A series parallel PV storage independent microgrid and its decentralized control

Transcription

1 Received: 27 January 208 Revised: 6 July 208 Accepted: 4 August 208 DOI: 0.002/etep.275 RESEARCH ARTICLE A series parallel PV storage independent microgrid and its decentralized control Lang Li,2 Yao Sun,2 Zhangjie Liu,2 Xufeng Yuan 3 Mei Su,2 Xiaochao Hou,2 Minghui Zheng 4 School of Information Science and Engineering, Central South University, Changsha, China 2 Hunan Provincial Key Laboratory of Power Electronics Equipment and Grid, Hunan, China 3 Electrical Engineering School, Guizhou University, Guiyang, China 4 Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, New York, USA Correspondence Zhangjie Liu, School of Information Science and Engineering, Central South University, Changsha 40083, China. liuzhangjie@csu.edu.cn Funding information National Natural Science Foundation of China, Grant/Award Number: 66223; Joint Research Fund of Chinese Ministry of Education, Grant/Award Number: 64A ; Natural Science Foundation of Hunan Province of China, Grant/ Award Number: 206JJ09 Summary This paper presents a series parallel photovoltaic (PV) storage independent microgrid (MG), where low voltage distributed generation units could be connected to power grid conveniently. Moreover, it provides a promising way to form medium voltage MG. For this proposed structure, a decentralized control is proposed. The PV units are controlled by a maximum power point tracking (MPPT) based droop control, where MPPT of the PV units is achieved. Storage units are regulated via a droop controller dependent its reactive power polarity and take the responsibility of keeping the power supply demand balance of the MG. Since all these controllers make decisions only based on their local information, the proposed control is reliable and cost effective. Moreover, the small signal stability of the system is proved. And simulation results are presented to verify the effectiveness. KEYWORDS decentralized scheme, droop control, microgrid, PV storage MG INTRODUCTION Recently, interests have been concentrated on integrating renewable energy sources into modern power distribution systems. -3 Among the renewable sources, photovoltaic (PV) has become one of the major distributed generators (DGs) due to environmental concerns and continuous decrease in the price. 4-6 The concept of PV storage microgrid (MG) is an effective way to integrate PV sources. 7-9 When the support from the public utility is unavailable, storages are employed as a critical element to maintain the power balance due to the intermittent nature of PVs. In the islanded PV storage MG, the proper power sharing among PVs and storages and maintaining voltage/frequency regulation of system are two major tasks. The PV storage MG, based on its connection way, can be classified into three typical configurations,0-2 : parallel (Figure A,B), series (Figure C), and series parallel (Figure D,E). Based on the allocation of PVs and storages, it can be classified into two typical structures 3-6 : PV storage integrated system (Figure A,C,D) and PV storage independent system (Figure B,E). The parallel PV storage integrated MG shown in Figure A, as the earlier research, storage unit and PV inverter are integrated together to form an ideal DG. It is usually controlled by decentralized approaches based on droop concept In these techniques, all the DGs are controlled as voltage sources to maintain the voltage/frequency regulation and power balance of Int Trans Electr Energ Syst. 208;e wileyonlibrary.com/journal/etep 208 John Wiley & Sons, Ltd. of6

