(PDF) A Decentralized Master-slave Method for Three-Phase Cascaded H-bridge System

IEEE Journal of Emerging and Selected Topics in Industrial Electronics A Decentralized Master-slave Method for Three-Phase Cascaded H-bridge System Guangze Shi, Hua Han, Yao Sun, Member, IEEE, Junlan Ou, Abderezak Lashab, Senior Member, IEEE and Josep M. Guerrero, Fellow, IEEE Abstract- For the cascaded H-bridge system, the existing which will become the future development trend [2] ,[3] . decentralized methods are mainly focused on single-phase According to the interfacing power electronic converters systems. Designing a decentralized method for the three-phase topology, the DG system can be formed by parallel-connected cascaded H-bridge system is more challenging, particularly, for or cascade-connected. The parallel-type form is evolved from achieving phase-to-phase frequency synchronization and three- the operation pattern of traditional power systems, which has phase voltage balance without real-time communication among been intensively investigated, including microgrids, distributed each distributed generation (DG). To achieve this, this paper energy storage systems, and so on. Existing control methods for proposes a decentralized master-slave method for three-phase parallel-type DG systems emphasize droop control and virtual cascaded H-bridge systems, where no communication is synchronous generator (VSG) control, which are inspired by the required for each local control unit of DGs. The overall control physical characteristics of synchronous generators. The droop is divided into two parts. One part is for the master DG to realize control is often employed to achieve autonomous power sharing phase synchronization among Phase-A, B, and C, which is among DGs, based on the condition of the line impedance being designed by modifying the power factor angle through droop mainly inductive [4] ,[5] . Alternatively, different from control. The other part is for slave DGs to achieve frequency synchronous generators, the DGs system has either small or synchronization of the modules in each phase, which adopts the power factor angle droop control. Small signal stability analysis even no inertia and damping property to support grid frequency based on fast and slow time scale separation is carried out for stability and even deteriorate it. The VSG control strategies are the proposed method. Last, the Control-hardware-in-loop proposed in [6] ,[7] to overcome these problems, which make (CHIL) tests and experimental results prove the effectiveness the system more stable and reliable under frequency fluctuation. of the proposed method. However, most types of DGs have the characteristics of low- Index Terms—Decentralized control, Voltage balance, level voltage. So the parallel topology is difficult to operate in Cascaded H-bridge system. a high-voltage-level application. The cascaded H-bridge topology offers an effective way for I. INTRODUCTION high voltage level application because it can easily boost DGs have attracted more and more attention to alleviate the voltage without bulky transformers and complex circuits. This energy crisis due to their merits of scalability, cleanness, easy topology can acquire simple voltage-scaling properties without integration, and high flexibility. In recent years, the market of expensive and bulky transformers. The cascaded H-bridge DGs has been growing rapidly and will project to account for system (CHS) has been originally investigated for static more than half of global electricity production by 2035 [1] . The synchronous compensators (STATCOMs) and motor drives [8] power electronic converter offers an effective way to connect -[10] . Recently some authors developed this series topology multiple DGs, which can provide suitable power flow control into new energy applications for integrating low-voltage DGs and voltage regulation for each DG. Thus the power-electronic- (e.g. photovoltaics (PVs), energy storages) to medium/high enabled power system will obtain more operational flexibility voltage power networks [11] -[14] . For instance, the maximum and controllability comparable to conventional power systems, allowed PV string voltage is 1kV in Europe, which makes the transformerless configuration using conventional converters Manuscript received November 22, 2022; revised March 07, May 05 and July unattainable. Furthermore, reviewing the existing literature, the 21; accepted August 22. This work was supported in part by the National CHSs usually operate with centralized control strategies [15] - Science Fund for Distinguished Young Scholars under Grant 62125308, and [17] because it is easy to design the controller and achieve the National Natural Science Foundation of China under Grant 52177205, synchronized modulation. In this control structure, as shown in 61933011, and 62192754. (Corresponding author: Hua Han.) G. Shi is with the School of Artificial Intelligence and Advanced Computing, Table I, the central controller is responsible for all control signal Hunan University of Technology and Business, Changsha, China, and also generation and measurement collection. But when the number with the School of Automation, Central South University, Changsha, China. of cascaded modules increases, the increasing computational (e-mail:

) H. Han, J. Ou and Y. Sun are with the School of Automation, Central South effort becomes a heavy burden for the central controller, if all University and the Hunan Provincial Key Laboratory of Power Electronics control tasks are still centralized into a single central controller. Equipment and Gird, Changsha 410083, China (e-mail:

, Meanwhile, the switch commands are transmitted at every

,

) A. Lashab and J. M. Guerrero are with the Department of Energy Technology, switching cycle, which should rely on high bandwidth Aalborg University, 9220 Aalborg (e-mail:

;

