Power System Model Reduction with Grid-Connected Photovoltaic Systems Based on Hankel Norm Approximation

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1 This paper was presente at the 6th Solar Integration Workshop an publishe in the workshop s proceeings. Power System Moel Reuction with Gri-Connecte Photovoltaic Systems Base on Hankel Norm Approximation Imran Maqbool, Gustav Lammert 2, Anton Ishchenko 3, Martin Braun,2 Fraunhofer IWES, Kassel, Germany 2 University of Kassel, Kassel, Germany 3 Phase to Phase BV, Arnhem, Netherlans imran maqbool@yahoo.com Abstract In orer to ensure stability of the system, transmission an istribution stuies nee to be performe. These simulation stuies can be time consuming ue to high complexity of the system. This paper investigates moel reuction on a power system with photovoltaic generation. This is one by ecreasing the complexity of the etaile moel, which leas to a smaller simulation time, but still maintaining the accuracy of the moel. Therefore, the Hankel norm approximation is chosen, which applies controllability an observability properties of the system to compute the singular values. These singular values are helpful for etermining the orer an accuracy of the reuce system. For the verification of the Hankel norm approximation, a small four-bus test system is stuie in MATLAB/Simulink. The results of the reuce linearize moel are compare with the original nonlinear moel an show goo accuracy. Inex Terms Dynamic equivalents, ynamic equivalencing, moel orer reuction, moal moel reuction, reuce-orer moel, Singular Value Decomposition (SVD) methos, power system stability. I. INTRODUCTION In orer to operate power systems reliably an to control them properly, power system stability stuies must be carrie out. In these stability stuies a ynamic response of the system is accurately stuie on computer software simulation tools through specific moels of the power system. Furthermore, in recent ecaes, power systems have change rapily. Power systems are expaning in both, size an complexity ue to ever increasing eman of electricity. Moreover, the integration of Distribute Generators (DGs) further increases the complexity of the power system. Due to this increase, moelling an simulation of power systems in etail has become more ifficult as it presents a huge computational buren. Therefore, much efforts have been put into eveloping efficient computational techniques for large complex systems. One of the best technique is to reuce the complexity of a large power system to a computationally feasible size of an equivalent system, which retains the ynamic characteristics of the power system with reasonable accuracy. The process of reucing a large power system network into an equivalent moel for ynamic system stuies is calle ynamic equivalencing or moel reuction []. In general, ynamic power system equivalents can be classifie into three major groups: ) coherency base methos; 2) measurement base methos; an 3) strictly mathematical methos [2]. In coherency base methos, the group of generators which ten to oscillate together (similar rotor angle characteristics) in the event of a isturbance are replace by an equivalent generator. Such a group of generators are calle coherent generators []. There are ifferent methos to ientify coherent groups of generators, e.g., the weak link metho [3] or the slow coherency metho [4]. A brief overview of these methos can be foun in [5]. Coherency base methos give a goo nonlinear equivalent of the system. The main isavantage of these methos is that they are only applicable for synchronous generators as they use rotor angle properties of the generators [6]. Measurement base reuction methos use either real-time measurements from phasor measurement units in the power system or a simulate response of the power system in orer to create a reuce moel of the system. These methos can be further ivie into two groups: Artificial Neural Network (ANN) methos an system ientification methos. In [7] ANNs are use to create a nonlinear reuce moel of the power system with fuel cells an microturbines as DGs. A black box moel is use in [8] to get the reuce moel of the system with small synchronous