2 2of6 LI ET AL. FIGURE Diagram of PV storage MG. A, parallel PV storage integrated system; B, parallel PV storage independent system; C, series PVstorage integrated system; D, series parallel PV storage integrated system, and E, series parallel PV storage independent system. MG, microgrid; PV, photovoltaic system. However, storage units in the PV storage DG need to be charged and discharged frequently due to the power output fluctuation of PVs, which leads to low efficiency and short lifespan. The parallel PV storage independent MG shown in Figure B becomes popular at the present researches, in which PVs and storages work as the separate DGs. In order to realize the voltage/frequency regulation of system, PVs and storages are preferred to behave as voltage sources. Du et al 2 designed a voltage source controller of PV inverter in which its voltage stability of dc bus is guaranteed. Elrayyah et al 22 presented an autonomous method, in which dp/dv is embedded into the droop scheme by using a proportional integral (PI) controller to realize maximum power point tracking (MPPT) and voltage/frequency regulation. Further, Elrayyah et al 23 introduced a universal controller, in which MPPT and dc bus voltage regulation are taken into account. Vandoorn et al 24 proposed another strategy that combines V g /V dc droop and P/V g droop together to ensure proper power sharing and to limit voltage variation. Due to the low voltage characteristic of PV and storage unit, it is difficult to connect with the medium /high voltage power networks directly. The series PV storage integrated MG shown in Figure C, which consists of cascaded H bridge (CHB) converters, has the capability of supplying a higher voltage levels The CHB converter is considered as one potential candidate for PV power generation. Yu et al, 30 Zhang et al, 3 and Xiao et al 32 have conducted lots of research on grid connected PV generation system; however, the studies on the islanded mode are less. Recently, He et al 2 studied series PV storage integrated MG in islanded mode, and an inverse power factor droop control has been proposed.

3 LI ET AL. 3of6 However, this control method is only suitable for resistance inductance loads. Sun et al 0 introduced an f P/Q method under the resistance inductance and resistance capacitance loads. In order to integrate the advantages of the parallel structure and the series structure, He et al introduced a seriesparallel PV storage integrated MG shown in Figure D as a combination of Figure A and C, in which the characteristic of PVs is not considered. Because the power balance control methods applied to the parallel or series MG cannot be directly performed in a series parallel system, a hierarchical power regulation method is presented. However, this method needs the communication networks, which increases costs and the failure probability due to communication failure. To the best of our knowledge, the decentralized control applied to the series parallel MG has been not reported. To address the above concerns, we introduce a new series parallel PV storage independent structure shown in Figure E as an expansion of Figure B and C, which holds the PV storage independent characteristics. The cascaded converter topology are emerging in middle and high level voltage storage system application due to its merits in using low voltage semiconductor switches, producing low output voltage harmonic distortion and inherent modularity that can acquire simple voltage scaling properties without expensive and bulky step up transformers. 7,30,33,34 It is applicable for large scale PV penetration. In addition, a fully decentralized control approach is proposed for this series parallel PV storage independent structure to ensure the power balance and voltage/frequency regulation of system. The main features of the proposed scheme are summarized as follows: ) It has increased reliability and decreased capital costs, as the scheme only needs the local information. 2) MPPT of each PV unit is ensured. 3) Synchronization of PV inverters and storage inverters can be guaranteed under both inductive and capacitive loads. The contributions of this paper are twofold. First, we introduce a new series parallel PV storage independent MG. Second, a fully decentralized control approach is proposed for this series parallel PV storage independent MG to ensure the power balance and voltage/frequency regulation of system. The rest of the paper is organized as follows. Section 2 shows the proposed series parallel PV storage independent MG. The proposed decentralized control frame for PVs and storages is presented in section 3. The stability analysis of system and sensibility analysis of parameters are carried out in section 4. Then, the simulation validations in section 5 are presented to verify the effectiveness and performance of the proposed scheme. Finally, the paper is concluded in section 6. 2 PROPOSED PV STORAGE MG 2. Structure of PV storage MG Figure 2 shows the configuration of the proposed series parallel PV storage independent MG in the islanded mode. The separate low voltage unit of PV or storage is series connected as a string generation module. Then, these string FIGURE 2 Diagram of the proposed series parallel PV storage independent MG. MG, microgrid; PV, photovoltaic