). communication. 1 IEEE Journal of Emerging and Selected Topics in Industrial Electronics TABLE I COMPARISON WITH THE EXISTING METHODS Features [15] -[17] [19] -[22] [23] -[32] Proposed method #N #N #N #N #N #N #N #N #N #N ZloadA ZloadB ZloadC ZloadA ZloadB ZloadC Zload Operation #2 #2 #2 #2 #2 #2 #2 #2 #2 #2 ZloadA ZloadB ZloadC VA,B,C VA,B,C Configuration #1 #1 #1 Central #1 #1 #1 Master #1 controller controller #1 #1 #1 VA,B,C A B C A B C A B C Control Centralized Distributed Decentralized Decentralized Configuration Computation High Middle Low Low load In-phase synchronization without Synchronization based on a Synchronization Synchronization Synchronization based on communication central controller and high without comm Mechanism communication Phase-to-phase synchronization communication unication based on PCC information Central controller-all units Communication Master-slave Central controller- \ Master-PCC sensor Links Master-PCC sensor local&PCC sensors Communication High High Low Low Dependency Resilient to communication Low Low High High failure Voltage Balance Ensured Ensured Not ensured Ensured Feasibility under Feasible Feasible Unfeasible Feasible Unbalance Load To reduce the computational effort, the distributed control equilibrium-points problems. For grid-connection mode, [26] methods are preferred whose control structure is shown in Table proposes a robust hybrid voltage/current synchronous control I. In this structure, the fast time scale tasks, like PWM scheme that ensures communication-free grid connection of generators, voltage/current controllers [19] ,[20] , and so on, are CHSs with a low communication burden. In [27] , an ω-sinφ implemented by local controllers, while the central controller droop control is proposed, which can substantially weaken the acts as the master to coordinate each local controller. For effect of network pressure fluctuations on system stability. example, the authors in [21] propose a distributed control Moreover, a series of decentralized control methods are method to reduce the data transmission from the central proposed for some specific applications, such as PVs [28] , controller to the local controllers. In [22] , a distributed control energy storage systems [29] , STATCOM [30] , series- considering communication delay and local control loop connected H-bridge rectifiers [31] , etc. In these methods, stability is proposed. However, when the DGs are except for frequency synchronization, more control objectives geographically far apart, the communication burden for are taken into consideration for each application's requirements, acquiring data and transmitting the control commands increases. such as maximum power point (MPPT), battery state of charge Moreover, the possible failure of the communication network (SOC) balancing, reactive power sharing, etc. Recently, to jeopardizes the stability and reliability of the CHS. achieve unified control for both grid-connected and islanded Inspired by droop control for paralleled-connecting systems, modes, the power factor angle droop control is proposed in [32] , the researchers gradually focus on the decentralized control which directly uses the power factor angles as feedback to methods [23] -[25] that can generate their own achieve frequency synchronization. However, the voltage/frequency references directly within each local aforementioned decentralized methods are only focusing on a controller based on the local measurement. In decentralized single-phase CHS. which are not straightforwardly applied to structures, each DG can generate its own voltage/frequency three-phase CHS, The three-phase CHS has more complex and references directly within each local controller based on the challenging control objectives, particularly achieving phase-to- local measurement. Moreover, the system control can be phase frequency synchronization and three-phase voltage localized, and each DG can be coordinated without balance with only the local controller, and no real-time communication. Thus, the communication dependency and communication among each DG module. computational load can be reduced simultaneously. In [23] , an To fill this research gap, this paper proposes a decentralized inverse power factor droop control scheme is first proposed to master-slave method for three-phase CHSs. In the proposed achieve frequency synchronization and proportional power control scheme, the DG near the point of common coupling sharing autonomously. The f- P/Q control [24] is designed with (PCC) is selected as the master of each phase, and the remaining considering different types of loads which can expand the n-1 DG units in each phase are regarded as slaves. The master application scope. In [25] , the unique-equilibrium-point DG should not only realize the frequency synchronization with decentralized method is proposed to overcome the multiple- the slave DGs but also ensure the phase-to-phase frequency 2 IEEE Journal of Emerging and Selected Topics in Industrial Electronics synchronization and three-phase voltage balance. The slaves in phase system but also the frequency between phases need to be each phase can independently synchronize with the master unit synchronized at first. based on the local controller. Thus, there is no communication  Ai =  Aj = Bi = Bj =Ci = Cj (2) link between the control units of the master and the slaves, which can reduce communication dependency and system costs. What’s more, the phases of each phase voltage need to be Compared with the traditional control approaches of three- 120 degrees from each other and the voltage amplitude of each phase CHS, as listed in Table I, the proposed control shows the phase should be equal, which can be expressed as following outstanding features: 1) The power control can be done locally in each module. Thus, the computation effort will 2 2  A =  B +  = C −  (3) be hugely reduced. 2) There is no communication between each 3 3 local control unit, which will reduce the impact of the communication vulnerability risks over the control VA = VB = VC (4) performance. 3) Compared with the decentralized methods of single-phase CHS, the proposed control method can achieve Obviously, compared with the single-phase system, the phase-to-phase and intra-phase frequency synchronizations control objectives for the three-phase one are more challenging. simultaneously. Meanwhile, the three-phase voltage balance is 2) Reasons for the challenges above VC VC guaranteed even under unbalanced load conditions. Simulation and experimental results have been conducted to validate the  AB  120 VA Phase feasibility of the proposed control.  