE), University of Chinese Academy of Sciences, No. 19(A) Yuquan Road, Shijingshan District, Beijing 100049, China Correspondence:
[email protected]; Tel.: +86-010-8254-7111
Received: 28 December 2017; Accepted: 31 January 2018; Published: 27 February 2018
Abstract: Multi-terminal AC/DC interconnection will be an important form of future distribution networks. In a multi-terminal AC/DC system, if scheduled power for the AC/DC converter exceeds limits this may result in instability of the DC network. In order to overcome these limitations and avoid an unstable situation during coordinated control, this paper proposes a general active stabilization method for a low-voltage multi-terminal AC/DC hybrid system. First, the typical coordinated control modes for a hybrid system are analyzed. Second, a multi-level active stabilization controller, using the Lyapunov method, is introduced, and a feedback law allowing large signal stability is proposed. Finally, a system simulation model is further established, and the proposed active stabilization method is tested and verified. Study results show that only low stabilizing power with a slight influence on the DC network dynamic can improve the system’s stability and ensure stable system voltage. Keywords: multi-terminal AC/DC; coordinated control; flexible DC; stability; active stabilization
1. Introduction With the rapid development and widespread application of new energy sources, new materials, and power electronics technology, users’ requirements in terms of power quality, reliability, and operational efficiency are increasing constantly and, as a result, the existing AC distribution network is facing considerable challenges in many areas, such as the diversification of electricity demand, large-scale distributed-generation access, the complex coordinated control of power flow, and so on. On the one hand, the type and quantity of electrical equipment in the distribution network has changed, and a large number of electric vehicles (EV), energy-storage systems, LED lights, and other DC devices are being widely used [1–3]. On the other hand, if distributed generation, such as the photovoltaic (PV) fuel cell, adopts the DC grid-connected interface, some conversion sections can be reduced, and the overall operational efficiency can be improved [4]. Consequently, these trends make the development of DC-distribution technology inevitable. It is noteworthy that AC equipment is still the main form of power consumption in the distribution network, and that the access of DC equipment will lead to the long-term coexistence of AC and DC loads. Therefore, an AC/DC hybrid system will become an important form of power distribution in the future [5–9]. In the United States, the Center for Power Electronics Systems (CPES) of Virginia Polytechnic Institute and State University has built a hybrid distribution system based on AC/DC stratified connections [10]. The University of North Carolina has presented a future renewable-electric energy-delivery and management system (FREEDM), with a 400 V DC network and a 120 V AC
Energies 2018, 11, 502; doi:10.3390/en11030502
www.mdpi.com/journal/energies
Energies 2018, 11, 502
2 of 20
network, which connects to the external power grid through an intelligent energy management (IEM) interface [11]. Moreover, the micro-grid of Osaka University in Japan [12] and the European universal and flexible power-management system (UNIFLEX) [13] have also put forward AC/DC power-distribution systems. Coordinated control is an important foundation for the stable operation of a multi-terminal AC/DC hybrid system [14]. How to achieve the coordinated operation of various types of distributed generation, loads, energy storage, and AC/DC converters has become one of the technological challenges of AC/DC hybrid system development. To date, there have been some achievements in the field of coordinated control in a multi-terminal AC/DC hybrid system. In [15], coordination-control algorithms in a hybrid AC/DC micro-grid for converters, such as a PV panel, wind-turbine generator (WTG) with a double-fed induction generator (DFIG), and battery, are studied, and have been modeled and verified using Simulink in MATLAB. In [16], a hybrid structure for an AC grid-connected micro-grid, with a DC connection based on back-to-back (B2B) converters, is studied, a control scheme for the utility-interfacing voltage-source converter (VSC) and DC micro sources is proposed, and different operating scenarios, even faults inside or outside the micro-grid, are also investigated. In [17], typical operation modes for low-voltage (LV) AC/DC micro-grids are proposed, and a coordination-control method of utility-interfacing VSC, storage energy, PV, and direct-driven WTG with a permanent magnet synchronous generator (PMSG) under each operation mode, are put forward. In [18], an improved virtual-impedance control method is proposed for bi-directional power converters in hybrid AC/DC microgrids operated in island mode, in order to reduce the circulating current and for accurate power-sharing. In [19], a new droop control scheme is investigated for a hybrid microgrid formed by multiple AC and DC sub-grids, in order to ensure active power-sharing and the autonomous operation of the hybrid microgrid. In [20], the power-sharing control issues in hybrid AC/DC