generators an oublyfe inuction generators use as DGs. In [9], [] grey box moelling techniques are use to obtain the reuce moel of the system. The main avantage of these methos is that they are useful when etails of the system are not known. These methos can be applie to istribution networks with large amount of DGs. For small to meium size systems these methos take more computational time then other methos as they nee a large ata set from simulations of preefine isturbances to create a ynamic equivalent moel. Strictly mathematical methos use existing techniques from control theory, which can be applie for the reuction of ifferent large-scale physical systems. These methos mostly have a strong mathematical basis an aim to approximate the input-output behavior of the consiere system. Most of

2 these mathematical methos use for moel reuction focus on linear systems, which in many cases accurately represent the physical system [6]. Depening on the external area properties that have to be retaine in the reuce moel, ifferent mathematical moel reuction methos can be applie. In [6] Hankel norm approximation is applie to reuce the system, whereas in [] an extension of balance truncation is use. Both of these methos belong to the family of Singular Value Decomposition (SVD). In [6], [2] Krylov methos are use to reuce large power systems. As these methos require a linear moel of the system, they are not vali when the operating point of the system is shifte far away from the original system operating point. It shoul be note that these methos can not properly represent the system for highly nonlinear phenomena [6]. As Hankel norm approximation is the most efficient metho out of the SVD base methos [6], it is consiere for the investigation in this paper. The paper is organize as follows. Section II provies an introuction to Hankel norm approximation. Section III escribes the test system incluing the PhotoVoltaic (PV) system on which the Hankel norm approximation is applie. In Section IV the results of the moel reuction are illustrate. Finally, the conclusions are presente in Section V. II. HANKEL NORM APPROXIMATION Hankel norm is the measure of how much energy can be transferre from the past inputs to the future outputs. The reuce orer moel base on Hankel norm attempts to reuce the system in such a way that Hankel norm of the reuce system is minimize an thus the worst case error [3]. The algorithm of Hankel norm approximation is shown in Fig. an can be ivie into 5 steps: ) State space of the system 2) Balancing the state space 3) Partitioning the state space 4) Fining all pass of the system 5) Extracting the reuce orer system A. State space of the system Hankel norm approximation is applicable only to linear (or linearize) systems. Therefore, consier the linear ynamical system in state space form: x(t) = Ax(t) Bu(t) t () y(t) = Cx(t) Du(t) (2) where x(t) is the state vector, u(t) is the vector of inputs an y(t) is the vector of outputs. Further, assume that there are n states, m inputs an p outputs. Hankel norm approximation uses two important properties of the ynamic system, namely controllability an observability. The controllability is the ability of the system to reach from the zero state x o to any other state in finite amount of State space of the system x(t) = Ax(t) Bu(t) t y(t) = Cx(t) Du(t) W c W o Balancing the state space z = Ãz Bu t y = Cz Du W c = W o = iag{σ, σ 2,..., σ n} Partitioning the state space z t = Ãz Ã2z 2 B u z 2 t = Ã2z Ã22z 2 B 2 u y = C z C 2 z 2 Du [ ] Σ W c = W o = Σ 2 Σ = iag{σ,..., σ k, σ kr,..., σ n} Σ 2 = σ k I r Fining all pass of the system  = Γ ( σ kãt 2 Σ ÃΣ σ k CT U B T ) ˆB = Γ ( Σ B σ k CT U ) Ĉ = C Σ σ k U B T ˆD = D σ k U. Fin T Extracting the reuce orer system A k =  B k = ˆB C k = Ĉ D k = ˆD G(s) Gk (s) = σ k 2. Perform similarity transformation x = T z. Select k 2. Determine r 3. Partition balance system states. Calculate Γ = Σ 2 σ2 k I 2. If m > p then replace: (Ã, B, C, D) (ÃT, C T, B T, D T ) 3. Solve B 2 C T 2 U =. Extract stable part of all pass ilation: Â, ˆB, Ĉ, ˆD 2. If m > p then replace: (Â, ˆB, Ĉ, ˆD ) (ÂT, ĈT, ˆB T, ˆD T ) Fig.. Methoology of Hankel norm approximation algorithm [6].