AB = 120 Disturbance II. MOTIVATION VA In this section, the motivation of this paper is discussed. The first is to introduce the three-phase CHS configuration. Fig. 1 presents the main circuit configuration of the three-phase CHS. VB VB Each phase (i.e. phases A, B, and C) is composed of n cascaded  AB =  BC =  CA = 120  AB   BC   CA  120 DGs with interfacing H-bridge converters with LC filter. The Fig. 2 Voltage angle unbalance diagrams. output voltages of the i-th converter in phases A, B, and C are There are two factors that will lead to voltage unbalance denoted as VAi e j𝛿Ai , VBi e jδBi an d VCi e jδCi , respectively. problems. The first one is voltage angle disturbance. As shown The coupling point voltages of phases A, B, and C are in Fig.2, due to the lack of phase balance constraint, the three- represented by VA e jδA , VB e jδB an d VC e jδC , respectively. phase voltage phases of the system are prone to unbalance under The relationship between the coupling point voltage and the external disturbances. voltages of all DG sources is expressed as follows: The second factor is unbalanced loads. In general, the load is n not absolutely balanced. When the system load is unbalanced, Vx e j x = Vxi e j xi x = A, B or C (1) the decentralized control relying only on local information will i =1 cause asynchronous frequency between phase subsystems. It is easy to see that the three-phase CHS is comprised of Taking the power factor angle droop control as an example, the three single-phase CHSs. There are interactions between in- frequency of each phase sub-system is a one-to-one phase and phase-to-phase units, which makes the coupling correspondence with the load impedance angle of mechanism of the three-phase system more complex. The corresponding phase sub-systems. system must ensure the three-phase voltage balance based on xi = 0 − mx xi =0 − mx x (5) frequency synchronization. This means that three-phase systems face more challenges. where x represents the load impedance angle in the x-th phase. VAe j A From (5), it is obvious that unbalance loads will lead to DG#1 Converter Lf VA1e j A1 frequency asynchronous among phase sub-systems.  A   B  C   Ai  Bi  Ci Cf (6) Loads Phase A ... III. PROPOSED CONTROL METHOD Phase B Phase C O To overcome the problems above, this paper proposes a DG#n Converter Lf Cf VAn e j An decentralized master-slave control framework, as shown in Fig. 3. In this control framework, the three master units of each N phase are controlled by a modified power factor control, which takes charge of coordinating the phase difference of each phase. Fig. 1 Configuration of three-phase CHS. The remaining units work autonomously and rely on local 1) More complex control objectives information. The detailed design and analysis of the proposed Different from single-phase CHS, the most important method are presented as follows. objective is to guarantee the three-phase voltage balance. To achieve this, not only the frequency of DG unit in each single- 3 IEEE Journal of Emerging and Selected Topics in Industrial Electronics Vxi = V * n (9) #N #N #N  Vxi sin (   xi dt ) x =A  ZloadA ZloadB ZloadC Vxi _ ref = Vxi sin (   xi dt − 2 3) x=B (10)  Vxi sin (   xi dt + 2 3) x =C #2 #2 #2 where ωxi, φxi Vxi and Vxi_ref are the angular frequency reference, #1 #1 #1 VA,B,C power factor angle, voltage amplitude reference and voltage A B C reference of the i-th DG in phase x, respectively. ω0 represents Fig.3 Proposed decentralized master-slave control framework the value of ω with no load. V* is the nominal voltage amplitude value of the phase voltage. mx is the adjustment droop A. Design Principle coefficient for the DGs in phase x. It is noted that there is a In this section, the design principle of the proposed control is frequency deviation in steady state. To make the system discussed. To guarantee three-phase voltage balance, the core frequency vary in an acceptable range [49.5~50.5], the of the control design is to realize the controllability of frequency maximum value of the droop coefficient is derived as follows and voltage of each single-phase. Due to the restraints of the mx _ max = 2 ( f max − f min ) / (max − min ) = 2 (11) load and the power factor angle droop control, changing one of the DG power factor angles and voltage amplitude can adjust As for the master DG unit, the power loop control is designed each phase system frequency and voltage. The voltage vector by modifying the power factor droop control, which is diagrams variation with changing one DG power factor angle expressed as follows φxm and voltage amplitude Vxm is described in Fig. 4. xL = 0 − mx ( xL −  xm )  Vxm φxm Vxm  (12) VxL = u xm V n  Vxm * φxm Vxi φxm Vxi φxi VxL sin (   xL dt ) φxi Vxi x= A  x φxi VxL _ ref = VxL sin (  xL dt −2 3) x = B (13) x x  φxm =φxi =x φxm φxi ωx φxm φxi ωx  xL (  xL V sin  dt + 2 3 ) x = C (a) (b) where ωxl, and VxL_ref are the angular frequency reference and Fig. 4 Voltage vector diagrams variation with small fluctuation on DG power voltage reference of the master DG in x phase, respectively. xL0 factor angle. represents the initial voltage angle. The ∆φxm and uxm are the To analyze the phase frequency controllability, a small frequency modification and voltage reference modification, fluctuation is added to the DG power factor angle. It is assumed respectively. that the load impedance angle has no change before and after The modification has two aspects. One is for the three-phase the small disturbance. From Fig. 4 (a), when the power factor voltage angle balance, which is provided by the frequency angle of one DG increases, the remaining units will decrease modification ∆φxm. Another one is for restoring the phase their power factor angle. Thus, the frequency of the x-th phase voltage amplitude in steady state, because the corresponding system will increase. Similarity in Fig. 4 (b), decreasing the phase voltage amplitude will decrease when changing the power factor angle of one DG will lead to a decrease in the x-th master power factor angles. The frequency modification and phase system frequency. Thus the phase frequency is adjustable. voltage reference modification are designed as follows Moreover, according to the cosine theorem, the voltage  k  amplitude of each phase is derived.  