microgrids are discussed, the drawbacks of conventional voltage droop methods are described, and a new frequency droop-based strategy is proposed to share power in hybrid microgrids. This research mainly takes the DC bus voltage signal (DBS) as the judgment criterion in order to propose a hierarchical or coordinated control strategy based on different operating system modes, and adjusts each converter to ensure power balance under various conditions. However, this method can only make DC voltage maintain the ideal reference value when the utility grid is normal; in other cases, DC voltage will deviate from the ideal reference point, which obeys a differential regulation. Nevertheless, a DC network contains a lot of constant power load (CPL) in practical applications, and CPL has negative impedance characteristics. Therefore, DC voltage deviating from the ideal reference point may aggravate DC-voltage fluctuations, and even lead to the collapse of the whole system [21,22]. Tools allowing large signal-stability analysis of a DC-power system with CPLs, such as the Takagi-Sugeno (TS) multi-modeling, block-diagonalized quadratic Lyapunov function, Brayton-Moser’s mixed potential function, and reverse-trajectory tracking, have been introduced [22], and a general active control method is proposed for multi-CPL DC power networks in order to ensure the system is stabilized at an operating point, which would otherwise be unstable [23]. However, the above studies mainly focus on AC/DC systems with low power and multi-loads. There is still a lack of effective coordinated control methods for a multi-terminal AC/DC hybrid system with multi-sources, high power and higher voltage. Using a multi-terminal AC/DC hybrid system based on flexible LV DC power distribution, this paper establishes an electrical equivalent model and the state-space model required for coordinated control. An active stabilization-control method using the Lyapunov theory is then proposed, and feedback laws are designed to ensure system global stability, a wide operational boundary with lower control cost, stable voltage of the DC network, and the normal operation of each piece of equipment in some operating conditions that would otherwise be unstable. This paper is structured as follows: Section 2 outlines the typical structure and coordinated control mode of the LV multi-terminal AC/DC hybrid system. Section 3 carries out system-stability analysis in order to study the stable boundary. Section 4 proposes the multi-level active stabilization control method. Section 5 shows the corresponding simulation results and analysis. Conclusions are drawn in Section 6.
Energies 2018, 11, 502 Energies 2018, 11, x FOR PEER REVIEW
3 of 20 3 of 20
2. 2. Coordinated Coordinated Control ControlMode Modeof ofLow-Voltage Low-Voltage(LV) (LV)Multi-Terminal Multi-TerminalAC/DC AC/DCHybrid HybridSystem System 2.1. 2.1. System System Structure Structure The structure of of aaLV LVmulti-terminal multi-terminalAC/DC AC/DC hybrid system is shown in Figure 1. The typical typical structure hybrid system is shown in Figure 1. The The utility-interfacing VSC is the interconnection interface between the AC and DC systems. The AC utility-interfacing VSC is the interconnection interface between the AC and DC systems. The AC side side of utility-interfacing each utility-interfacing is connected to some AC node in the of each VSC is VSC connected to some AC line orAC AC line nodeor located in thelocated corresponding corresponding ACthe system, andofthe DCutility-interfacing side of each utility-interfacing VSC is to the It DCis AC system, and DC side each VSC is connected to connected the DC system. system. It is noteworthy that AC systems do not exist with interconnections directly between each noteworthy that AC systems do not exist with interconnections directly between each other, and each other, and each has thevoltage independent voltage and frequency support by utility its internal AC system hasAC thesystem independent and frequency support provided byprovided its internal grid, utility grid, respectively. The AC system can absorb power from the DC system or inject power respectively. The AC system can absorb power from the DC system or inject power into theinto DC the DC system the utility-interfacing VSC, according to theinstructions power instructions the system throughthrough the utility-interfacing VSC, according to the power from thefrom dispatch dispatch agency.onBased on theexchange power exchange DC system, load-balancing and agency. Based the power betweenbetween each ACeach andAC DCand system, load-balancing and powerpower-flow optimization among multiple AC systems can be achieved. flow optimization among multiple AC systems can be achieved. The TheDC DCsides sides of ofthe theVSC1, VSC1,VSC2, VSC2, and and VSC3 VSC3 are areconnected connected to to each each other other through through the the DC DC system, system, which build the themulti-terminal multi-terminalinterconnection interconnection structure. A device, DC device, such as WTG, a PV, WTG, which can can build structure. A DC such as a PV, EV, or EV, or battery energy-storage system (BESS) is usually integrated into the DC system through the battery energy-storage system (BESS) is usually integrated into the DC system through the DC/DC DC/DC converter. converter.