3 time by the action of some input u(t). In orer to reach this final state a limite amount of energy ε c is require [4]: ε c = x T o W c x o (3) where W c is calle the controllability gramian an efine as: W c = e At BB T e ATt t (4) In other wors, the controllability provies a quantitative measure of how closely the inputs an the states are couple. The secon important property of the system is observability. States of a system are internal variables an in general it might be impossible to irectly measure them. At the same time, the outputs can be measure rather easily. The observability is the ability of the system to uniquely etermine state x o through the time from the measure outputs y(t). A certain amount of output energy is generate by the state x o for a natural response (zero input) [4]: ε o = x T o W o x o (5) where W o is calle the observability gramian an efine as: W o = e ATt C T Ce At t (6) In other wors, the observability gramian provies a quantitative measure of how closely the outputs an the states are couple. Usually, (4) an (6) are not use for computing the gramians as they involve matrix exponentials an integrals. Instea, gramians are compute by solving the Lyapunov equations [4]: B. Balancing the state space AW c W c A T = BB T (7) A T W o W o A = C T C (8) In orer to apply moel reuction to the system, it shoul be in balance state space representation. Thus, the states of the system are such that the egree of controllability an egree of observability of each state is the same. Mathematically, balancing methos consist of the simultaneous iagonalization of controllability an observability gramians. This can be achieve by the use of similarity transformations which change the state an the system matrices but retain the same inputoutput behavior [4]. Suppose the following equation: x = T z (9) where T is a non-singular, constant matrix. Substituting this expression into () an (2) yiels: z t = T AT z T Bu () y = CT z Du () with à = T AT, B = T B, C = CT an D = D leas to: z t = Ãz Bu (2) y = Cz Du (3) It is possible to fin such a similarity transformation that changes the system in balance form an iagonalize the gramians: W c = W o = iag{σ, σ 2,..., σ n } (4) where σ, σ 2,..., σ n are real an positive numbers calle Hankel singular values. The amount of Hankel singular values is equal to the amount of states of the system. Furthermore, it is assume that these values are orere in such way that σ σ 2... σ n by moifying the rows an columns of the matrices of the system. C. Partitioning the state space After balancing the system, the orer of the reuce system k is selecte using Hankel singular values of the system. If the general situation is consiere, it is possible that several Hankel singular values with an inex starting from k are equal, i.e., σ k > σ k = σ k2 =... = σ kr > σ kr... σ n >, where r is the number of equal Hankel singular values with an inex starting from k. As seen, if σ k σ k2 then r =. After selecting k an etermining r, the system can be transforme to a partially balance form with the controllability an observability gramians [5]: [ ] Σ W c = W o = (5) Σ 2 Σ = iag{σ, σ 2,..., σ k, σ kr,..., σ n } (6) Σ 2 = σ k I r (7) where I r is the ientity matrix with imension r. The system matrices Ã, B, C, D can be partitione in the same way as the gramians: z t = Ãz Ã2z 2 B u (8) z 2 t = Ã2z Ã22z 2 B 2 u (9) y = C z C 2 z 2 Du (2) where the vector z has the imension n r an the vector z 2 has the imension r. D. Fining all pass of the system All pass is a key concept use in Hankel norm approximation. Suppose the transfer function of the original system efine by () an (2) is G(s). Then there exists a system with the transfer function Ĝ(s) such that the gain of the error transfer function G(s) Ĝ(s) is constant for all frequencies an it is sai to be all pass. In orer to fin all pass of the system, efine a iagonal non-singular matrix [5]: Γ = Σ 2 σ 2 ki (2)

