Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid

energies

Article Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid

Yong Shi, Dong Xu *, Jianhui Su, Ning Liu, Hongru Yu and Huadian Xu Anhui New Energy Utilization and Energy Saving Laboratory, School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China; [email protected] (Y.S.); [email protected] (J.S.); [email protected] (N.L.); [email protected] (H.Y.); [email protected] (H.X.) * Correspondence: [email protected]; Tel.: +86-0551-6290-4042

Received: 8 May 2019; Accepted: 27 May 2019; Published: 29 May 2019

Abstract: The voltage and frequency response model of microgrid is significant for its application in the design of secondary voltage frequency controller and system stability analysis. However, most models developed for this aspect are complex in structure due to the difficult mechanism modeling process and are only suitable for offline identification. To solve these problems, this paper proposes a black-box modeling method to identify the voltage and frequency response model of microgrid online. Firstly, the microgrid system is set as a two-input, two-output black-box system and can be modeled only by data sampled at the input and output ports. Therefore, the simplicity of modeling steps can be guaranteed. Meanwhile, the recursive damped least squares method is used to realize the online model identification of the microgrid system, so that the model parameters can be adjusted with the change of the microgrid operating structure, which makes the model more adaptable. The paper analyzes the black-box modeling process of the microgrid system in detail, and the microgrid platform, including 100 kW rated power inverters, is employed to validate the analysis and experimental results.

Keywords: microgrid; recursive damped least squares; black-box modeling; online identiﬁcation

1. Introduction Since the microgrid technology can realize flexible and efficient applications of distributed power generations, more and more attention has been paid to its research [1,2]. However, as the scale of microgrid increases, its large-signal behavior becomes complex at the system level analysis. To deal with these problems, microgrid dynamic modeling is one of the most effective methods. At present, microgrid modeling methods mainly include two types: mechanism modeling and identification modeling [3]. The mechanism modeling is carried out on the premise of mastering microgrid control method and structure, and therefore the model has high precision accordingly. Using the constructed model, the small/large signal stability analysis can be performed and the high frequency response characteristics of microgrid system can be studied as well [4,5]. However, with the widespread use of commercial converters, the detailed information relating to converters’ topology, parameters, and control methods is hardly to be obtained due to the commercial confidentiality and other reasons, which makes the mechanism modeling method extremely difficult to carry out. Compared with the mechanism modeling, the identification modeling method only needs to sample the input and output data to complete the identification modeling process, instead of mastering the detailed information of microgrid structure and control method. The identified model neglects to some extent the high frequency response characteristics, yet it is simple and convenient to use, thus increasing attention has been paid to it.

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Identification modeling method was initially applied in the grid-connected photovoltaic (PV) distributed generation (DG) system modeling [6], and gradually extended to distributed power supplies and distributed power controllers [7–9]. For example, References [10,11] studied identification modeling method of DC/DC converter, and nonlinear models were selected to describe converters’ output behaviors. In Reference [12], the black-box modeling process for a single three-phase inverter was proposed, and the dynamic parameters of the system were obtained through the step response. Moreover, the model also took the effects of cross-coupling into account, which further improved the accuracy of the model. In Reference [13], the parameters of the switched reluctance generator were identified, and the influence of nonlinear factors was considered as well, such as PI regulator and clamping functions. Currently, most of the research on microgrid identification modeling is aimed at the certain single converter integrated into the microgrid, such as DC/DC converter, photovoltaic inverter, and so on. By contrast, there are few studies on black-box identification modeling of microgrid at system level. Even for the identification modeling of the whole microgrid, the first step is to build non-linear models of the microgrid components, and then combine them into the microgrid model which is complex in model structure. For instance, Reference [14] described the relationships between the input and output voltage current of each DG by multi-segment nonlinear functions and then combined them together to build up the microgrid model. However, as the scale of the microgrid increases, the adaptability of this method is greatly limited. Reference [15] adopted the non-linear autoregressive method to construct the DGs integrated into the microgrid model by sampling the voltage data at the input port and the fault current at the output port, with up to 11 parameters to be identified. On the other hand, most identification modeling methods are suitable for fixed microgrid structure, and off-line identification strategies can be employed to model microgrid system [16]. Although under some circumstances (such as similar power supply characteristics, operation mode is basically fixed, etc.), off-line identification has certain applicability, but for the common microgrid structure, there is a random change in power output and load switching, which makes the applicability of this method greatly being challenged. If the online identification method is adopted in identifying microgrid model, the real time update of the model parameters can make up for this deficiency. Nowadays, on-line identification methods are usually adopted in asynchronous motor parameters’ identification [17,18], while application of these methods being adopted in microgrid system modeling has not yet been found in literatures. Therefore, this paper proposes a novel black-box modeling method of microgrid system from a brand-new perspective. The model can be identified only by the voltage frequency data sampled at the point of common coupling (PCC) and the total active and reactive power references data obtained at the input port. The main features are as follows:

1. This method has wide adaptability and is not speciﬁc to a particular structure of microgrid. When the structure of microgrid changes randomly, this method is still applicable. 2. The internal structure of the microgrid system and the types of loads are not considered at all. The method only pays attention to the response characteristics observed at PCC to ensure that the model structure is simple and easy to be applied. 3. The proposed method is not conﬁned to the analysis of relationship between power references and the voltage frequency, it is a general idea that can be applied in analyzing other port data of microgrid. 4. Meanwhile, this method is also applicable to all kinds of microgrid systems, such as AC microgrids, DC microgrids, AC/DC hybrid microgrids, and microgrids with various types of electrical equipment and loads (like rotating generators based DG, inverter connected distributed energy resource (DER), controllable and uncontrollable loads, etc.). Energies 2019, 12, 2049 3 of 16 Energies 2019, 12, x FOR PEER REVIEW 3 of 17

93 2.2. Topology andand ModelingModeling RequirementsRequirements ofof MicrogridMicrogrid