xm =  k P + I   ( j −  x ) (14) 2  s  j, j  x  n   n  Vx2 =   Vxi  + Vxm2 + 2   Vi Vxm cos ( xm −  xi ) (7)  i =1,i  m   i =1,i  m  u xm = ( 2n − 1) + ( n − 1)2 cos2 (  xm ) − ( n − 1) cos (  xm ) (15) From (7), it is obvious that the voltage amplitude of each where Ω={A,B,C}. kP and kI are the frequency adjustment phase can be regulated by modification of the power factor coefficient. δ’A=δA. δ’B=δB+2π/3. δ’C=δC-2π/3. angle and voltage amplitude of one DG. 2) Double voltage&current loop control B. Proposed Method Design The double voltage&current loop control is used to track In this paper, the overall control diagram is comprised of the voltage reference generated by power loop control. In three-layer feedback loop control: power loop control, voltage this paper, the proportional-resonant (PR) control [33] is loop control, and current loop control. adopted for double voltage&current loop control, because it 1) Power loop control is more suitable for tracking AC variable reference than PI The power loop control is designed based on the power factor control. The PR control voltage and current are designed as angle droop control, which takes the local power factor angle as follows feedback to modify the local frequency [32] . As for the slave 2k    s Gv ( s ) = k pV + 2 rV cV (16) DG unit, the power loop control is to achieve frequency s + 2cV  s + r2 synchronization among phase sub-system units, which is 2k    s designed as follows Gi ( s ) = k pI + 2 rI cI (17) xi = 0 − mxi xi (8) s + 2cI  s + r2 4 IEEE Journal of Emerging and Selected Topics in Industrial Electronics where kpV, krV, kpI and krI are the proportional and resonant Moreover, it is easily obtained that the phase angle difference parameters of the double voltage&current loop control, of the three-phase voltage is 120 degrees according to the respectively. the ωr, ωcV and ωcI are the resonant frequency Proportional-Integral (PI) controller, which is expressed as and the cut-off frequency of the double voltage&current loop follows control  A = B = C   A =  B + 2 / 3 =  C − 2 / 3 (22) Fig. 5 shows the simplified control block diagram of the Also, the system should satisfy the power balance of supply closed-loop system. (a) slave controller, (b) master controller. and demand, so we can obtain As for the control block of the slave DG unit shown in Fig. 5(a),  u xmV * ( n −1)V * the local power factor angle is calculated as the feedback of the Vx cos  Load = cos  xL + n cos  xi n power control, which can modify the frequency reference value  (23) V sin  u xmV * ( n −1)V * to regulate the output active and reactive powers. The following = sin  + sin  is the double voltage&current loop control, which is to track the  x Load n xL n xi voltage reference generated by the power loop control. The Square the two formulas of (23) and then add them together, ( ) 2 feedback signals of the double voltage& current loop control  * + ( n − 1) + 2u xm ( n − 1) cos ( xL −  xi ) (24) 2 are capacitance voltage and inductance current, respectively. Vx2 =  V  u xm 2  n  Fig. 5(b) represents the simplified control block diagram of All DGs in each phase share the same frequency. Based on the master controller. It is clear to see that the voltage and the proposed control, the following formula (25) is obtained. current loop control are the same as slave control. Different from the slave controller, the power loop control for the master  xL −  xi =  xm (25) controller is modified by phase voltage angle δA,B,C, because the Substituting (15) and (25) into (24), it is derived that the master control has responsibility for phase frequency voltage amplitude of each phase is the same. synchronization and system voltage regulation. Specifically, a Vx = V * (26) frequency modification is calculated by a consensus algorithm From (22) and (26), the voltage balance of three-phase CHSs based on the phase voltage angle, which is to achieve phase-to- can be achieved in steady state. phase frequency synchronization. Also, a voltage modification The derivation of equation (22) is expressed as follows is estimated by the vector triangular relation, which is to  A = B = C (27) compensate for the voltage decrease caused by frequency Combining (18) and (27), the frequency synchronization of modification. all units in the different phases is achieved ωref,xi δref ,xi Vref,xi sinδref,xi iLref,xi Filter Inductors Filter Capacitors  AL =  Ai =  A = BL = Bi = B = CL = Ci = C (28) Vxi  1/s sin  Gv(s)  Gi(s) Gpwm(s) 1/(Ls+R) 1/Cs ω* V* 1/n Vref,xi PR PR iL,xi IV. STABILITY ANALYSIS mx φxi atan2 Pxi Vxi According to the above analysis, the dynamic model of Qxi Proposed Control iload Control Model Physical Model three-phase CHS can be summarized as follows (a) & =  = − m ( −  )  xL xL 0 x xL xm ωref,xL δref ,xL & Filter Inductors Filter Capacitors Vref,xL sinδref,xL iLref,xL  xi = 0 − mx xi VxL ω*  1/s sin  Gv(s)  Gi(s) Gpwm(s) 1/(Ls+R) 1/Cs (29)  Vref,xL PR PR ( ) 1/n & xm =k P  j  j −  x + k I  j ( j −  x ) iL,xL V* Eq.(14) φxm uxm Eq.(15) & & δA,B,C Control Model Physical Model φxL PxL VxL mx  atan2 QxL iload Proposed Control A. Reduced Fast System Stability Analysis (b) Power Loop Control Voltage Loop Control Current Loop Control Since the dynamic characteristics of the power factor GV ( s ) = 0.01 + 8s GI ( s ) = 0.01 + 4s GPWM ( s ) = 1 controllers are much faster than the proposed modification, the s 2 + s + ( 2  50 ) s 2 + s + ( 2  50 ) 1.5 / f c + 1 studied system with the proposed control can be regarded as a 2 2 Fig. 5 Simplified control block diagram of the closed-loop system. (a) slave two-time scale system. According to singular perturbation controller, (b) master controller theory [34] , when analyzing the reduced fast system, the slow C. Steady-state analysis variables can be considered as constant parameters. From (29), In the steady state, based on the power factor angle droop the small signal model of the reduced fast system can be controller [32] , the frequency synchronization between the expressed as master and slave can be obtained. &% = − m % xL = xi = x x = A, B, C (18)  xL x xL  &% In order to regulate the system voltage conveniently, it is  xi = − mx% xi (30)    xm =  xm c necessary for the droop coefficients to be the same for the units  c in the same phase. mAL = mA 2 = L = mAn (19) u = f  xm  xm ( c ) where the superscript c represents that the variable is regarded mBL = mB 2 = L = mBn (20) as a constant value. mCL = mC 2 = L = mCn (21) 5 IEEE Journal of Emerging and Selected Topics in Industrial Electronics The power factor angle φxi is calculated based on local active pole λ2(Y) will move away from the imaginary axis with power and reactive power information, which is derived as increasing the mx, which means that the response speed of the follows proposed control will be improved. However, the larger droop Q  n Vxi sin ( xi − xj + x ) coefficient mx will lead to an unallowable frequency deviation.  