4 If the number of inputs is more than the number of outputs (m > p) then replace Ã, B, C, D by ÃT, C T, B T, D T. The next step is to fin a unitary matrix U by solving: B 2 C T 2 U = (22) Then, the all pass ilation of the system is efine by: Â = Γ ( σ kãt 2 Σ Ã Σ σ k CT U B T ) (23) ˆB = Γ ( Σ B σ k CT U ) (24) Ĉ = C Σ σ k U B T (25) ˆD = D σ k U (26) E. Extracting the reuce orer system The all pass of the system is usually unstable an only contains k stable poles [4]. Therefore, the stable part of the all pass ilation Â, ˆB, Ĉ, ˆD has to be etermine. This stable part is the reuce orer moel of the system. If the number of inputs is more than the number of outputs (m > p), replace Â, ˆB, Ĉ, ˆD by ÂT, ĈT, ˆB T, ˆD T. This is the last step of the algorithm. One of the important features of the Hankel norm metho is that the error of the approximation is boune: G(s) G k (s) = σ k (27) where G k (s) is the transfer function of the k-th orer reuce moel. It means that the largest error for all frequencies is equal to the k Hankel singular value. This a priori knowlege of the achievable errors allows to select a suitable value of k. It is crucial to a that the Hankel norm approximation can be performe only by means of algebraic operations with the state space matrices. Therefore, it is fast, reliable an numerically simple [4]. A. Test system III. MODELLING OF THE TEST SYSTEM In power system moel reuction, the system is usually ivie into two subsystems, i.e., stuy area an external area, as seen Fig. 2. The stuy area is the part of the system which is of great interest. This area is retaine in the power system moel reuction process an all the isturbances an configuration changes happen in it. Therefore, the moelling of the stuy area is one in etail. The external area is the rest of the power system containing all the external generators, loas etc. The external area can be a neighboring area, which oes not effect the ynamics External area (reuce linear moel) Photovoltaic system Stuy area (etaile nonlinear moel) Fig. 2. Test system for performing the Hankel norm approximation. of the stuy area very much. Hence, the external area can be reuce to get a simplifie equivalent moel of the system. Both subsystems are connecte to each other by lines an bounary buses. The test system use in this stuy is relatively simple in orer to focus on the basic steps of the Hankel norm approximation. A four-bus system is employe in MATLAB/Simulink as given in Fig. 2. The system consists of a PV system with a coupling branch connecte to an infinite bus through a transformer an a feeer branch. The stuy area contains the infinite bus, transformer an the feeer branch an it will be subjecte to the isturbance. The external area consists of the PV system an the coupling branch, connecte to the bounary bus (bus 2). The parameters of the test system are given in Table I. B. Photovoltaic system The PV system use in this stuy is base [6]. In aition, for bulk power system ynamic stuies generic PV system moels, such as [7], can be use. The control block iagram of the inverter is shown in Fig. 3 an the parameters are given in Table I. The power references an the voltages at the point of connection are use to set references for the current controllers. The inverter is also provie with a DC link voltage from the PV array. These variables constitute the inputs to the moel whereas the currents at point of connection are treate as outputs. The control is performe in a rotating reference frame (q-omain) an the reference angle for the abc-q transformation is provie by the Phase-Locke Loop (PLL). The measure input voltage at the point of connection is transforme to the q-omain using the transformation matrix by the following equation [6]: v O cos θ cos ( ) θ 2π 3 cos ( ) θ 2π 3 2 v Oq = 3 sin θ sin ( ) θ 2π 3 sin ( ) θ 2π 3 v O v Oa v Ob v Oc (28) The PLL use here is given in Fig. 4. It is base on aligning in close-loop control the angle of the q-transformation such that the voltage at the connection point has no q-axis component [6]. A PI controller acts on the alignment error to set the rotational frequency (29) an that frequency is integrate to give the transformation angle (3): ω = KP PLL v Oq KI PLL v Oq t (29) θ = ω t (3) The power controller shown in Fig. 5 is use in open-loop control to calculate the reference currents in q-form with the ai of the reference active an reactive power an the voltage