94 2.1.2.1. Typical TopologyTopology ofof MicrogridMicrogrid SystemSystem 95 TheThe typical microgrid topologytopology isis shownshown inin FigureFigure1 1.. TheThe systemsystem commonlycommonly comprises comprises DGs DGs such such 96 asas photovoltaicphotovoltaic andand windwind powerpower generations,generations, as wellwell asas large-capacitylarge-capacity energyenergy storagestorage devicesdevices thatthat 97 maintainmaintain the stability stability of of the the microgrid microgrid voltage voltage and and frequency. frequency. These These devices devices areare connected connected to the to theAC 98 ACbus busvia viathe theinverters. inverters. Some Some active active and and reactive reactive loads loads are are also also included included in in the the microgrid microgrid system. 99 EquipmentEquipment withwith communicationcommunication functionsfunctions suchsuch asas speciﬁcspecific DGsDGs andand smartsmart switchesswitches can be connected 100 toto thethe microgrid centralcentral controllercontroller (MGCC)(MGCC) whichwhich cancan alsoalso sendsend controlcontrol instructionsinstructions toto DGsDGs andand smartsmart 101 switchesswitches inin turn.turn. When the smart switch at PCC is disconnected, disconnected, th thee whole microgrid system operates 102 inin islandedislanded mode.mode. At this time, the microgrid can ad adoptopt a peer-to-peer control structure, andand then use 103 thethe controlcontrol strategystrategy suchsuch asas droopdroop controlcontrol [[19]19] andand virtualvirtual synchronoussynchronous machinemachine controlcontrol [[20]20] toto formform 104 thethe voltagevoltage andand frequencyfrequency ofof thethe microgrid.microgrid.

105 Figure 1. Topological Structure of Microgrid System. 106 Figure 1. Topological Structure of Microgrid System. 2.2. Modeling Requirements 107 2.2. Modeling Requirements When studying the coordinated control of the microgrid system, usually the first step is to model 108 and analyzeWhen studying the DGs the included coordinated in the microgridcontrol of the system, microgrid which system, is an important usually the part first before step is the to overall model 109 modelingand analyze of the microgrid. DGs included A rule in of the thumb microgrid is to usesystem, the mechanismwhich is an important modeling methodpart before to equivalentthe overall 110 multiplemodeling energy of microgrid. storage A inverters rule of thumb of the sameis to use type the into mechanism one inverter modeling so that method the microgrid to equivalent model 111 canmultiple be greatly energy simplified. storage inverters However, of the in most same microgrid type into systems,one inverter besides so that the the energy microgrid storage model devices, can 112 therebe greatly are various simplified. types However, of distributed in most power microgrid devices, systems, which besides affect the the system energy characteristics storage devices, as there well. 113 Theseare various factors types cannot of distributed be ignored, power which leaddevices, to the whic modelh affect being the too system complex. characteristics Without the as informationwell. These 114 offactors the internal cannot structurebe ignored, of the which microgrid lead to system, the model the microgrid being too model complex. can beWithout equivalent the information to a black-box of 115 modelthe internal and identification structure of the modeling microgrid method system, can the be adoptedmicrogrid to model model can the be microgrid equivalent system. to a black-box 116 modelAccording and identification to the black-box modeling model method of microgrid can be adopted as shown to in model Figure the2, the microgrid parameters system. optimization 117 of secondaryAccording voltage to the and black-box frequency model controller of canmicrogri be carriedd as outshown with lessin Figure complexity. 2, the The parameters black-box 118 modeloptimization of microgrid of secondary is considered voltage asand part frequency of the open-loopcontroller can transfer be carried function out ofwith the less overall complexity. control 119 structure,The black-box and themodel corresponding of microgrid root is considered locus can beas obtainedpart of the according open-loop to tr theansfer open-loop function transfer of the 120 function.overall control Hence, structure, the performance and the corresponding of microgrid systemroot locus can can be analyzedbe obtained and according the parameters to the open- of PI 121 regulatorsloop transfer can befunction. optimized Hence, as well. the performance of microgrid system can be analyzed and the 122 parameters of PI regulators can be optimized as well.

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123

124123 Figure 2. Microgrid Black-Box Model for Secondary Frequency and Voltage Regulation Control 125 Structure.Figure 2. Microgrid Black-Box Model for Secondary Frequency and Voltage Regulation Control Structure. 124 Figure 2. Microgrid Black-Box Model for Secondary Frequency and Voltage Regulation Control 126125 3.3. Identification IdentiﬁcationStructure. of of Black-Box Black-Box Model Model

127126 3.1.3.3.1. Identification The The Structure Structure of of IdentificationBlack-Box Identiﬁcation Model Model Model According to the modeling requirements discussed above, the equivalent structure of microgrid 128127 3.1. TheAccording Structure to of the Identification modeling requirementsModel discussed above, the equivalent structure of microgrid 129 modelmodel can bebe obtained obtained as as shown shown in Figurein Figure3. Instead 3. Instead of analyzing of analyzing the internal the internal structure structure of the microgrid, of the 130128 microgrid,the black-boxAccording the modelblack-box to the can modeling model be directly can requirements be constructed directly discussedconstructed by sampling above, by thesampling the active equivalent the and active reactive structure and power reactive of microgrid references power 131129 referencesmodeldata at can the data inputbe obtainedat portsthe input and as ports theshown frequency and in theFigure frequency and 3. voltage Instead and amplitude voltageof analyzing amplitude data atthe the internaldata output at the ports.structure output ports.of the 130 microgrid, the black-box model can be directly constructed by sampling the active and reactive power 131 references data at the input ports and the frequency and voltage amplitude data at the output ports.