4 4of6 LI ET AL. generation modules are parallel connected to form a higher voltage PV storage MG without needs of step up transformer. 2 In order to optimize storage allocation, realize centralized management and reduce the transmission loss, 8-2 storage units are installed near the load side. 35,36 In other words, this system is composed of only one string centralized energy storage system (ESS), which is directly interfaced to the point of common coupling (PCC). This string ESS is incorporated with m storage units. Meanwhile, the system also includes D string PV modules, D {, 2, 3, }. The k th string PV module is made of n_k PV units, n_k {, 2, 3, }. 2.2 Equivalent model For convenience, the simplified equivalent circuit of the proposed MG is shown in Figure 3, where V k P i e jδk P i and V Sj e jδ Sj are the output voltage of the ith PV unit of the k th string module and the jth storage unit, i {, 2,, n_k}, j {, 2,, m}; V k P e jδk P is the total output voltage of the k th string PV module; V S e jδ S is the output voltage of the string ESS module. Z k and θ k are the modules and angle of impedance, respectively. Z load and θ load are the modules and angle of load demands, respectively. From Figure 3, the output real power P k P i and the reactive power Qk P i of the ith PV unit of the kth string module are derived as follows: P k P i þ jqk P i ¼ V k P i e jδk P i V k P e jδk P V S e jδ S = Z k e jδ *; k () where V k P e jδk P ; V S e jδ S are rewritten as V k P e n k jδk P ¼ V k P j e jδk P j ; (2) V S e jδ S ¼ m V S i e jδ S i : (3) Usually, the impedance is mainly inductive in medium /high voltage power system. From () to (3), the power transmission characteristics are given as follows: P k P i ¼ V k P i Z k m V S j sin δ k P i δ S j n k! V k P j sin δk P i δk P j ; (4) FIGURE 3 Equivalent circuit of the proposed PV storage MG. MG, microgrid; PCC, point of common coupling; PV, photovoltaic

5 LI ET AL. 5of6 Q k P i ¼ V k P i Z k n k V k P j cos δk P i δk P j m V S j cos δ k P i δ! S j : (5) From (4) to (5), it can be concluded that the output power of each PV unit can be regulated by changing their output voltage amplitudes and the phase angle differences. Similarly, the output real power P S_j and the reactive power Q S_j of the jth ESS unit are derived as follows: P S j þ jq S j ¼ V S j e jδ V S j S e jδ S D jz load je jδ load k¼!! V k P e * jδk P V S e jδ S : (6) e jδ k Z k Then, the power transmission characteristics of the jth ESS unit are expressed as P S j ¼ V S j j Z load n k k¼ V S j D j m V S i cos δ S j δ S i þ δ load þ V k P i sin δ S j δ k P i V S j D Z k k¼ m V S i sin δ S j δ S i Z k ; (7) Q S j ¼ V S j j V S Z load j m n k j D k¼ V S i sin δ S j δ S i þ δ load V k P i cos δ S j δ k P i þ V S j D Z k k¼ m V S i cos δ S j δ S i Z k : (8) From (7) to (8), the output power of each storage unit is related to their output voltage amplitudes, the phase angle differences, and the load characteristic. 3 PROPOSED DECENTRALIZED CONTROL The decentralized control, used in parallel connection MG and series connection MG, cannot directly apply to the islanded series parallel PV storage independent MG. In this section, the control principle of the PV and storage inverter based on its local information will be introduced. 3. Controller of PV inverter The targets of the PV inverter include () realizing MPPT and (2) working as voltage sources and keeping synchronization with other inverters. The proposed algorithm of ith PV inverter of the k th string module is expressed as ω k P i ¼ ω* P k P þ k I s P k Pf i Pk a i ; (9) V k P i ¼ V * n k : (0) where ω k Pi and ω* P are reference angular frequency and nominal angular frequency, respectively. Pk Pf i is the filtered output active power. P k ai is the available active power given by the front end MPPT controller.33 k P and k I are the proportional and integral coefficients of the PI controller. V * is the nominal voltage including the voltage drops. The control block diagram of the PV inverter is shown in Figure 4A.