1
2
3
5
6
8 21
AC system2
4 7
9
10
11
12
13
14
15
16
17
18
19
20
24
25
26
22
23
...
28 29 ...
27
VSC2
VSC1 DC line 1
DC bus DC line 2
3
8 9 10 16 17
7 15 1 18
19
21
22
25
24
20 23
5
DC/DC converter
6 11 12 13 14
VSC3 PV/WTG/BESS DC line 3 DC system 5
26
... 12 18
17
4
...
AC system1
4
...
2
2
8
3 7
6
10
9 1
11
16 15 20 19 22 21
14
13 ...
AC system3
Figure1.1.Structure Structureof ofaalow-voltage low-voltage(LV) (LV)multi-terminal multi-terminalAC/DC AC/DC hybrid hybridsystem. system. Figure
2.2. Coordinated Coordinated Control ControlMode Mode 2.2. Theutility-interfacing utility-interfacingVSC VSCusually usuallyadopts adoptsUUdc and and Q Q control, control, P P and and Q Q control control and and droop droop control, control, The dc andthe thecorresponding correspondingdetailed detailedcontrol controlstrategies strategiesare areshown shownininFigure Figure2.2. and Take VSC1 shown in Figure 1 as an example; its P and Q control formedby bythe theouter outerpower power Take VSC1 shown in Figure 1 as an example; its P and Q control isisformed loop and inner-currents loop to realize active/reactive power-tracking scheduling, where P and Q, loop and inner-currents loop to realize active/reactive power-tracking scheduling, where P and Q, respectively,represent represent actual and reactive ofPtheand VSC; ref and Qref respectively, thethe actual activeactive powerpower and reactive power ofpower the VSC; QrefPrespectively ref respectively represent the reference active power and reactive power of the VSC (subscript represent the reference active power and reactive power of the VSC (subscript ref indicates the referenceref indicates theinreference of variables in this article); vabc three-phase represents the three-phase AC and voltages of variables this article); vabc represents the actual ACactual voltages of the VSC; iabc of the VSC;the and iabc represents theAC actual three-phase AC U currents of the VSC. Udc and Q control is represents actual three-phase currents of the VSC. dc and Q control is formed by the outer formed by loop the outer DC voltage loop and and is responsible for providing DC voltage and inner-currents loop, andinner-currents is responsibleloop, for providing constant voltage for the constant voltage for the DC network, where U dc,ref and Udc respectively represent the reference and DC network, where Udc,ref and Udc respectively represent the reference and actual voltage of the DC actual voltage of the DC network; idref and iqref, respectively, represent the d-q axis reference of threephase AC currents of the VSC (subscripts d, q respectively indicate the d axis reference and q axis reference of variables in this article); and id and iq respectively represent the d-q axis value of iabc.
Energies 2018, 11, 502
4 of 20
network; idref and iqref , respectively, represent the d-q axis reference of three-phase AC currents of the VSC (subscripts d , qPEER respectively indicate the d axis reference and q axis reference of variables in4 of this Energies 2018, 11, x FOR REVIEW 20 article); and id and iq respectively represent the d-q axis value of iabc . Droop control is formed by the outer voltageisdroop loop loop, and is responsible controlling the voltage DroopDC control formed byand theinner-currents outer DC voltage droop loop and for inner-currents loop, andfor is the DC network while sharing loads. responsible for controlling the voltage for the DC network while sharing loads.
+ Pref
Droop coefficient
P
Udc
+
Droop control idref
PI
+ iqref Udc,ref
idref,iqref
Udc&Q control + idref PI iqref
Udc,ref
dq
PWM
dq
abc
P&Q control
Pref, Qref Calculate
abc
Udc
+ id,iq
PI
P, Q -
+ PI PWM
vabc Electrical connection points
iabc va vb vc
ia ib ic
Udc
Figure 2. Typical control strategies of utility-interfacing voltage-source converter (VSC). Figure 2. Typical control strategies of utility-interfacing voltage-source converter (VSC).