xi = arctan xi = arctan nj =1 (31)  j =1Vxi cos( xi − xj + x ) Thus the value of mx should take both response speed and Pxi frequency deviation into account. The small signal equation of (30) is derived as Combined with the frequency deviation requirements (11), % = x  j =1, j  L ( % − % ) B n it is usually necessary to choose the droop coefficient as large xL xL xj (32) Ax as possible within the available range. However, the droop ( %xi = Cx (%xi − %xL ) + Bx  j =1, j  L (%xi − %xj ) n ) Ax (33) coefficient is not normally chosen to be the maximum value, because a margin is required depending on the practical  c2 2 c s (  Ax = u xm + ( n − 1) + 2 ( n − 1) u xm cos  xL −  xs s ) requirements. Thus in this paper, the value of the droop coefficient is selected as follows  c ( s s )  Bx = u xm cos  xs −  xL + ( n − 1)  (34) mx = 0.9  mx _ max = 1.8 (43)   x  xm xm ( ( C = u c 2 + ( n − 1) u c cos  s −  s  xs xL  )) B. Reduced Slow System Convergence Analysis When analyzing a slower system, the dynamic of fast where the superscript s represents steady state, ~ represents the variables is considered as getting into steady state. Thus, the small signal variable. small signal model of the reduced slow system is derived as Substituting (31)~(34) to (30), and rewriting it in a matrix follows. form gives  =  =  s %& = − Y  % (35)  xL xi x  s  A A M Z 0 0    x =  0 − m x xi =0 − mx ( x −  xi )  (44) Y= 0 0  (36)  ( ) & xm = k P  j  j −  x + k I  j ( j −  x ) M BZB & &  0 0 M C ZC  where E below represents the identity matrix. where ∆φxi represents slave power factor angle variation. M x = mx E (37) Combine formula (24) and rewrite formula (26), + ( n − 1) − 2uxm ( n − 1) cos ( −  xm ) 2  ( n −1) Bx − x B L − x B  n2 = uxm 2 (45)  Ax Ax Ax   − Cx ( n − 2 ) Bx + C x L Bx  Z x =  Ax −  (38) - φxm  M Ax M O Ax M  - φxi n uxm  − Cx − x B L ( n − 2 ) Bx + C x  φxi  Ax Ax Ax  φxm n-1 According to the M-matrix theorem, the eigenvalues of Y can Fig. 6 Relationship between the master modification ∆φxm and slave power be derived as the following factor angle variation ∆φxi.  ( Y ) =  ( M A Z A ) ,  ( M B Z B ) ,  ( MC ZC ) (39) According to the cosine theorem, the above triangle figure (Fig. 6) can be obtained, which reveals the relationship between First, the MxZx is the diagonally dominant matrix, the the master modification ∆φxm and the slave power factor angle characteristic of which is derived as follows variation ∆φxi. Based on the Law of Sines, the following can be n obtained mx Z xii   mx Z xij (40)  xi = arcsin  xm sin (  xm )  u j =1, j i (46)  n  From (40), it can be observed that the gerschgorin region is Combining (23), yields, on the right half plane. According to the gerschgorin circle % xi = Dx % xm (47) theorem, the eigenvalues of MxZx fall within the gerschgorin regions. So it can be obtained that Dx =  sxm u s ( n −1) sin 2  xm s  ( ) s 2 + uxm s cos  xm  / n − uxm sin  xm 2 s s (48)  (MxZx )  0 (41)  xm ( )  u + n −1 cos  s xm From (41), the matrix Y is a positive semi-definite with The ∆φxm should lie in (-π/2, π/2), so the Dx is a constant eigenvalues 0 =λ1 (Y) <λ2 (Y) ≤…≤λ3n (Y). The unique zero greater than 0. Then the small signal model equation can be eigenvalue is corresponding to rotational symmetry, which does derived as follows not affect the system stability [35] . Therefore, the stability of %& = D m %  x x x xm  % (49) the reduced fast system is proved. It is clear to see that λ2 (Y) is the dominant pole that    xm & = k P ( j ) D j m j % jm − Dx mx %xm + k I  (% − % ) j j x determines the rate of convergence. Rewrite (49) in matrix form, 2 ( Y ) = mx max ( i ( Z x ) i ( Z x )  0 ) (42)  δ%&   0 DM   δ% x   x =   (50) According to (42), as mx is larger than zero, the dominant  φ%& xm   −k I L −k P DML   φ% xm  6 IEEE Journal of Emerging and Selected Topics in Industrial Electronics where stability and dynamic performance of the system is clarified D = diag DA , DB , DC 33 (51) by the root locus analysis. From this, an approximate range of the control parameters selection can be obtained. In M = diag mA , mB , mC 33 (52) addition, it is noted that the parameters of eigenvalue analysis are the same as CHIL parameters.  2 −1 −1 Fig. 7 shows the root locus changing as the coefficient kP L =  −1 2 −1 (53) varies from 0 to 3 while the kI is set to be 2.5. As shown in Fig.  −1 −1 2  7, all eigenvalues lie on the left half plane, which can prove the Assuming that kP and kI are greater than zero, the stability of the system. When kP increases, the conjugate poles Lyapunov energy function is defined as follows move towards the real axis, which means that the oscillations 1 1 will be decreased. To achieve better dynamic performance [36] , V = Δφ% Tm ( D  M ) Δφ% m + kI δ% T LT δ%  0 T (54) the damping ratio is better to be constrained in [0.4, 0.8]. Thus, 2 2 in this paper, the selection range of the coefficient kP is [0.9, The derivation of (48) is expressed as follows 1.8]. dV = −k P Δφ% Tm ( D  M ) L  D  M  Δφ% m  0 Fig. 8 shows the root locus changing as the coefficient kI T (55) dt varies from 0.1 to 6 while the kP is set to be 1.3. From Fig. 8, According to the Lyapunov energy function method, if a the stability of the system is proved by that all eigenvalues lie given system can find a positive definite energy function V on the left half plane. When ki increases, the dominant poles and satisfy dV/dt<0, then the total energy of the system will move away from the imaginary axis, which means that the gradually decrease and eventually reach a stable state. It is oscillations will be increased. According to the aforementioned damping ratio range, the coefficient ki is better to be constrained obvious that the reduced slow system is stable from formulas in [1.2, 5.2]. (54) and (55). The larger the control coefficient m and kp will lead to the transient energy decreasing faster, the faster the V. CHIL TESTS transient energy decrease, which accelerates the system getting into the steady-state operation. DSP- 2.5 0.9 0.8 0.7 0.56 0.4 0.2 TMS320F28335 2 OPRT4510 Control Desk 1.5 0.955 kP :0 → 3 PWM Signals 1 0.988 kI = 2.5; Imaginary 0.5 Analog 5 4 3 2 1 0 Output -0.5 Digital Input 0.988 kp=1.8 -1 Sampling -1.5 0.955 kp=0.9 -2 0.9 0.8 0.7 0.56 0.4 0.2 Fig. 9 Control-hardware-in-loop (CHIL) platform structure -2.5 -5 -4 -3 -2 -1 0 1 TABLE II Real CHIL PARAMETERS Fig. 7 Root locus change as kP increases from 0 to 3. 4 Symbol Item Value 0.54 0.4 0.3 0.21 0.13 0.06 V* Voltage reference 311V 3 0.7 ki=5.2 kI :0.1 → 6 ω* Frequency reference 314rad m Droop control coefficient 1.8 2 kP = 1.3; fc PWM frequency 10kHz n DG number in each phase 3 Imaginary 0.9 ki=1.2 1 Lf Filter inductance 1.6mH 0 Cf Filter capacitor 40uF Frequency modification proportional kP 1.3 -1 0.9 coefficient Frequency modification integral kI 2.5 -2 coefficient 0.7 kpV Proportional coefficient of voltage control 0.01 -3 krV Resonant coefficient of voltage control 8 -4 0.54 0.4 0.3 0.21 0.13 0.06 ωcV Cut-off frequency of voltage control 0.