6 6of6 LI ET AL. FIGURE 4 Diagram of local controller A, PV unit and B, ESS unit. ESS, energy storage system; PV, photovoltaic 3.2 Controller of ESS unit The targets of the ESS inverter include () maintaining the power supply demands balance of system and (2) working as voltage sources and realizing the frequency synchronization. The controller of jth ESS unit is given as follows: ω S j ¼ ω * S þ K S sgn Q Sf j PSf j ; () V S j ¼ V * m : (2) where ω S_j and ω * S are reference angular frequency and nominal angular frequency, respectively. P Sf_j is the filtered output active power. sgn( ) is a signum function. K S is a positive power sharing coefficient, which could control the angular frequency within the allowable ranges [ω min, ω max ], ω max, and ω min are the allowed maximum and minimum angular frequency of MG. ω * S ¼ ð ω max þ ω min Þ=2, and K S P Sf_ j [0, (ω max ω min )/2]. The schematic diagram of the controller for ESS unit is depicted in Figure 4B. According to (9), (0), (), and (2), every inverter make decisions only based on its own information; thus, the decentralized manner is realized. 4 STABILITY ANALYSIS In this section, the small signal stability of the PV storage MG will be investigated by root locus technique

7 LI ET AL. 7of6 4. PV units modeling Equation 9 is rewritten as _ω k P i ¼ k P P _ k Pf i k I P k Pf i Pk a i : (3) Assume that ω s is the MG synchronous angle frequency in the steady state. Let δ s = ω s dt,anddenote e δ k Pi ¼ δk Pi δ s,then _eδ k P i ¼ ωk P i ω s: (4) P k Pf i is the average terms of Pk P i obtained by a low pass filter _P k Pf i ¼ Pk P i Pk Pf i ω c ; (5) where ω c is the cutoff frequency of the low pass filter. Because the dc link voltage dynamics of each PV unit are slower compared with other dynamics, according to the idea of time scale separation, P k a i could be viewed as a constant. Then, the small signal model of (4) and (3) to (5) around the stable operating point is given as follows: ΔP k P i ¼ V * 2 n kz k n k n k m m cos δ k P i δ S j cos δ k P i δk P j k Δ e δ P i Δe δ S j Δ e δ k P i Δe δ k! P j ; (6) Δ _ω k P i ¼ k PΔ _P k Pf i k IΔP k Pf i ; (7) Δ _ e δ k P i ¼ Δωk P i ; (8) Δ _P k Pf i ¼ ω cδp k P i ω cδp k Pf i : (9) 4.2 ESS units modeling Similarly, the ESS units are modeled as _P Sf j ¼ P S j P Sf j ωc ; (20) _eδ S j ¼ K S sgn Q Sf j PSf j : (2) Then, the small signal model of (7) and (20) to (2) is given by ΔP S j ¼ V * 0 m D k¼ m V * n k cos δ S j δ k P i n kz k sin δ S j δ S i þ δ load jz load j Δ e δ S j Δ e δ k P i V * 2 þ D k¼ m m 2 cos δ S j δ S i Z k C A Δe δ S j Δ e δ S i ; (22)

8 8of6 LI ET AL. Δ _P Sf j ¼ ω c ΔP S j ω c ΔP Sf j ; (23) Δ e _ δs j ¼ K S sgn Q Sf j ΔPSf j : (24) 4.3 Entire system modeling Combining the small signal models of PV and ESS units, model of the entire system is expressed as (25) _X ¼ AX; (25) where the state variable vector X and the system matrix A are shown in Appendix A. 4.4 Eigenvalue analysis The root locus method is adopted to perform the stability analysis. According to the simulation system described in section 5, the effect of different k P, k I, K S,reactanceX, and load reactance X load on root locus diagrams are studied in this section. ) Root locus with respect to k P. Figure 5 shows the root locus diagram as k P increases from 0 to 2e 3, with resistance R = 0.2Ω, X = 0.628Ω, load resistance R load = 2.5Ω, X load = 3.4Ω, k I =2e 3, and K S =e 4. As seen, when k P is small, λ ~λ 2 lie on the right half plane, and the system is unstable. While k P is great than.e 4, all the poles lie on the left half plane. In other words, the system becomes stable in this case. 2) Root locus with respect to k I. Figure 6 shows the root locus diagram as k I change from 0 to 0.4 with R = 0.2Ω, X = 0.628Ω, R load = 2.5Ω, X load = 3.4Ω, K S =e 4, k P =e 3. As seen, some poles depart from the left half plane and move into the right half plane, and the system starts to lose its stability when k I = That is to say a large k I is adverse to stability in this system. 3) Root locus with respect to K S. Figure 7 shows the root locus diagram as K S increases from.e 3 to.e 3, with the parameters: R =0.2Ω, X = 0.628Ω, R load =2.5Ω, X load =3.4Ω, k P =e 3, and k I =2e 3. When K S increases from.e 3 to 0 (every unit uses the traditional droop control scheme), the eigenvalues are always in the right FIGURE 5 Root locus ask P increases from 0 to 2e 3 FIGURE 6 Root locus as k I increases from 0 to 0.4