Based on the aforementioned control strategies, the coordinated control of the LV multi-terminal Based on the aforementioned control the coordinated control ofmode. the LVFor multi-terminal AC/DC hybrid system can be divided intostrategies, master–slave mode or peer-to-peer the master– AC/DC hybrid system can be divided into master–slave mode or peer-to-peer mode. the slave mode, one VSC adopts the Udc and Q control to be master VSC in order to provide stable For voltage master–slave mode, one VSC adopts the Udc and adopt Q control be master VSCto in be order to VSCs provide of the DC network; meanwhile, the other VSCs the to P and Q control slave in stable order voltage of the DC network; meanwhile, the other VSCs adopt the P and Q control to be slave VSCs to adjust active/reactive power, accepting and tracking the power-scheduling command, in order to adjust active/reactive power,each accepting and tracking the power-scheduling command, respectively. For the peer-to-peer mode, VSC adopts droop control in order to provide voltage respectively. For load-sharing the peer-to-peer mode, each VSC adopts droop control in order to provide voltage support and DC collectively. support DCmainly load-sharing Thisand paper studiescollectively. the master-slave mode, so the voltage of the DC network is controlled This paper mainly studies the master-slave mode, so the voltage of the DC network is controlled by one master VSC, and the other slave VSCs track the power-scheduling command, avoiding voltage by one master VSC, and the other slave VSCs track the power-scheduling command, avoiding voltage regulation by multiple VSCs, and the coupling between individual controls. regulation by multiple VSCs, and the coupling between individual controls. 3. System Stability Analysis 3. System Stability Analysis 3.1. Equivalent 3.1. Equivalent Structure Structure At present, the master-slave master-slave mode mode of of LV LV multi-terminal multi-terminalAC/DC AC/DC hybrid At present, the hybrid systems systems is is more more common common in practical applications. Take Figure 1 as an example; under this mode, if VSC1 is the master VSC VSC to to in practical applications. Take Figure 1 as an example; under this mode, if VSC1 is the master provide constant DC voltage, and VSC2 and VSC3 are the slave VSCs accepting power scheduling, provide constant DC voltage, and VSC2 and VSC3 are the slave VSCs accepting power scheduling, assuming that the power loss of VSC2, VSC3 and the DC/DC converter can be neglected, the system equivalent structure can be presented in Figure 3. The DC line (rm, Lm) connects the master VSC and DC bus; DC lines (rs1, Ls1 and rs2, Ls2) connect each slave VSC and the DC bus.
Energies 2018, 11, 502
5 of 20
assuming that the power loss of VSC2, VSC3 and the DC/DC converter can be neglected, the system equivalent structure can be presented in Figure 3. The DC line (rm , Lm ) connects the master VSC and Energies 2018, x FOR PEER 5 of 20 DC bus; DC11,lines (rs1 , Ls1REVIEW and rs2 , Ls2 ) connect each slave VSC and the DC bus.
rm Lm im Master VSC
Um idc + DC/DC converter
Pdc
Udc Cdc
Figure 3. Equivalent structure of LV multi-terminal AC/DC hybrid system. Figure 3. Equivalent structure of LV multi-terminal AC/DC hybrid system.
3.2. Stable Operation Boundary of Voltage-Source Converter (VSC) 3.2. Stable Operation Boundary of Voltage-Source Converter (VSC) The system shown in Figure 3 can be modelled as follows: The system shown in Figure 3 can be modelled as follows: dim = U m − rm im − U dc Ldi m Lm dtmdt= Um − rm im − Udc Ls1Ldidts1dis1==UUdc − rs1 is1 − Us1 s1 dc − rs1is1 − U s1 Ls2 dis2dt= U − rs2 is2 − Us2 dc dtdi (1) dUdcs2 Pdc C = i L U = − ris2s1is2−−iU mdc − s2s2− Udc dc s2 dt d t Cs1 dUs1 = is1 − Ps1 (1) dtdU dc Us1 Pdc CdU Cs2 dcdts2dt= i=s2im−− UPis1s2s2− is2 − U dc dU s1 Ps1 = is1 − the variables of master VSC, slave VSC1, slave Cs1 Subscripts m , s1 , s2 , and dc respectively indicate dt U VSC2, and DC/DC converter in this article. Um and is1m , respectively, represent the DC voltage and dU s 2 Ps2 is2, −respectively, current of the master VSC. Us1 , is1 , CCs1s2, and =Ps1 represent the DC voltage, current, dt U slave capacitor and actual active power of the VSC1. Us2s2 , is2 , Cs2 , and Ps2 , respectively, represent the DC voltage, current, actual active powerthe of the slave VSC2. idc , CVSC, Pdc, VSC1, respectively, m, s1,capacitor, s2, and dc and respectively indicate variables of master slave slave Subscripts dc , and represent the DC current, capacitor, and actual active power of the DC/DC converter. VSC2, and DC/DC converter in this article. Um and im, respectively, represent