132 Figure 3. 133132 Figure 3. Black-boxBlack-box model model structure structure of of microgrid. microgrid. In theory, if the line impedance and the output impedance of the inverters included in the 134133 In theory, if the line impedanceFigure 3. Black-box and the model output structure impedance of microgrid. of the inverters included in the microgrid system are purely inductive, the relationship between the frequency and the active power, 135 microgrid system are purely inductive, the relationship between the frequency and the active power, as well as the relationship between the voltage and the reactive power are completely decoupled. 136134 as wellIn astheory, the relationship if the line impedancebetween the and voltage the output and the impedance reactive power of the are inverters completely included decoupled. in the However, in practical microgrid systems, the line impedance and the output impedance are not purely 137135 However,microgrid systemin practical are purely microgrid inductive, systems, the relationthe lineship impedance between andthe frequencythe output and impedance the active arepower, not inductive, so the coupling term should be considered in the black-box model. Therefore, there are 138136 purelyas well inductive, as the relationship so the coupling between term the should voltage be consideredand the reactive in the powerblack-box are model.completely Therefore, decoupled. there 4 parameters to be identiﬁed in the black-box model as shown in Equation (1): 139137 areHowever, 4 parameters in practical to be identifiedmicrogrid in systems, the black-box the line model impedance as shown and in Equationthe output (1): impedance are not 138 purely inductive, so the coupling term! should be considered in! the black-box! model. Therefore, there fPCC GsG()f P(s) GsG ()f Q(s) P0total 139 are 4 parameters to be identifiedfP PCCin the =black-boxfP model fQ as  shown 0total  in Equation (1): =  (1) EPCC GsGsG()EP(s) G ()EQ(s ) Q 0total (1) EQPCCEP EQ  0total  fPPCCGsfP() Gs fQ ()  0total  =  where P0total and Q0total refer to the totalGsGs active() and () reactive  power  references in the microgrid(1) 140 where P0total and Q0total refer toEQ PCCthe total EPactive and EQ reactive  0total power  references in the microgrid respectively, and according to the power droop coeﬃcients of each DG, MGCC assigns the total power 141 respectively, and according to the power droop coefficients of each DG, MGCC assigns the total 140 wherereference P0total values and PQ0total0total andreferQ 0totalto theto eachtotal DGactive according and reactive to Equation power (2): references in the microgrid 142 power reference values P0total and Q0total to each DG according to Equation (2): 141 respectively, and according to the power droop coefficients of each DG, MGCC assigns the total 142 power reference values P0total and Q0total to each DG according to Equation (2):

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mP==== mP mP mP Energies 2019, 12, 2049 11 22 33 ii 5 of 16  nQ==== nQ nQ nQ  11 2 2 33 ii  =+++ (2)  PPPPP0total 1 2 3  i   =+++  m PQQQQQ0total= m P 1= m 2P 3=  =i m P  1 1 2 2 3 3 i i  ···  n1Q1 = n2Q2 = n3Q3 = = niQi 143 i i ··· i i (2) where m and n refer to the active and reactive power droop coefficients of ith inverter, P and Q refer  P0total = P1 + P2 + P3 + Pi 144 to the active and reactive power references of ith inverter, and··· fPCC and EPCC are frequency and voltage  Q0total = Q1 + Q2 + Q3 + Qi 145 amplitudes of PCC. GfP(s) and GEP(s) are transfer functions of··· the active power reference with respect 146 whereto them frequencyi and ni refer and to the the voltage active and amplitude reactive of power PCC droop respectively, coefficients GfQ(s) of iandth inverter, GEQ(s) arePi and the Qtransferi refer 147 tofunctions the active of and the reactive reactive power power references reference ofwithith inverter,respect to and thef PCCfrequencyand EPCC andare the frequency voltage amplitude and voltage of 148 amplitudesPCC respectively. of PCC. GAmongfP(s) and them,GEP (s)GEP are(s) transferand GfQ functions(s) are added of the to active the model power specifically reference with in order respect to 149 toanalyze the frequency the cross-coupling and the voltage effect. amplitude of PCC respectively, GfQ(s) and GEQ(s) are the transfer 150 functionsAccording of the reactiveto Equation power (1), reference the microgrid with respect system to is theregarded frequency as a and two-input the voltage and amplitudetwo-output 151 ofblack-box PCC respectively. model. Without Among involving them, GEP the(s) characterist and GfQ(s) areics addedof a single to the DG, model the black-box specifically identification in order to 152 analyzemodel of the the cross-coupling whole microgrid effect. system can be obtained as shown in Figure 4. 153 AccordingBy changing to Equationthe total active (1), the and microgrid reactive power system references is regarded issued as a by two-input MGCC and and sampling two-output the 154 black-boxvoltage and model. frequency Without response involving data the from characteristics the smart switch, of a single the DG,voltage the black-boxand frequency identification response 155 modelmodel of of the microgrid whole microgrid can be constructed system can by be on-line obtained identification as shown in method. Figure4 .

GsfP ()

GsEP ()

GsfQ ()

GsEQ () 156 Figure 4. Equivalent black-box model of voltage frequency response of microgrid. 157 Figure 4. Equivalent black-box model of voltage frequency response of microgrid. By changing the total active and reactive power references issued by MGCC and sampling the 158 3.2. Identification Experiments Design voltage and frequency response data from the smart switch, the voltage and frequency response model 159 of microgridSince the can step be experiment constructed process by on-line is easy identification to realize and method. has good identification performance, the 160 step response experiment is adopted to identify the transfer function of the model. In order to 161 3.2.simplify Identification the identification Experiments process Design of the transfer functions mentioned above, this work divides 162 modeledSince transfer the step functions experiment into process two groups. is easy As to sh realizeown in and Figure has good 5, one identification group carries performance, out the step 163 theexperiment step response by performing experiment a step is adoptedon the active to identify power thereference transfer valu functione, the other of the group model. carries In out order the 164 tostep simplify experiment the identification by performing process a step of on the the transfer reactive functions power reference mentioned value. above, this work divides modeled transfer functions into two groups. As shown in Figure5, one group carries out the step Active power Reactive power experiment by performing a step on the active power reference value, the other group carries out the reference step reference step step experimentP by performing a step on the reactive powerQ reference value. 0total () fPCC 0total fPCC GsfP GsfQ () 3.2.1. Active Power Reference Step The active power reference step experiment is applied to identify G (s) and G (s). The Q E fP EEP 0total is set to 0, while the total active() power referencePCC is regarded as the input of the stepPCC experiment by GsEP GsEQ () performing a step on P0total. At the same time, the f PCC1 and EPCC1 are sampled at PCC as the output of the step experiment.

(a) fPCC (b) G f P(s) = Q =0 P0total 0total (3) EPCC 165 Figure 5. Input power references stepGEP experiments(s) = (a) Active= power references step; (b) Reactive P0total Q0total 0 166 power references step.