9 LI ET AL. 9of6 half plane. Thus, the traditional droop scheme cannot directly apply to the series parallel system. When K S increases from 0 to.e 3, the eigenvalues move from the left half plane to the right half plane. Thus, the system is stable when K S (0, e 3). 4) Root locus with respect to X. In case of R = 0.2Ω, R load = 2.5Ω, X load = 3.4Ω, k P =e 3, k I =2e 3, and K S =e 4, the root locus diagram is depicted in Figure 8 as X increases from 0.Ω to Ω. The eigenvalues always stay at the left half plane; thus, the system is stable when X [0.Ω,Ω]. 5) Root locus respect to X load. In this test, R = 0.2Ω, X = 0.628Ω, R load = 2.5Ω, k P =e 3, k I =2e 3, K S =e 4, and let X load changes from 4Ω to 4Ω; the corresponding root locus diagram is depicted in Figure 9. When X load [ 4, 0] Ω, the eigenvalues λ 7 ~λ 0 are shown in Figure 9A. Otherwise, λ 7 ~λ 0 are depicted in Figure 9B as X load (0, 4] Ω. As seen, the system is stable under both resistance capacitance and resistance inductance. FIGURE 7 Root locus as K S increases from.e 3 to.e 3 FIGURE 8 Root locus as X increases from 0.Ω to Ω FIGURE 9 Root locus as X load changes from 4Ω to 4Ω FIGURE 0 Simulation model of a PV storage MG

10 0 of 6 LI ET AL. TABLE Parameters for simulations Item Symbol Value Unit Line resistance/inductance R /L 0.2/2e 3 Ω/H Voltage reference V * 3 V Frequency reference f * S =f * P 50/5 Hz Sharing coefficients K S e 4 rad/(w s) PI control coefficients k P /k I e 3/2e 3 Abbreviation: PI, proportional integral. FIGURE Simulation results of case. A, load demands; B, frequency; C, active power; and D, reactive power

11 LI ET AL. of 6 FIGURE 2 near second Load voltage waveforms FIGURE 3 Simulation results of case 2. A, load demands; B, frequency; C, active power; and D, reactive power

12 2 of 6 LI ET AL. 5 SIMULATION RESULTS The simulation studies of proposed PV storage MG and control scheme are performed on MATLAB/Simulink platform. A PV storage MG with two storages (ESS# and ESS#2) and two PVs (PV# and PV#2) is built, which is shown in Figure 0. The simulation parameters are listed in Table, and the allowable frequency ranges are set in the interval [49 Hz, 5 Hz] Case : Simulation with PV power fluctuation To test the effect of PV power fluctuation on the MG system, deliberate fluctuations are imposed on PV# and PV#2. The load demands are shown in Figure A, and the output active power of PVs fluctuates randomly every 0.5 seconds. The frequency curves are depicted in Figure B, which are also controlled within the feasible ranges [49 Hz, 5 Hz]. From Figure C, the output active power of storages is increased and decreased with the varying output active power of PVs to balance the system power demands. The reactive power allocations among DGs are shown in Figure D. The simulation waveforms of load voltage near t = second are shown in Figure 2. From this test, the maximum utilization of renewable sources and power balance of system are always achieved. 5.2 Case 2: Simulation with load change under resistance inductance load In this case, the effect of load changes on system operation is tested. The output active power of PV# and PV#2 are set to 000 and 50 W, respectively. And the load demands change at second and recover at 2 seconds as shown in Figure 3A. The resulted frequencies are shown in Figure 3B, which converges to a constant quickly with the load change. Figure 3C shows the output power of each unit. Clearly, the deviation between output power of PV units FIGURE 4 Simulation results of case 3. A, load demands; B, frequency; and C, active power