the DC voltage and Slave slaveVSC. VSC2, haverepresent operational constraints current of VSC1, the master Us1and , is1,the Cs1,DC/DC and Ps1,converter respectively, the boundary DC voltage, current, between other, active and cannot will remain stable when they capacitor each and actual powerbeofadjusted the slavearbitrarily. VSC1. Us2, The is2, Csystem s2, and P s2, respectively, represent the cooperate with each other in a stable region; otherwise, it will be unstable. The stable operational DC voltage, current, capacitor, and actual active power of the slave VSC2. idc, Cdc, and Pdc, respectively, boundaries slave VSC1 and slave three conditions (Pdcconverter. = −30 kW, Pdc = 0 kW, represent theofDC current, capacitor, andVSC2 actual under active power of the DC/DC Pdc =Slave 30 kW) are shown in Figure based on Equation withoperational necessary parameters shown in VSC1, slave VSC2, and4,the DC/DC converter(1) have boundary constraints Table 1 (P , P , P , and P represent the rated power of the master VCS, between each other, and cannot arbitrarily. The system will remain stable whenslave they m,rated s1,rated s2,rated be adjusted dc,rated VSC1, slavewith VSC2, and DC/DC respectively). cooperate each other in a converter, stable region; otherwise, it will be unstable. The stable operational In Figure 4a, the DC/DC converter a power supply outputs power boundaries of slave VSC1 and slave VSC2works underas three conditions (Pand dc = −30 kW, Pactive dc = 0 kW, Pdcwith = 30 P −30 kW. in If P theonstable operation boundary ofparameters slave VSC1, represented Ps1.stab kW) shown Figure 4, based Equation (1) with necessary shown in Tableas 1 (P m,rated,, s2 changes, dc =are varies maximum stablepower operation of slave slave VSC1, VSC1,slave represented as Ps1,rated, accordingly, Ps2,rated, and Pand dc,ratedthe represent the rated of theboundary master VCS, VSC2, and DC/DC converter, respectively).
Energies 2018, 11, 502 Energies 2018, 11, x FOR PEER REVIEW
6 of 20
6 of 20
Table 1. Parameters of equivalent structure of LV multi-terminal AC/DC hybrid system.
Ps1.stabmax is obtained (Ps1.stabmax ≈ 95 kW) when Ps2 = 0; at the same time, if Ps1 changes, the stable operation boundary of slaveSymbol VSC2, represented , varies accordingly, and the maximum stable Value as Ps2.stab Symbol Value rm represented 0.0091 Ωas Ps2.stabmax Ls2 0.04469 mH(Ps2.stabmax ≈ 270 kW) when operation boundary of slave VSC2, , is obtained Lm 0.04469 mH Cs2 1050 μF Ps1 = 0. Cdc 1000 μF Um 800 V Table 1. Parameters ofrequivalent structure multi-terminal AC/DC hybrid system. s1 0.0182 Ω of LV Pm,rated 500 kW Ls1 0.0894 mH Ps1,rated 100 kW Symbol Cs1 Value Symbol 300 kW Value 500 μF Ps2,rated rs2 0.009ΩΩ Pdc,ratedLs2 80 kW 0.04469 mH rm 0.0091 Lm
Cs2
0.04469 mH
1050 µF
1000works µF Um supply and outputs 800 V active power with In Figure 4a, theCDC/DC converter as a power dc rs1 the stable0.0182 Ω boundary Pm,rated 500 kW Pdc = −30 kW. If Ps2 changes, operation of slave VSC1, represented as Ps1.stab, varies Ls1 0.0894 mH Ps1,rated 100 kW accordingly, and the maximum stable operation boundary of slave VSC1, represented as Ps1.stabmax is Cs1 500 µF Ps2,rated 300 kW obtained (Ps1.stabmax ≈ 95 s2 = 0; at the same time, if Ps1 changes, the stable operation rs2 kW) when P0.009 Pdc,rated Ω 80 kW boundary of slave VSC2, represented as Ps2.stab, varies accordingly, and the maximum stable operation boundary of slave VSC2, represented as Ps2.stabmax, is obtained (Ps2.stabmax ≈ 270 kW) when Ps1 = 0. When thethe output power of of thethe DC/DC Figure 4b4b shows that When output power DC/DCconverter convertergradually graduallyreduces reducestoto0 0kW, kW, Figure shows Ps1.stabmax has no obvious change and, meanwhile, P is decreased to about 260 kW. When s2.stabmax that Ps1.stabmax has no obvious change and, meanwhile, Ps2.stabmax is decreased to about 260 kW. Whenthe DC/DC converter is used as a load to absorb power with PdcPdc= =30 that PPs1.stabmax the DC/DC converter is used as a load to absorb power with 30kW, kW,Figure Figure4c 4c shows shows that s1.stabmax stillstill hashas nono significant change, while decreasedtotoabout about245 245kW. kW.InInsummary, summary,for fora DC/DC a DC/DC significant change, whilePs2.stabmax Ps2.stabmax isisdecreased converter, thethe output power decreases when it is used as a power supply,supply, or absorption power increases converter, output power decreases when it is used as a power or absorption power increases as load, P s1.stabmax , is affected less, but P s2.stabmax will be decreased, obviously. as load, Ps1.stabmax , is affected less, but Ps2.stabmax will be decreased, obviously.