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==== mP11 mP 22 mP 33 mPii  nQ==== nQ nQ nQ  11 2 2 33 ii  =+++ (2)  PPPPP0total 1 2 3  i  =+++  QQQQQ0total 1 2 3  i

143 where mi and ni refer to the active and reactive power droop coefficients of ith inverter, Pi and Qi refer 144 to the active and reactive power references of ith inverter, and fPCC and EPCC are frequency and voltage 145 amplitudes of PCC. GfP(s) and GEP(s) are transfer functions of the active power reference with respect 146 to the frequency and the voltage amplitude of PCC respectively, GfQ(s) and GEQ(s) are the transfer 147 functions of the reactive power reference with respect to the frequency and the voltage amplitude of 148 PCC respectively. Among them, GEP(s) and GfQ(s) are added to the model specifically in order to 149 analyze the cross-coupling effect. 150 According to Equation (1), the microgrid system is regarded as a two-input and two-output 151 black-box model. Without involving the characteristics of a single DG, the black-box identification 152 model of the whole microgrid system can be obtained as shown in Figure 4. 153 By changing the total active and reactive power references issued by MGCC and sampling the 154 voltage and frequency response data from the smart switch, the voltage and frequency response 155 model of microgrid can be constructed by on-line identification method.

GsfP ()

GsEP ()

GsfQ ()

GsEQ () 156

157 Figure 4. Equivalent black-box model of voltage frequency response of microgrid.

158 3.2. Identification Experiments Design 159 Since the step experiment process is easy to realize and has good identification performance, the 160 step response experiment is adopted to identify the transfer function of the model. In order to 161 simplify the identification process of the transfer functions mentioned above, this work divides 162 modeled transfer functions into two groups. As shown in Figure 5, one group carries out the step 163 Energiesexperiment2019, 12 ,by 2049 performing a step on the active power reference value, the other group carries out6 of the 16 164 step experiment by performing a step on the reactive power reference value.

Active power Reactive power reference step reference step P Q 0total () fPCC 0total fPCC GsfP GsfQ ()

E () PCC EPCC GsEP GsEQ ()

(a) (b)

165 FigureFigure 5. 5.Input Input power power references references step step experiments experiments (a) Active (a) Active power power references references step; (b step;) Reactive (b) Reactive power 166 referencespower references step. step.

3.2.2. Reactive Power Reference Step

The reactive power reference step experiment is applied to identify GfQ(s) and GEQ(s). The P0total is set to 0, while the total reactive power reference is regarded as the input of the step experiment by performing a step on Q0total. Meanwhile, the f PCC2 and EPCC2 are sampled at PCC as the output of the step experiment.

fPCC G f Q(s) = P =0 Q0total 0total (4) EPCC G (s) = P = EQ Q0total 0total 0

3.3. Identification Strategy and Process By sampling the input and output data of the microgrid system, the transfer function expressions of the microgrid model shown in Figure4 can be obtained through an identification algorithm. The existing identification algorithms usually use the recursive least squares method to identify the system on-line and keep updating the parameters in real time. However, as the covariance matrix decreases in the recursive process, the parameters are prone to explode [21]. In order to enhance the stability of the recursive process, a damping term can be added to suppress the parameter explosion. Therefore, the recursive damped least squares method is used in this paper as the identification algorithm. Moreover, in the classical control method for discrete systems, the bilinear transformation can ensure that the continuous system and the discrete system have the same stability and the same steady-state gain [22]. Therefore, the bilinear transformation is chosen to discretize the transfer functions shown in Equations (2) and (3), and the recursive damped least squares method is used for the parameter identification. This paper takes GfP(s) as an example and analyzes the identification process in detail, other transfer functions GEP(s), GfQ(s) and GEQ(s) can adopt the same process.

3.3.1. Data Preprocessing

The ﬁrst task is to preprocess the sampled data P0total and EPCC1. It mainly includes two steps:

1. Offset removal: The black-box model of the microgrid system includes the steady-state component and the dynamic component. But the parameters to be identified only involve the system dynamics of the system model, thus it is necessary to remove the steady-state components, i.e., the steady-state values of the input and output signals (before the step is done) respectively. 2. Measurement prefiltering: Since the sampling data contains some ripple and noise caused by the switching frequency, prefiltering input and output data should be considered before the recursive process to avoid the identification accuracy being affected. Energies 2019, 12, 2049 7 of 16

3.3.2. Model Order and Initial Value Selection Secondly, the number of coefficients of each polynomial (transfer function order) and the initial values of parameters should be determined. The first n data to be identified (in order to reduce the computation load, the selected data should not be too much) can be selected, and the least squares algorithm can be used to test a certain transfer function order iteratively. By comparing the fitness between the model output and the measured output in different orders, the order of the transfer function GfP(s) can be determined and remains constant in the process of recursion. The fitting performance can be calculated as follows:  s  N  P 2   (y(k) y)    N  k=1 −  1 X f =  y = y(k) Best fit 100% 1 s  (5) ×  − N  N  P 2  K=1  (y(k) yˆ(k))   k=1 −  where y(k) is the measured output of microgrid, yˆ(k) is the model output, y is the average value of the measured output. The higher the f Best fit, the better the model can reproduce the output characteristics of the microgrid port. Considering that the high model order affects the computing speed significantly, so as long as the fitness meets the requirements, choosing a lower transfer function order is recommended. Then, the order of the numerator of GfP(s) is set to 1, while that of the denominator is 2, the corresponding expression of GfP(s) can be obtained as

fPCC(s) b s + b G (s) = = 1 2 (6) f P 2 P0total(s) s + a1s + a2

The transfer function can be discretized by the bilinear transformation, as shown in Equation (7):

1 2 B1 + B2z− + B3z− G f P = G f P(s) Tz 1 = (7) (s= +− ) 1 2 2z 1 1 + A1z− + A2z− where  2  A1 = T /2 + 2a2 a1T /∆  − −  2  A2 = T /4 + a1T/2 + a2 /∆   B = (b + b T/2)/∆  1 2 1 (8)   B2 = (2b2 b1T)/∆  −  B = (b + b T/2)/∆  3 2 1  2 ∆ = T /4 + a1T/2