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 kpI Proportional coefficient of current control 0.1 Real krI Resonant coefficient of current control 4 Fig. 8 Root locus change as kI increases from 0.1 to 6. ωcI Proportional coefficient of voltage control 0.5 To further discuss the selection range of kP and kI, the ωr Resonant frequency of PR control 314rad eigenvalue analysis of reduced slow system mode is To verify the proposed control, a 3×3 cascaded H-bridge presented. The influence of the control parameters on the converters-based DGs system is designed in the CHIL platform, 7 IEEE Journal of Emerging and Selected Topics in Industrial Electronics as shown in Fig. 9. The main circuit including the three-phase The CHIL test results are shown in Fig. 11. To better illustrate cascaded H-bridge system, the LC filter, the line impedance, the effectiveness, the different initial phase angles of converters and the load are emulated in the OPAL-RT4510 simulator. The are set at the beginning, 0 rad to the DG units in phase A, 0.15 controller is implemented in the DSP-TMS320F28335 control rad to the DG units in phase B, -0.15 rad to the DG units in board, and the sampling frequency is set to be 10kHz. The phase C. The RL load is set to 4+4jΩ. In the beginning, due to experimental data is recorded by the host machine of OPAL- the different initial phase angles, the voltage unbalance reaches RT4510. The topology of the system and the detailed control about 8.6%, which is out of the allowable range (≤4%). After diagram are presented in Fig. 10. The main circuit and control the proposed control starts, the voltage unbalance decreases parameters are listed in Table II. evidently, and changes to zero eventually. Thus, the three-phase voltage balance is achieved in steady state. During the control Factor Droop Control * φ working, the system can achieve voltage balance, frequency Modified Power ʃ  m  V * φm synchronization, and power sharing, which proves the um Equation Equation feasibility of the proposed control. (14) (13) atan B. Case II: Performance under unbalance load P Q 49.9 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 Inner Frequency Control Power Unbalance Data regulation (Hz) Loop Calculasion 49.8 load starts + Lf Cf - (a) 49.7 Master control Switch 3 Active power Lf Phase C system Phase B system 2 (kW) + Unbalance - Cf 1 load starts Local (b) 0 Controller Common 3 Reactive power load 2 (kVar) + Lf Cf - Unbalance 1 load starts Inner v0 Data regulation (c) 0 Control iL v0 i0 -90° 15 Loops Unbalance rate Unbalance Unbalance rate(%) 10 load starts * Power Factor V LPF LPF 5 Reference Droop Control Voltage φ Q Power  m atan Calculation (d) 0 0 1 2 3 4 5 6 * P Time(s) Slaver control without communication with Master Fig.12 Case II results: (a) frequency, (b) active power, (c) reactive power, (d) Fig. 10 Topology of the system and detailed control diagram. voltage unbalance rate A. Case I: Different initial phase angle In this section, the feasibility of the proposed control under unbalance RL load is verified and the CHIL test results are 50.5 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 presented in Fig. 12. In this case, the load will change from Frequency balance state to unbalance state. In detail, loads of phase A, B (Hz) 50 49.5 and C are changed from 4+4jΩ, 4+4jΩ and 4+4jΩ to 4+4jΩ, (a) 49 Control starts 5+4jΩ, and 4+5jΩ. After load change, the voltage unbalance is 3 Control starts increased first and then decreased to zero, which means the Active power 2.5 voltage balance has already been achieved. Also, the active and (kW) reactive power dispatch is changing due to unbalance load, as 2 shown in Fig. 12 (b) and (c). Although the power distribution (b) 1.5 has changed, the frequencies keep synchronized eventually 3 Reactive power Control starts after short fluctuation. Therefore, the proposed control works 2.5 well under unbalance load. (kVar) 2 C. Case III: load changing and load characteristic changing (c) 1.5 Case III aims to illustrate the robustness of the proposed 15 Unbalance rate control under load changing and load characteristic changing. Unbalance 10 In the beginning, the system works under a balance three-phase rate(%) 5 load value (8+2jΩ for each phase). At t=3s, a large three-phase Control starts 0 balance load about 6+2jΩ is switched on. After load increases, (d) 0 1 2 3 4 5 6 as shown in Fig. 13 (a), the frequency is getting into another Time(s) steady state point after a small disturbance. Moreover, the Fig. 11 Case I results: (a) frequency, (b) active power, (c) reactive power, (d) voltage unbalance rate. voltage unbalance rate and power sharing are still ensured. At 8 IEEE Journal of Emerging and Selected Topics in Industrial Electronics t=11s, the load for phase C is changed from RL load about 50.5 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 Frequency 8+2jΩ to RC load about 8-2jΩ. That is to say, the load changes 49.85 (Hz) 49.8 49.75 from balanced state to unbalance state and there are existing 50 49.7 1 2 3 4 different types of load in different phases. From Fig. 13 (a) and Frequent-variation load (a) 49.5 (d), despite large disturbances in frequency and voltage 4 unbalance rate, the frequency synchronization is eventually Active power 3 achieved and the unbalanced rate reduces to 0 finally. In (kW) 2 addition, the phase voltage amplitude returns to the normal 1 Frequent-variation load rated value after a small disturbance. All in all, under different (b) 0 dynamic conditions, the CHIL results prove the robustness of 4 Reactive power the proposed control. 