Ps1 (kW)
Unstable region
Stable region
Ps2 (kW)
(a)
Ps1 (kW)
Unstable region
Stable region
Ps2 (kW)
(b) Figure 4. Cont.
Energies 2018, 11, 502 Energies 2018, 11, x FOR PEER REVIEW
7 of 20 7 of 20
Unstable region
7 of 20
Ps1 (kW)
Energies 2018, 11, x FOR PEER REVIEW
Stable region
Ps1 (kW)
Unstable region
Stable region
Ps2 (kW)
(c) Figure 4. Stable operation boundary of slave VSC1 and slave VSC2 under three conditions: (a) Pdc = Figure 4. Stable operation boundary of slave VSC1 and slave VSC2 under three conditions: (a) −30 kW; (b) Pdc = 0 kW; and (c) Pdc = 30 kW. Pdc = −30 kW; (b) Pdc = 0 kW; and (c) Pdc = 30 kW.P (kW) s2
(c)
3.3. Instability Analysis 3.3. Instability Analysis Figure 4. Stable operation boundary of slave VSC1 and slave VSC2 under three conditions: (a) Pdc = In order to research system stability, a simulation model of the LV multi-terminal AC/DC hybrid 0 kW; and (c) Pdc = 30akW. −30 (b) Pdc =system In order tokW; research simulation model of the LV multi-terminal AC/DC hybrid system depicted in Figure 3 isstability, established by MATLAB/Simulink (R2014a, The MathWorks, Inc., system depicted in Figure 3 is established by MATLAB/Simulink (R2014a, The MathWorks, Inc., Natick, USA), and the specific simulation parameters are shown in Table 1. The operational 3.3.MA, Instability Analysis Natick, MA, USA), and the specific simulation parameters are shown in Table 1. The operational condition with relatively less stability margin in Section 3.2 is selected to show the influence of the In order to research system stability, a simulation model of the LV multi-terminal AC/DC hybrid condition with relatively less stability margin in Section 3.2 is selected to show theadopts influence of the operation point change on system stability. In the simulation model, the master VCS Udc and system depicted in Figure 3 is established by MATLAB/Simulink (R2014a, The MathWorks, Inc., operation point change on system stability. In the simulation model, the master VCS adopts U and Q control with U m = 800 V; slave VSC1 and slave VSC2, respectively, adopt P and Q control with dc Natick, MA, USA), and the specific simulation parameters are shown in Table 1. The operational Ps1 Q control with U = 800 V; slave VSC1 and slave VSC2, respectively, adopt P and Q control with = 50 Kw and P s2 = 50 kW; and the DC/DC converter is used as a load with P dc = 30 kW. The rated linem relatively less stability margin in Section 3.2 is selected to show the influence of the condition with Ps1to-line = 50operation kW and point Ps2 =rated 50 kW; theof DC/DC is used asand athe load with Pdcadopts 30380 kW. The voltage and frequency AC system AC system 2, AC system 3= are and rated 50 change onand system stability. Inconverter the1, simulation model, master VCS UdcV and Q control with Um rated = 800 V; slave VSC1 VSC2, and system Q control3 with Ps1 V and Hz, respectively. line-to-line voltage and frequency ofand ACslave system 1, respectively, AC system adopt 