3.3.3. Online Recursive Algorithm Then the corresponding diﬀerence equation of Equation (7) is:

f (k) = A f (k 1) A f (k 2) + B P (k) + B P (k 1) + B P (k 2) (9) PCC − 1 PCC − − 2 PCC − 1 0total 2 0total − 3 0total − where f PCC (k) and P0total (k) are discrete values of f PCC and P0total at time k. Let k = 3, 4,..., m, m is the total number of samples during the time of recursive computation. Equation (9) can be rewritten in the form of a matrix:

Fm = Hmθ (10)

T T T where Fm = [ f (3), f (4), , f (m)] , Hm = [ϕ , ϕ , , ϕm] , θ = [A , A , B , B , B ] , ϕm = PCC PCC ··· PCC 3 4 ··· 1 2 1 2 3 [ f (m 1), f (m 2), P (m), P (m 1), P (m 2)]. PCC − PCC − total total − total − Energies 2019, 12, 2049 8 of 16

_ The idea of the recursive damped least squares method is to revise the estimated value θ _ _ recursively so that the minimum sum of squared errors between Fm and F m = Hm θm can be obtained:

_ _ T _ J(θ) = (Fm Hm θm) (Fm Hm θm) = min (11) − − _ where J(θ) is the objective function. As discussed above, adding damping to the algorithm can suppress parameter explosion, so a damping term is added on the basis of equation (11):

_ _ T _ _ _ T_ _ T J(θ) = (Fm Hm θm) (Fm Hm θm) + µ θm θm 1 θm θm 1 (12) − − − − − − where µ is the damping factor, which is related to the linearity of the identiﬁcation model. This paper adopts the adaptive damping factor method to adjust the value of µ. Firstly, the matrix T HnHn is employed to determine the initial value of µ:

T µ = 0.01 mean(HnH ) (13) 0 × n where the function mean is to compute the average value of the matrix. The linearity of the microgrid black-box model can be quantitatively evaluated by the ratio of the reduction and the change of the objective function J in the iterative process [22]:

_ _ J(θm) J(θm 1) λ = _ − _ − (14) eJ(θm) J(θm 1) − − _ _ where J(θm) is the calculated value of the objective function J at time m, eJ(θm) is the second-order Taylor series expansion of the objective function J at time m:

_ 2 _ _ _ _ _ J00 (θn) _ _ eJ(θm) = J(θn) + J0(θn)(θm θn) + (θm θn) (15) − 2 − where J0 is the ﬁrst derivative of J, J00 is the second derivative of J. Therefore, the damping factor can be corrected using the value calculated in Equation (14):

1. Under the condition of λ < 0.3, it shows that the objective function J exhibits a linearity reduction trend during the recursive process, µ is needed to increase in the subsequent recursive process to ensure better frequency and voltage identiﬁcation 2. Under the condition of 0.3 < λ < 0.7, it indicates that the objective function J has a small change in linearity during the recursive process, therefore the damping factor is close to the best value and there is no need to change it. 3. Under the condition of λ > 0.7, a smaller damping factor µ should be chosen.

In order to achieve the adaptive correction of damping factor during the identiﬁcation process, Proportional coeﬃcient η is employed, whose value is generally between 2 and 10, as shown below:

_ _ J(θm) J(θm 1) η = − − + 2 (16) ( f (m) ϕ(m) x ) x PCC − · k · k when µ is needed to be increased, µ can be taken as

µ = µ η (17) 0 · Energies 2019, 12, 2049 9 of 16

Energies 2019, 12, x FOR PEER REVIEW 9 of 17 on the contrary, when µ is needed to be decreased, µ can be taken as μμ=⋅η 0 (17) µ = µ0/η (18) 253 on the contrary, when μ is needed to be decreased, μ can be taken as According to the extremum principle, theμμ= minimumη value of J can be obtained by taking the 0 (18) differentiation to the right part of Equation (12). Through combing Equations (10)–(12), the recursive equation254 basedAccording on recursive to the extremum damped leastprinciple, square the methodminimum is value derived of J ascan below: be obtained by taking 255 the differentiation to the right part of Equation (12). Through combing Equations (10)–(12), the 256 recursive equation basedh on recursive damped leasti 1 square method is derived as below:  = + T + T −  Pm+1 µI Hm 1Hm 1 ϕmϕm  − h − i −1 h i (19)  ˆ = ˆ =++μ TTˆ +ˆϕϕ + T ( ) ˆ θm+1PIHHmmmmmθ+−−111m µPm+1 θm θm 1 Pm+1hm+1 fPCC m ϕm+1θm  − − − (19) θθˆˆ=+μ  θθ ˆˆ − +T  −ϕ θ ˆ  mmmmmmm++−+++11111PCC1PPhf ()m mm  Ever since a set of observation data ϕm is added, Pm and θˆm are revised once, therefore new estimation257 valuesEver since of parametersa set of observation are derived data 𝝋 andm is on-lineadded, P identificationm and 𝜽m are revised of parameters once, therefore are realized. The258 identificationnew estimation procedure values isof shownparameters in Figure are derive6. Whered and εon-lineis the identificati allowableon range of parameters of precision are error, N is259 the totalrealized. number The ofidentification samples at procedure the current is shown moment. in Figure When 6. the Where recursive ε is the algorithm allowable computerange of to the sampling260 precision moment error, or theN is parameters the total number estimation of samples does at not the change, current the moment. recursive When process the recursive is terminated 261 algorithm compute to the sampling moment or the parameters estimation does not change, the until the new sampling data arrives. After obtaining the parameters of the transfer function G (z), 262 recursive process is terminated until the new sampling data arrives. After obtaining the fP the263 bilinear parameters inverse of transformationthe transfer function can G befP(z carried), the bilinear out to inverse convert transformation the transfer can function be carried from out discrete domain264 to to convert continuous the transfer domain. function from discrete domain to continuous domain.