3 (kVar) 50.1 Load Load Load 2 Frequency increases decreases characteristic 50 (Hz) changes 1 Frequent-variation load 49.9 0 (c) (a) 49.8 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 10 6 Unbalance Load Frequent-variation load Unbalance rate Active power rate(%) Load characteristic increases 4 changes 5 (kW) Load 2 decreases (d) 0 (b) 0 0 2 4 6 8 10 400 Time(s) System voltage Load 200 VA VB VC Load characteristic Fig. 15 Case IV results of frequent-variation unbalance load: (a) frequency, increases (b) active power, (c) reactive power, (d) voltage unbalance rate. (V) changes Amplitude 0 fluctuation -200 To further discuss the dynamic performance of the proposed (c) -400 method under frequent-variation load, two CHIL tests of the 15 Unbalance rate Load characteristic proposed control under frequent-variation balance load and Unbalance unbalance load are carried out respectively. The frequent- rate (%) 10 changes 5 variation load Zfrl is comprised of a resistor and an inductor 0 connected in series, where Zfrl=Rfrl+jXfrl. The resistance and (d) 0 4 8 12 16 inductance of the load are randomly selected between 10 and Time (s) 20. Moreover, the load impedance values changed frequently, Fig.13 Case III results: (a) frequency, (b) active power, (c) system voltage, (d) which is refreshed by a 20Hz pulse trigger signal. voltage unbalance rate. Fig. 14 describes the CHIL result of the proposed control D. Case IV: performance under frequent-variation load under a frequent-variation balance load. After inserting a 50.5 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 frequent-variation balance load, the system has the ability to track rapidly changing loads, which proves the proposed control Frequency 49.8 (Hz) 50 49.7 can achieve good dynamic performance. All units can still 1 2 3 4 guarantee frequency synchronization and the active power of Frequent-variation load (a) 49.5 each unit can still maintain approximately equal distribution. 3 Moreover, the unbalance rate can be reduced to 0. Active power 2.5 The CHIL result of the proposed control under a frequent- (kW) 2 Frequent-variation load variation unbalance load is shown in Fig. 15. As for frequent variation unbalance load, the resistances of the three-phase (b) 1.5 loads is still randomly selected between 10 and 20. The 3 inductances of the load in phase A, B and C are randomly Reactive power 2.5 selected according to the following equation. (kVar) 2 Frequent-variation load  X frlA = random 10, 20  1.5  X frlB = random 0,10 (56) (c)  10  X frlC = random 20,30 Unbalance rate Unbalance When the frequent-variation unbalance load is switched on, rate(%) 5 Frequent-variation load there are disturbances in frequency and voltage unbalance rate. It is clear to see from the partially enlarged view of Fig.15 (a) (d) 0 0 2 4 6 8 10 that the frequency can still converge after fluctuating. Moreover, Time(s) although the unbalance rate cannot be reduced to 0 due to the Fig. 14 Case IV results of frequent-variation balance load: (a) frequency, (b) frequent-variation unbalance load, it can still be guaranteed to active power, (c) reactive power, (d) voltage unbalance rate. be within the safe range. All in all, despite working under frequent-variation loads, the proposed control method is still effective. 9 IEEE Journal of Emerging and Selected Topics in Industrial Electronics E. Case V: Comparison between the three-phase CHS with results and waveforms of three-phase with neutral-point and without neutral-point connection connection, respectively. Fig. 18 and 19 present the CHIL test 50.0 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 results and waveforms of three-phase without neutral-point Frequency connection, respectively. (Hz) 49.8 50.0 A#1 A#2 A#3 B#1 B#2 B#3 C#1 C#2 C#3 Frequency (Hz) Unbalance load (a) 49.6 49.8 5 Active power 4 Unbalance load (a) 49.6 (kW) 3 5 Active power 2 4 (kW) (b) Unbalance load 1 3 Reactive power 4 2 3 Unbalance load (kVar) (b) 1 2 Reactive power 4 1 3 (kVar) Unbalance load (c) 0 2 5 Unbalance 4 Unbalance rate 1 rate(%) Unbalance load 3 (c) 0 2 5 Unbalance 1 4 rate(%) Unbalance load (d) 0 3 0 2 4 6 8 10 Time(s) 2 1 Unbalance load Fig. 16 Case V results of three-phase CHS with neutral-point connection: (a) (d) 0 frequency, (b) active power, (c) reactive power, (d) voltage unbalance rate. 0 2 4 6 8 10 Time(s) Fig. 18 Case V results of three-phase CHS without neutral-point connection: (a) frequency, (b) active power, (c) reactive power, (d) voltage unbalance rate. (a) Load Voltage (V) 150V/div (a) Load Voltage (V) 150V/div 20ms/div 20ms/div (b) Load Current (A) (b) Load Current (A) 50A/div 50A/div 20ms/div 100 100 10 10 20ms/div 50 50 5 5 73.4 7.82 0.2 100 100 10 7.74 0.1 50 50 5 Three-phase Positive-sequence Negative-sequence Zero-sequence 71.3 9.24 current component component component 0.14 Fig. 17 Case V waveforms of three-phase CHS with neutral-point connection. Three-phase Positive-sequence Negative-sequence Zero-sequence In this case, a comparison CHIL test has been added for current component component component three-phase CHS with and without neutral-point connection. To Fig. 19 Case V waveforms of three-phase CHS without neutral-point discuss their difference, the case of adding an unbalance load is connection. selected. In the beginning, the system works under a balance When working under balance load before 4s, the frequency three-phase load value (4+4jΩ for each phase). At t=4s, a three- synchronization, power sharing, and three-phase voltage can be phase un balance load (9Ω, 12Ω and 36Ω for phase A, B, and achieved in both two comparison cases. Moreover, in steady C, respectively) is added. Fig. 16 and 17 are the CHIL test state, the two comparison cases can share the same value of the 10 IEEE Journal of Emerging and Selected Topics in Industrial Electronics steady-state frequency, active power and reactive power. As for The hardware experimental tests are conducted at the balance load, there is no difference for the three-phase CHS Intelligent Microgrid Lab at Aalborg University [37] . The with and without neutral-point connection. Thus, the feasibility experimental setup is shown in Fig. 20, which consists of six under balance load without neutral-point connection is verified. DC power supplies, six inverters, two dSPACE controllers, and After t=4s, the system adds an unbalance load. The variation of AC loads. In these experiment tests, each phase system is frequency, active and reactive power are different between the comprised of two cascaded DGs. The dSPACE 1006 platform three-phase CHS with and without neutral-point connection. is programmed, where the program is compiled under But in both comparison cases, as shown in Fig. 16 (a) (b) and Matlab/Simulink. The experimental parameters are listed in Fig. 18 (a) (b), the frequency synchronization and three-phase Table III. voltage balance can be achieved after fluctuation. Moreover, it 50 A#1 A#2 B#1 B#2 C#1 C#2 Frequency can be seen from the waveform figure (Fig. 17 and Fig. 19) that (Hz) 49.9 the voltage can still maintain balance and the current can be Load quickly tracked after the unbalanced load is cut in. Due to the (a) 49.8 increase different structure, the steady-state currents are also different. 