2, andP AC are 380 = 50shown Kw and s2 = 50 kW; DC/DC point converter as a P load Pdc =is30inkW. rated line- the As inPFigure 4c, and the the operation (Ps1is=used 50 kW, s2 = with 50 kW) theThe stable region; 50 Hz, respectively. to-line voltage and rated frequency of AC system 1, AC system 2, and AC system 3 are 380 V and 50 and operation point s1 = 804c, kW, Ps2operation = 50 kW) is also (P in s1 the= stable closeistoinunstable region; As shown in (P Figure the point 50 kW,region, Ps2 = but 50 kW) the stable region; the Hz, respectively. the operation point (P s1 kW, = 90 kW, P s2 =kW) 50 kW) is already in the unstable region. operation point (P = 80 P = 50 is also in the stable region, but close to unstable region; and s1 in Figure 4c, s2the operation point (Ps1 = 50 kW, Ps2 = 50 kW) is in the stable region; the As shown operation point (Ps1 = 50 kW, Ps2 = Figure 5point describes the simulation waveform of Udc when the operation (P(P kW,PPs2s2 = kW) 50 kW) is in already in region, thethe unstable region. operation point 80 kW, = 50 is also the stable but close to unstable region; and s1s1== 90 50 Figure kW) is changed to (P s1 = 80 kW, Ps2 = 50 kW) and (Ps1 = 90 kW, Ps2 = 50 kW) at t = 2 s. As shown in the operation point (P s1 =simulation 90 kW, Ps2 = 50 kW) is already indc thewhen unstable region. 5 describes the waveform of U the operation point (Ps1 = 50 kW, 5a,Figure Udc has a short-term and smallwaveform oscillations Ps1 changes, canPs2quickly operation pointP(Ps1=but = 50 50 it kW, 5 describes Udc when Ps2Figure = 50 kW) is changed to the (Ps1simulation = 80 kW,amplitude Ps2 = 50of kW) andthe (Pwhen = 90 kW, kW) at= t = 2 s. s1 s2 restore a steady state. As shown in Figure 5b, when P s1 is changed to 90 kW, U dc cannot be restored 50 kW) is changed to (Ps1 = 80 kW, Ps2 = 50 kW) and (Ps1 = 90 kW, Ps2 = 50 kW) at t = 2 s. As shown in As shown in Figure 5a, Udc has a short-term and small amplitude oscillations when Ps1 changes, but it Figurevoltage 5a, Udc has a short-term and small amplitude oscillations Ps1ischanges, but it can quickly to a stable after the short-time damping oscillation, andwhen there then a divergent oscillation can quickly restore a steady state. As shown in Figure 5b, when P is changed to 90 kW, U cannot s1 90 kW, Udc cannot be restored dc restore a steady state. As shown in Figure 5b, when Ps1 is changed to until instability of the voltage. be restored to a stable the short-time dampingand oscillation, and there is then a divergent to a stable voltagevoltage after theafter short-time damping oscillation, there is then a divergent oscillation oscillation until instability the voltage. until instability of the of voltage. 1000
900 1000
800
900
800
700
500
600
300
805 800
805
500
U (V)
400
810 810
U (V)
600
U (V)
U (V)
700
400
795 800 790 795 790
300
1.95
200
1.95
200
2
2.05
Time2.05 (s)
2
2.1 2.1
Time (s)
100 100
0 1.5
0 1.5
1.6
1.6
1.7
1.7
1.8
1.8
1.9
1.9
2
Time (s)
2
Time (s)
(a) (a) Figure 5. Cont.