θˆ Pnn,

−1  =+μ TT +ϕϕ PIHHmmmmm+−−111  θθˆˆ=+μPPhf θθ ˆˆ − +T ()m −ϕ θ ˆ   mmmmmmm++−+++11111PCC1  mm 

θθˆˆ()mm−− ( 1) maxii < ε ? ∀i θˆ − i (1)m

265

266 FigureFigure 6. Flowchart6. Flowchart ofof identiﬁcationidentification procedure. procedure.

4. Experiments and Model Validation In this paper, a series of model identiﬁcation experiments are carried out on a microgrid test platform to verify the accuracy of the proposed modeling method. The controllers of the two inverters integrated into the microgrid use MCU (TMS320F28335, Texas Instruments, Inc, Dallas, TX, USA), Energies 2019, 12, x FOR PEER REVIEW 10 of 17

267 4. Experiments and Model Validation

268 EnergiesIn 2019this, 12paper,, 2049 a series of model identification experiments are carried out on a microgrid10 test of 16 269 platform to verify the accuracy of the proposed modeling method. The controllers of the two inverters 270 integrated into the microgrid use MCU (TMS320F28335, Texas Instruments, Inc, Dallas, TX, USA), 271 thethe pulse pulse width width modulation modulation (PWM) (PWM) carrier carrier frequenc frequencyy of of each each controller controller is is set set to to 6 6 kHz, kHz, and and the the 272 integratedintegrated three-phasethree-phase resistiveresistive load load is is conﬁgured configured to 30to kW.30 kW. Both Both inverters inverters operate operate in P-f droopin P-f controldroop 6 4 mode. Active and reactive power droop coeﬃcients are chosen as 4 10 and 4 10−6 respectively.−4 273 control mode. Active and reactive power droop coefficients are chosen× − as 4 × ×10 −and 4 × 10 274 respectively.Under the normal Under operation the normal of the operation experimental of the platform, experimental the input platform, and output the datainput are and sampled output by data the 275 areYokogawa sampled recorder. by the Yokogawa All devices reco arerder. connected All devices to the are AC connected Bus and communication to the AC Bus lines,and communication the structure of 276 lines,the microgrid the structure test platformof the microgrid is shown test in Figureplatform7. is shown in Figure 7.

277 Figure 7. The structure of microgrid test platform. 278 Figure 7. The structure of microgrid test platform. 4.1. Identification Experiments of GfP(s) and GEP(s) 279 4.1. Identification Experiments of GfP(s) and GEP(s) To carry out the identification experiments of GfP(s) and GEP(s), parallel inverters and the 280 three-phaseTo carry active out the load identification are integrated experiments into the microgrid.of GfP(s) and After GEP(s the), parallel system inverters reaches and a steady the three- state, 281 phasethe active active and load reactive are integrated power references into the aremicrogrid. set to 0 byAfter MGCC the system at first. reaches Then the a steady active powerstate, the reference active 282 andis changed reactive to power 80 kW references at 9 s, and are set set to 0to again 0 by at MGCC 15.7 s. at During first. Then this experiment, the active power the reactive reference power is 283 changedreference to remains 80 kW at at 0. 9 Figures, and8 setshows to 0 the again identification at 15.7 s. During data P0total this, Eexperiment,PCC1, and f PCC1the reactivesampled power by the 284 referencewaveform remains recorder at at 0. PCC. Figure 8 shows the identification data P0total, EPCC1, and fPCC1 sampled by the 285 waveformAccording recorder to the at identificationPCC. process shown in Figure6, the sampled data P0total and f PCC1 can be used to identify the model transfer function GfP(s).

8 2 4.229 10− s 0.001s + 5.0184 G f P(s) = · − (20) s2 + 945.97s + 2.5172 105 · By applying the same input on both the model and the microgrid system, the ﬁtting performance can be obtained as shown in Figure9. Since only the dynamic component is analyzed, the steady-state component contained in each measurement has to be removed. As it can be seen, the model output matches the microgrid output properly, and f Best ﬁt reaches 95.36.

Energies 2019, 12, x FOR PEER REVIEW 11 of 17

Energies 2019, 12, 2049 11 of 16 Energies 2019, 12, x FOR PEER REVIEW 11 of 17

286

287 Figure 8. Active power step sample data for identification (a) Active power references; (b) Reactive 288 power references; (c) PCC frequency; (d) PCC voltage amplitude.

289 According to the identification process shown in Figure 6, the sampled data P0total and fPCC1 290 can be used to identify the model transfer function GfP(s). ⋅−+−82 = 4.229 10ss 0.001 5.0184 GsfP () (20) s25++⋅945.97s 2.5172 10 291 By applying the same input on both the model and the microgrid system, the fitting performance 286292 can be obtained as shown in Figure 9. Since only the dynamic component is analyzed, the steady- 293 state Figurecomponent 8. Active contained power step in each sample measurement data for identiﬁcation has to be (a )removed. Active power As references;it can be seen, (b) Reactive the model 287 Figure 8. Active power step sample data for identification (a) Active power references; (b) Reactive 294 outputpower matches references; the microgrid (c) PCC frequency; output properly, (d) PCC voltage and fBest amplitude. fit reaches 95.36. 288 power references; (c) PCC frequency; (d) PCC voltage amplitude.

289 According to the identification process shown in Figure 6, the sampled data P0total and fPCC1 290 can be used to identify the model transfer function GfP(s). ⋅−+−82 = 4.229 10ss 0.001 5.0184 GsfP () (20) s25++⋅945.97s 2.5172 10 291 By applying the same input on both the model and the microgrid system, the fitting performance 292295 can be obtained as shown in Figure 9. Since only the dynamic component is analyzed, the steady- 293 state componentFigure 9. Fitting contained curves of in the each transfer measurement function model has obtained to be removed. from the active As it power can be step seen, experiment. the model 296 Figure 9. Fitting curves of the transfer function model obtained from the active power step experiment. 294 output matches the microgrid output properly, and fBest fit reaches 95.36. Similarly, using the sampled data P0total and EPCC1, the transfer function GEP(s) can be obtained:

0.06107s2 + 64.24s + 152.9 GEP(s) = (21) s3 + 1249s2 + 3.95 105s + 4.933 108 · ·

4.2. Identiﬁcation Experiments of GfQ(s) and GEQ(s)

To carry out the identiﬁcation experiments of GfQ(s) and GEQ(s), parallel inverters and the three-phase reactive load are integrated into the microgrid. After the system reaches a steady state, 295