13 LI ET AL. 3 of 6 and load demands is balanced by ESS units. The waveforms of reactive power are shown in Figure 3D. As seen, the system has a fast transient response with the load change. 5.3 Case 3: Simulation with switching from resistance inductance to resistancecapacitance load The simulation is implemented under the resistance inductance and resistance capacitance load. The load schedules shown in Figure 4A are inductive and capacitive in the interval [0 s, 2 s] and [2 s, 4 s], respectively. The available active power of PV# and PV#2 is set to 800 and 000 W, respectively. The frequencies are shown in Figure 4B, which exceed 50 Hz in the first interval because of using the inverse droop, and vice versa in the second interval. The waveforms of active power are depicted in Figure 4C. According to the results, the proposed scheme can achieve the frequency synchronization autonomously under the inductive and capacitive loads. 5.4 Case 4: Simulation with charge and discharge of ESS To test the effect of the storages charge and discharge on the MG system, the active power load demand is set to 2480 W. Meanwhile, the total available active power of PVs is set 200 and 2900 W in the interval [0 s, 2 s] and [2 s, 4 s], respectively. As shown in Figure 5A, the total available active power of PVs falls short of the load demands in [0 s, 2 s] and oversupplies in [2 s, 4 s]. The frequency curves are shown in Figure 5B. The active power of DGs is depicted in Figure 5C, in which storages are discharged and charged in [0 s, 2 s] and in [2 s, 4 s], respectively. Therefore, the charge and discharge operation of storages is realized to improve the utilization efficiency of PVs. FIGURE 5 Simulation results of case 4. A, load demands; B, frequency; and C, active power

14 4 of 6 LI ET AL. 6 CONCLUSION A series parallel PV storage independent MG and its decentralized control are presented in this study. The structure of the proposed MG general generalizes the concept of MG to a certain extent. It will benefit the development of the medium/high MG. Moreover, the proposed control scheme provides technical guidance for the decentralized control technology development of series PVs or energy storage, especially when the number of the series units is large. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China under Grants 66223, and the Joint Research Fund of Chinese Ministry of Education under Grant 64A , and the Natural Science Foundation of Hunan Province of China under Grant 206JJ09. ORCID Lang Li Yao Sun Zhangjie Liu REFERENCES. Sun Y, Hou X, Yang J, Han H, Su M, Guerrero JM. New perspectives on droop control in AC microgrid. IEEE Trans Ind Electron. 207;64(7): Wang X, Dougal RA, Zhang J. Cycle by cycle error reduction droop method to improve power sharing in low voltage microgrid. Int Trans Electr Energy Syst. 208;28(4) Guo L, Su J, Lai J, Wang Y. Research on power scheduling strategy for microgrid in islanding mode. Int Trans Electr Energy Syst. 207;28(2) Mahmood H, Michaelson D, Jiang J. A power management strategy for PV/battery hybrid systems in islanded microgrids. IEEE J Emerg Sel Top Power Electron. 204;2(4): Yan X, Abbes D, Francois B. Uncertainty analysis for day ahead power reserve quantification in an urban microgrid including PV generators. Renew Energy. 207;06: Vazquez G, Martinez Rodriguez PR, Escobar G, Sosa JM, Martinez Mendez R. A PWM method for single phase cascade multilevel inverters to reduce leakage ground current in transformerless PV systems. Int Trans Electr Energy Syst. 206;26(): Zhang L, Sun K, Li Y, Lu X, Zhao J. A distributed power control of series connected module integrated inverters for PV grid tied applications. IEEE Trans Power Electron, no. early access Mahmood H, Michaelson D, Jiang J. Strategies for independent deployment and autonomous control of PV and battery units in islanded microgrids. IEEE J Emerg Sel Top Power Electron. 205;3(3): Wu H, Wang S, Zhao B, Zhu C. Energy management and control strategy of a grid connected PV/battery system. Int Trans Electr Energy Syst. 205;25(8): Yao S, Guangze S, Xing L, et al. An f P/Q droop control in cascaded type microgrid. IEEE Trans Power Syst. 208;33(): He J, Li YW, Wang C, Pan Y, Zhang C, Xing X. A hybrid microgrid with parallel and series connected micro converters. IEEE Trans Power Electron. 207;. 2. He J, Li Y, Liang B, Wang C. Inverse power factor droop control for decentralized power sharing in series connected microconvertersbased islanding microgrids. IEEE Trans Ind Electron. 207;64(9): Fakham H, Lu D, Francois B. Power control design of a battery charger in a hybrid active PV generator for load following applications. IEEE Trans Ind Electron. 200;58(): Bai W, Abedi MR, Lee KY. Distributed generation system control strategies with PV and fuel cell in microgrid operation. Control Eng Pract. 206;53: Karimi Y, Oraee H, Golsorkhi MS, Guerrero JM. Decentralized method for load sharing and power management in a PV/battery hybrid source islanded microgrid. IEEE Trans Power Electron. 207;32(5): Vega Garita V, Ramirez Elizondo L, Mouli GRC, Bauer P, Ieee. Review of residential PV storage architectures. In: 206 Ieee International Energy Conference, IEEE International Energy Conference, New York: Ieee, 206.