2.1
2.1
2.2
2.2
2.3
2.3
2.4
2.4
Energies 2018, 11, 502
8 of 20
Energies 2018, 11, x FOR PEER REVIEW
8 of 20
820
815
810
U (V)
805
800
795
790
785
780 1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
Time (s)
(b) Figure 5. Simulation waveform of Udc: (a) changing Ps1 from 50 kW to 80 kW at t = 2 s; and (b) changing
Figure 5. Simulation waveform of Udc : (a) changing Ps1 from 50 kW to 80 kW at t = 2 s; and (b) Ps1 from 50 kW to 90 kW at t = 2 s. changing Ps1 from 50 kW to 90 kW at t = 2 s. From the results shown in Figure 5, Ps1, Ps2, and Pdc have operation boundary constraints between each other, and in the process of their coordinated control, when the transmitted power of any one From the results shown in Figure 5, Ps1 , Ps2 , and Pdc have operation boundary constraints between exceeds the corresponding stable operation boundary, they will enter the unstable region, and DC each other, and in the process of their coordinated control, when the transmitted power of any one voltage could then be unstable, further affecting the stability of the whole system. This also means exceedsthat thethe corresponding stable operation they will enter thesystem unstable region, transmitted power of each VSC in anboundary, LV multi-terminal AC/DC hybrid will be limitedand DC voltage under couldoperation then be unstable, further affecting the stability of the whole system. This also boundary constraints, and a stability problem must be considered in order to means determine the power limitations of VSCs so as to avoid unstable control, which may cause operational that the transmitted power of each VSC in an LV multi-terminal AC/DC hybrid system will be failure or damage, and may even overreach these limited under operation boundary constraints, andlimitations. a stability problem must be considered in order to
determine the power limitations of VSCs so as to avoid unstable control, which may cause operational 4. Active Stabilization Control of LV Multi-Terminal AC/DC Hybrid System failure or damage, and may even overreach these limitations. 4.1. Stabilization Modeling
4. Active Stabilization Control of LV Multi-Terminal AC/DC Hybrid System The state variables in Equation (1) can be expressed as: ~
4.1. Stabilization Modeling
x* = xo + x
(2)
The state variables in Equation (1) can be expressed as:
where x = [im, is1, is2, Udc, Us1, Us2]T; superscript * indicates the current value of each variable; ~ superscript o indicates the steady-state value of each∼variable before a change or disturbance; x x∗ = xo + x indicates the variation of each variable. Considering the new state variable vector given by Equation (2), Equation (1) can be expressed by T Equation (3).
(2)
where x = [im , is1 , is2 , Udc , Us1 , Us2 ] ; superscript * indicates the current value of each variable; ∼ superscript o indicates the steady-state value of each variable before a change or disturbance; x indicates the variation of each variable. Considering the new state variable vector given by Equation (2), Equation (1) can be expressed by Equation (3).
∼
∼ 1 Lm Udc ∼ 1 Ls1 rs1 is1 − ∼ 1 Ls2 rs2 is2 −
= − L1m rm im −
dis2 dt
=
=
∼
∼ dUdc
dt
∼
di m dt ∼ dis1 dt
=
∼
dUs1 dt
=
∼
dUs2 dt
=
∼ 1 Ls1 Udc ∼ 1 Ls2 Udc ∼ 1 Cdc im
− − −
∼ 1 Cdc is1
−
∼ 1 Cs1 is1
+
∼ 1 Cs2 is2
+
∼ 1 Cdc is2
+
Us1 Ps1 ∼
o +U o Us1 s1 Us1
∼
Us2 Ps2
∼ o +U )U o (Us2 s2 s2
∼ 1 Udc Pdc ∼ Cdc o +U o Udc dc Udc
∼
1 Cs1 1 Cs2
∼ 1 Ls1 Us1 ∼ 1 Ls2 Us2
(3)
Energies 2018, 11, 502
9 of 20
Then, the system represented in (3) can be modeled by the linear model of Equation (4): ∼
dx ∼ = A x + Bu dt
(4) ∼
∼
∼
where u = [udc , us1 , us2 ]T ; and udc , us1 , and us2 , respectively, represent the input of Udc , Us1 , and Us2 . The Lyapunov equation can be represented as Equation (5) [24]: PA + AT P + Q = 0
(5)
where P and Q are positive-definite symmetric matrices, and P is a solution of the Lyapunov equation. The feedback law for active stabilization control is designed as follows:
udc,fb u0 = us1,fb us2,fb
0 0 0 = − 1 Cdc 0 0
0 0 0 0 1 Cs1
0
0 0 0 0 0 1 Cs2
T
∼ i m ∼ is1 ∼ is2 P ∼ U dc ∼ Us1
(6)
∼
Us2 ∼
where u0 = [udc,fb , us1,fb , us2,fb ]T ; and udc,fb , us1,fb , and us2,fb , respectively, represent the feedback of Udc , ∼
∼
Us1 , and Us2 . Proof of Theorem 1. The Lyapunov equation is represented as: V (x) = xT Px
(7)
therefore,
· dV (x) · T = xT Px + xT Px = xT A T P + PA x + u0 B T Px + xT PBu0 dx Since P is a symmetric matrix, so:
(8)
dV (x) = xT A T P + PA x + 2xT PBu0 dx
(9)
u0 = −BT Px
(10)
dV (x) = xT A T P + PA − 2PBB T P x dx
(11)
As u is chosen as: So:
It is well know that −2PBBT P represents negative-definite matrices because BBT and PP are both positive-definite matrices, and PA + AT P < 0 from (5), therefore:
Thus:
AT P + PA − 2PBBT P < 0
(12)
dV (x)