296 Figure 9. Fitting curves of the transfer function model obtained from the active power step experiment.

Energies 2019, 12, x FOR PEER REVIEW 12 of 17

297 Similarly, using the sampled data P0total and EPCC1, the transfer function GEP(s) can be obtained: 2 ++ = 0.06107ss 64.24 152.9 GsEP () (21) s32++⋅+⋅1249ss 3.95 10 5 4.933 10 8

298 4.2.Energies Identification2019, 12, 2049 Experiments of GfQ(s) and GEQ(s) 12 of 16

299 To carry out the identification experiments of GfQ(s) and GEQ(s), parallel inverters and the three-phase 300 reactivethe active load and are reactive integrated power into references the microgrid. are set toAfter 0 by the MGCC system at ﬁrst.reaches Then a steady the reactive state, powerthe active reference and 301 reactiveis changed power to references 80 kVar at are 9 s, set and to set0 by to MGCC 0 again at after first. 5.6Then s. Duringthe reactive this power experiment, reference the is active changed power to 302 80reference kVar at 9 remains s, and set at 0.to 0 Figure again 10after shows 5.6 s. the During identiﬁcation this experiment, data Q0total the active, EPCC2 power, and f referencePCC2 sampled remains by theat 303 0.waveform Figure 10 shows recorder the at identification PCC. data Q0total, EPCC2, and fPCC2 sampled by the waveform recorder at PCC.

304 Figure 10. Reactive power step sampling data for identiﬁcation (a) Active power references; (b) Reactive 305 Figure 10. Reactive power step sampling data for identification (a) Active power references; (b) power references; (c) PCC frequency; (d) PCC voltage amplitude. 306 Reactive power references; (c) PCC frequency; (d) PCC voltage amplitude.

According to the identification process shown in Figure6, the sampled data Q0total and EPCC2 can 307 According to the identification process shown in Figure 6, the sampled data Q0total and EPCC2 can be used to identify the model transfer function GEQ(s). 308 be used to identify the model transfer function GEQ(s). 5 2 1.0129⋅−+−5210− s 1.288s + 7826.65 GEQ(s=) =1.0129 10· ss 1.288− 7826.65 (22) GsEQ () s2 + 2760.77s + 2.545 106 (22) s26++⋅2760.77s 2.545 10 · By applying the same excitation on both the model and the microgrid system, the fitting performance can be obtained as shown in Figure 11. Clearly, the model output also matches the microgrid output, and f Best fit reaches 90.72. Similarly, using the sampled data Q0total and f PCC2, the transfer function GfQ(s) can be obtained as follows: 6 2 4 4 8.085 10− s + 2.219 10− s + 6.091 10− G f Q(s) = · · · (23) s3 + 8.47s2 + 710.6s + 5187 Energies 2019, 12, x FOR PEER REVIEW 13 of 17

309 By applying the same excitation on both the model and the microgrid system, the fitting 310 Energiesperformance2019, 12, 2049 can be obtained as shown in Figure 11. Clearly, the model output also matches13 the of 16 311 microgrid output, and fBest fit reaches 90.72.

312

313 FigureFigure 11. 11.Fitting Fitting curves curves of the of transfer the transfer function function model obtainedmodel obtained from the from reactive the powerreactive step power experiment. step 314 experiment. 4.3. Recursive Model Validation

315 AccordingSimilarly, to using the powerthe sampled references data Q step0total experiments,and fPCC2, the transfer the voltage function frequency GfQ(s) can response be obtained model as is 316 obtainedfollows: under the certain microgrid structure. Furthermore, in order to verify the adaptability of the recursive damped least squares method⋅+⋅+⋅−−−62 applied in the 4 model identification, 4 that is, the online = 8.085 10ss 2.219 10 6.091 10 GsfQ () (23) identification effect during the period ofs32 microgrid+++8.47ss 710.6 structure 5187 changing, several adjustments have 6 6 been made: active power droop coefficients of two inverters are changed to 4 10− and 8 10− 4 4× × respectively; reactive power droop coefficients are changed to 2 10− and 4 10− respectively; the PV 317 4.3. Recursive Model Validation × × inverter which is supplied by Chroma solar PV array simulators is added to the microgrid structure, 318 and theAccording maximum to output the power power references of the PV step inverter experime is setnts, to the 4 kW.voltage During frequency the power response references model stepis 319 experiment,obtained under the number the certain of inverters microgrid connected structure. to Furt thehermore, microgrid in areorder changed, to verify the the active adaptability and reactive of 320 loadsthe arerecursive set to didampedfferent parameters,least squares as method listed inapplied Table 1in. the model identification, that is, the online 321 identification effect during the period of microgrid structure changing, several adjustments have 322 been made: active power droop coefficientsTable 1. Experimental of two inverters parameters. are changed to 4 × 10−6 and 8 × 10−6 323 respectively; reactive power droop coefficients are changed to 2 × 10−4 and 4 × 10−4 respectively; 324 the PV inverterTime which is Totalsupplied Power byReferences Chroma solar PV Activearray simulators Load Reactiveis addedLoad to the microgrid

325 structure, andt 1the maximum output0/0 power of the PV inverter 0 is set to 4 kW. 0 During the power 326 references stept2 experiment, the number0/0 of inverters connected 12 kW to the microgrid 20 kVar are changed, the 327 active and reactivet3 loads are set60 to kW diff/40erent kVar parameters, as 12listed kW in Table 1. 20 kVar t4 100 kW/60 kVar 12 kW 20 kVar t 60 kW/40 kVar 12 kW 20 kVar 328 5 Table 1. Experimental parameters. t6 60 kW/40 kVar 8 kW 10 kVar Time t7 Total60 kW power/40 kVar references (INV1 shutdown) Active8 kW load 10 kVar Reactive load t 60 kW/40 kVar (INV1 start up) 8 kW 10 kVar t1 8 0/0 0 0 t9 0/0 8 kW 10 kVar t2 0/0 12 kW 20 kVar t10 0/0 0 0 t3 60 kW/40 kVar 12 kW 20 kVar t4 100 kW/60 kVar 12 kW 20 kVar The inputt5 and outputdata 60 kW/40 during kVar the experiment are recorded 12 kW by the waveform recorder, 20 kVar as shown in Figure t126 . 60 kW/40 kVar 8 kW 10 kVar Meanwhile,t7 the 60 kW/40 same experimental kVar (INV1 shutdown) processes as shown in Table 8 kW1 are carried out on 10 a correspondingkVar simulationt8 model 60under kW/40 the kVar MATLAB (INV1/ startSimulink up) environment 8 kW (2014a, The MathWorks, 10 kVar Inc, Natick, MA, USA),t9 which is constructed 0/0 according to the black-box 8 equivalentkW model shown 10 kVar in Figure4. Initial valuest10 of P and θˆ can be figured0/0 out according to the transfer 0 function parameters 0 obtained in Sections 3.2 and 4.1. The frequency and voltage output of the simulation model can be obtained by 329 using theThe recursive input and damped output leastdata squaresduring the method experiment for online are recorded identification. by the waveform recorder, as 330 shown in Figure 12. The experimental and simulation results are compared with each other, as shown in Figure 13. It can be seen that the simulation model is able to reproduce the system responses of the microgrid system during the experiment, especially when the frequency and voltage responses of the microgrid have step changes, the output of the simulation model tracks the changes accurately. Hence, the identification algorithm implemented in the simulation model is validated. Energies 2019, 12, 2049 14 of 16