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16 6 of 6 LI ET AL. APPENDIX A The variable vectors are h i where ΔP k Pf ¼ ΔPk Pf ΔP k k Pf nk, Δ e δ Δ e h δ S ¼ Δ e δ S Δ e i δ Sn. The system matrix A is rewritten as where h X ¼ ΔP k Pf Δ e δ k P Δω k P ΔP Sf Δ e i T; δ S (26) P ¼ Δ e δ k P Δe δ k Pnk, Δω k P ¼ Δωk P Δω k Pnk, ΔPSf ¼ ½ΔP Sf ΔP Sf n 2 3 A A A A A¼ A 3 A A 35 ; (27) A 42 0 A 44 A A 54 0 A ¼ ω c I; I ¼ diag½ n n ; 2 a V * 3 V * n k cos δk P δk P n k A 2 ¼ ω c n kz k ; V * n k cos δk P n k 5 δk P a n a i ¼ V * m m cos δ k P i δ V * k S j n k n cos δ k P i δk P j ; 2 cos δ k V * 2 P δ S cos δ k P δ 3 S m A 5 ¼ ω c 6 n kmz k ; cos δ k P n k δ S cos δ k P n k δ S m A 23 ¼ I; A 3 ¼ ðk P ω c k I ÞI; A 32 ¼ k P A 2 ; A 35 ¼ k P A 5 ; 2 cos δ S δ k V * 2 P cos δ S δ k 3 P n Α 42 ¼ ω c 6 n kmz k ; cos δ S m δ k P cos δ S m δ k P n A 44 ¼ ω c I; 2 3 V * V * n k cos δ S δ k P i m D k¼ n kz k V * 2 m jz load j m sinðδ S δ S i þ δ load Þ V * 2 jz load j sin δ ð S δ S m þ δ load Þ m 2 B þ D cosðδ S δ S m Þ A þ D m k¼ Z k B cosðδ S δ S i Þ A k¼ Z k A 45 ¼ ω c ; V * V * n k cos δ S m δ k P i 0 m D V * 2 jz load j sin ð δ k¼ n kz k V * 2 m 2 S m δ S þ δ load Þ 0 m 2 þ D C cosðδ S m δ S Þ A jz load j m sinðδ S m δ S i þ δ load Þ k¼ Z k þ D m 6 B cosðδ S m δ S i Þ C A 7 5 k¼ Z k 2 3 sgn Q Sf A 54 ¼ K S : sgn Q Sf m,