400 Active power To analyze their difference, the three-phase unbalance currents (W) are decomposed into positive-sequence, negative-sequence, 200 Load and zero-sequence currents based on symmetrical component 0 increase (b) method. As for the system with neutral-point connection, the 40 Reactive power system has negative-sequence and zero-sequence currents. As (Var) for the system without neutral-point connection, the system 20 Load only has negative-sequence current and no zero-sequence increase current. The main reason for this difference is that zero- (c) 0 sequence current generally only exists in systems with neutral- Unbalance 2 rate (%) Unbalance rate point connection or neutral-point grounding. All in all, when Load 1 increase under unbalance load, the feasibility of the system without neutral-point connection is proved. Although the proposed (d) 0 control method has some differences in steady-state 0 1 2 3 4 5 6 performance between the two comparison systems, the Time (s) frequency synchronization, three-phase voltage balance, and Fig. 21. Experimental results of case I. (a) frequency, (b) active power, (c) reactive power, (d) voltage unbalance rate. stable operation can be still achieved. (a) Load Voltage (V) VI. EXPERIMENTAL RESULTS DC Power Supply-1 DC Power dSpace Controller Supply-2 Phase C #1, #2 Phase A #1, #2 DC Power Supply-6 Phase B #1, #2 DC Power Supply-3 DC Power Supply #4, #5 Oscilloscope (b) Load Current (A) Control Desk #1 Control Desk #2 AC Loads Switch Fig. 20. The hardware experimental setups of the cascaded three-phase system. TABLE III EXPERIMENTAL PARAMETERS Fig. 22. Experimental waveforms of the proposed control under load changes. Symbol Item Value V* Voltage reference 200V A. Case I: Performance under load changes ω* Frequency reference 314rad The performance and waveform of the proposed control m Droop control coefficient 1.8 fc PWM frequency 10kHz under load change are demonstrated in Fig. 21 and Fig. 22. As Ts Sampling time 1e-4s shown in Fig. 21 (a), after the load increases, the frequencies of n DG number in each phase 2 all units increase a little after fluctuation. This is because the Lf Filter inductance 1.8mH increasing load characteristic is resistive, which will decrease Cf Filter capacitor 25uF Frequency modification proportional the power factor angle. Except that there is a small step response kP 1.3 during the load change, the voltage unbalance rate remains coefficient Frequency modification integral almost constant at zero. From Fig. 22, the voltage amplitudes of kI 2.5 coefficient each phase remain at 200V in steady state, where the dynamic 11 IEEE Journal of Emerging and Selected Topics in Industrial Electronics response of the current is fast. Moreover, frequency returns to 0 after a small fluctuation. Not only that, Fig. 26 synchronization, voltage balance, and power sharing are represents that the three-phase voltage can be kept balanced and guaranteed. In short, the above experimental results prove the the voltage amplitude can remain at the normal value, which effectiveness of the proposed control strategy in case of load means that the system has better voltage quality. changes. 50 A#1 A#2 B#1 B#2 C#1 C#2 Frequency B. Case II: Comparison between the proposed control and (Hz) 49.9 the traditional power factor angle droop control Unbalance load starts 50 A#1 A#2 B#1 B#2 C#1 C#2 (a) 49.8 Frequency 400 Active power (Hz) 49.9 Unbalance (W) load starts Frequency 200 asynchronization Unbalance (a) 49.8 load starts 400 0 Active power (b) 60 Reactive power Unbalance (W) 200 load starts 40 (Var) Unbalance load starts 0 20 (b) 60 0 Reactive power Unbalance (c) load starts 2 40 (Var) Unbalance Unbalance rate Unbalance load starts rate(%) 20 1 0 (c) 100 Unbalance (d) 0 Unbalance rate Unbalance load starts 0 1 2 3 4 5 6 rate(%) Time (s) 50 Fig. 25. Experimental results of the proposed control. (a) frequency, (b) active (d) 0 power, (c) reactive power, (d) voltage unbalance rate.. 0 1 2 3 4 5 6 (a) Load Voltage (V) Time (s) Fig. 23. Experimental results of traditional decentralized control. (a) frequency, (b) active power, (c) reactive power, (d) voltage unbalance rate. (a) Load Voltage (V) (b) Load Current (A) (b) Load Current (A) Fig. 26. Experimental waveforms of the proposed control under unbalance load. VII. CONCLUSIONS In this paper, a decentralized master-slave control method is Fig. 24. Experimental waveforms of the traditional decentralized control under proposed for the three-phase cascaded H-bridge system. In each unbalance load. phase sub-system, one DG is assigned to be the master, while In case of working under an unbalance load, an the rest are slaves. Different from the existing master-slave experimental comparison test between the proposed control and control framework, the proposed control can ensure the traditional power factor angle droop control is conducted. communication-less operation among all DGs. In detail, as for As shown in Fig. 23, when the unbalanced load is switched on, master DG, a modified power factor droop control is designed the frequency of each phase cannot achieve synchronization to guarantee the system voltage balance and keep the normal and the voltage unbalance rate increases rapidly. From Fig. 24, operation under an unbalanced load. Also, the slave DGs are it is clear to see that the three-phase voltages cannot be kept worked under individual power factor droop control with local balanced. Moreover, it suffers from voltage and current measurements, which ensures frequency synchronization in distortions. In contrast, as observed from Fig. 25, when each phase. Due to no communication between the local control introducing the unbalance load, the phase frequency units of the master and slave, the risk of communication failure synchronization can be achieved and the voltage unbalance rate and the cost are decreased, which also increases system 12 IEEE Journal of Emerging and Selected Topics in Industrial Electronics reliability to a certain extent. Moreover, the stability has been [21] S. Huang, R. Teodorescu, and L. Mathe, “Analysis of communication based distributed control of MMC for HVDC,” in Proc. 2013 15th Eur. proved based on the singular perturbations theory. Finally, the Conf. Power Electron. Appl., 2013, pp. 1–10. CHIL tests and experimental results have validated the [22] L. Mathe, P. D. Burlacu, and R. Teodorescu, “Control of a modular feasibility of the proposed method. multilevel converter with reduced internal data exchange,” IEEE Trans. Ind. Inform., vol. 13, no. 1, pp. 248–257, Feb. 2017. REFERENCES [23] J. He, Y. Li, B. 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