Energies 2019, 12, x FOR PEER REVIEW 14 of 17

331 Figure 12. Data waveforms sampled at input and output ports of the microgrid (a) Active power 332 Figure 12. Data waveforms sampled at input and output ports of the microgrid (a) Active power 333 references;Energiesreferences; 2019 (b, 12), Reactive x (FORb) Reactive PEER power REVIEW powe references;r references; ( c(c)) PCCPCC frequency; frequency; (d) ( dPCC) PCC voltage voltage amplitude. amplitude. 15 of 17

334 Meanwhile, the same experimental processes as shown in Table 1 are carried out on a 335 corresponding simulation model under the MATLAB/Simulink environment (2014a, The 336 MathWorks, Inc, Natick, MA, USA), which is constructed according to the black-box equivalent 337 model shown in Figure 4. Initial values of P and 𝜽 can be figured out according to the transfer 338 function parameters obtained in Section 3.2 and Section 4.1. The frequency and voltage output of the 339 simulation model can be obtained by using the recursive damped least squares method for online 340 identification. 341 The experimental and simulation results are compared with each other, as shown in Figure 13. 342 It can be seen that the simulation model is able to reproduce the system responses of the microgrid 343 system during the experiment, especially when the frequency and voltage responses of the microgrid 344 have step changes, the output of the simulation model tracks the changes accurately. Hence, the 345 identification algorithm implemented in the simulation model is validated.

346 Figure 13. Identification model and measured output curves (a) PCC frequency; (b) PCC voltage amplitude. 347 Figure 13. Identification model and measured output curves.

348 5. Conclusions 349 In this paper, the black-box modeling technique of voltage frequency response model of 350 microgrid system has been studied. In terms of the parameter identification algorithm of the recursive 351 damped least squares, the model can be identified and is suitable for performing simulations of 352 microgrid system on system level, especially when the internal information of a microgrid cannot be 353 accessed. The procedure of constructing the black-box model of microgrid is analyzed emphatically. 354 Compared with the traditional method of mechanism modeling, this method greatly simplifies the 355 modeling steps and provides a general idea for microgrid system modeling. 356 The output data obtained from the simulation model has been compared with the measured 357 output under the conditions of changing the microgrid structure. Depending on the recursive 358 algorithm, the identification model can reproduce the frequency and voltage characteristics of the 359 microgrid port accurately in all cases, which indicates that the black-box identification modeling 360 method has a wide range of adaptability to realize on-line identification of the microgrid system. 361 In future, in order to improve the identification accuracy, the influence of different 362 communication delays on the proposed identification strategy needs to be further studied. In 363 addition to this, how to apply the proposed identification model to a secondary frequency and 364 voltage regulation control system, so as to optimize parameters of secondary voltage and frequency 365 controllers, are also set as the future research directions.

366 Author Contributions: This paper was a collaborative effort between the authors. Y.S. provided the original idea 367 and reviewed the paper, D.X. and H.X. organized the manuscript and realized MATLAB-based simulation, H.Y. 368 carried out experiment validation, J.S. and N.L. provided professional advice and technical comments.

369 Funding: This research was funded by Double First Class Project for Independent Innovation and Social Service 370 Capabilities (45000-411104/012), National Key Research and Development Program of China (2017YFB0903503) 371 and the Basic Operating Expenses of Central Scientific Research (PA2018GDQT0021).

372 Conflicts of Interest: The authors declare no conflict of interest. 373

Energies 2019, 12, 2049 15 of 16

5. Conclusions In this paper, the black-box modeling technique of voltage frequency response model of microgrid system has been studied. In terms of the parameter identification algorithm of the recursive damped least squares, the model can be identified and is suitable for performing simulations of microgrid system on system level, especially when the internal information of a microgrid cannot be accessed. The procedure of constructing the black-box model of microgrid is analyzed emphatically. Compared with the traditional method of mechanism modeling, this method greatly simplifies the modeling steps and provides a general idea for microgrid system modeling. The output data obtained from the simulation model has been compared with the measured output under the conditions of changing the microgrid structure. Depending on the recursive algorithm, the identification model can reproduce the frequency and voltage characteristics of the microgrid port accurately in all cases, which indicates that the black-box identification modeling method has a wide range of adaptability to realize on-line identification of the microgrid system. In future, in order to improve the identification accuracy, the influence of different communication delays on the proposed identification strategy needs to be further studied. In addition to this, how to apply the proposed identification model to a secondary frequency and voltage regulation control system, so as to optimize parameters of secondary voltage and frequency controllers, are also set as the future research directions.

Author Contributions: This paper was a collaborative effort between the authors. Y.S. provided the original idea and reviewed the paper, D.X. and H.X. organized the manuscript and realized MATLAB-based simulation, H.Y. carried out experiment validation, J.S. and N.L. provided professional advice and technical comments. Funding: This research was funded by Double First Class Project for Independent Innovation and Social Service Capabilities (45000-411104/012), National Key Research and Development Program of China (2017YFB0903503) and the Basic Operating Expenses of Central Scientific Research (PA2018GDQT0021). Conflicts of Interest: The authors declare no conflict of interest.

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