5 TABLE I TEST SYSTEM PARAMETERS Description Symbol Value Unit Nominal phase voltage v n 4 [V] Gri frequency f 5 [Hz] Photovoltaic system rate apparent power S r, PV 2 [kva] DC bus voltage v DC [V] Coupling resistance R c.3 [Ω] Coupling inuctance L c.96 [mh] Filter resistance R.56 [Ω] Filter inuctance L.35 [mh] Filter capacitance C 5 [µf] Phase-Locke Loop proportional gain KP PLL 2. [ ] Phase-Locke Loop integral gain KI PLL 5 [/s] Current controller proportional gain K q P [ ] Current controller integral gain K q I 46 [/s] Feeer branch resistance R f.275 [Ω] Feeer branch inuctance L f.92 [mh] Transformer rate apparent power S r. [MVA] Transformer rate voltage high sie V r, HS 2 [kv] Transformer rate voltage low sie V r, LS.4 [kv] Transformer resistance R tr.28 [Ω] Transformer inuctance L tr.83 [mh] P* Q* VOabc Power controller i * Lq i err Lq V * normlq Current Controller Inverter VDC Vlq LC filter an coupling Fig. 3. Control block iagram of the photovoltaic system [6]. abc q VOq KP PLL Fig. 4. Control block iagram of the Phase-Locke Loop [6]. at the point of connection. The output reference currents are calculate as: KI PLL i * O = v OP * v Oq Q * vo 2 v2 Oq i * Oq = v OqP * v O Q * vo 2 v2 Oq ω VOq PLL ioq ilq ω VOq θ q q abc abc ioabc ilabc VOabc θ (3) (32) The filter inuctor currents are controlle in the current controller. Therefore, ajustment must be mae to the reference P* Q* V O V Oq V O -V Oq V Oq V O - P* Q* i * O i * Oq - i L i Lq - i i O i q i Oq Secon-orer low-pass filter with cut-off frequency ω c Fig. 5. Control block iagram of the power controller [6]. in orer to account for the capacitor current of the filter as: i Σ = i * O i C = i * O (i L i O ) (33) i Σ q = i * Oq i Cq = i * Oq (i Lq i Oq ) (34) A secon-orer low-pass filter is use to remove the harmonics an noises that may result from the istortion of the input voltage at the point of connection. The current controller references after filtering are given as: i * L = ωc 2 (i Σ i * t L) t 2ω c i * L (35) i * Lq = ωc 2 (i Σ q i * t Lq) t 2ω c i * Lq (36) where ω c is the cut-off frequency of the filter. The current controller use is given in Fig. 6. It uses two PI controllers together with cross-axis ecoupling terms an feeforwar terms for the connection voltage [6]. The equations for this controller are: v * I = v O ωli Lq K Pi err L K I v * Iq = v Oq ωli L K q P ierr Lq K q I i * L i * Lq i err L t (37) i err Lq t (38) It is assume that the switching frequency of the inverter is sufficiently high. Therefore, the inverter can be simplifie to a saturate voltage gain, as shown in Fig. 7. The last block of the moel contains a passive low-pass filter an a coupling impeance. The filter has important ynamic i L err i Lq err K P err q L K P q err q Lq K I K I q ωl Fig. 6. Control block iagram of the current controller [6]. i Lq ωl i L V O V Oq V * l V * lq V DC V * norml V * normlq

6 V * norml Saturation V sat V l.8.6 Nonlinear linear.4 V * normlq V q sat V DC V lq Fig. 7. Control block iagram of the inverter moel [6]. effects own to quite low frequencies [6]. The equations escribing phase a of the filter are: A. Linearization C v Ca t v Ia = i La R L i La t v Oa = v Ca L c i Oa t v Ca (39) i Oa R c (4) = i La i Oa (4) IV. MODEL REDUCTION OF THE SYSTEM As previously explaine, Hankel norm approximation is only applicable to linear (or linearize) systems. In orer to reuce the orer of the moel, the first step involves linearization. This has been performe for the PV system in the q-reference frame. The PLL is consiere as part of the conversion system an therefore, it is not linearize. The initial operating point (states x, inputs u an outputs y ) is efine base on the power flow results. The Jacobian matrices of the system at the operating point are etermine as: A = f x B = f x,u u x,u C = g D = g (42) x u x,u x,u whereas the original nonlinear system is represente by: t x(t) = f( x(t), u(t) ) (43) y(t) = g ( x(t), u(t) ) (44) Fig. 8. Comparison of the nonlinear an the linearize moel. Response of the photovoltaic system current of phase a at the bounary bus (bus 2) for a.8 p.u. voltage ip of. s. B. Moel reuction The linearize moel has 3 states in total, the same as the nonlinear moel. After linearization it is possible to transform the system into its balance form an fin its Hankel singular values. They are plotte in Fig. 9 on a logarithmic scale. From the analysis of the Hankel singular values, it appears that the balance system can be reuce rastically as most of the states are baly controllable an observable. Further simulations have shown that by using Hankel norm approximation, the system might be reuce to the 6th orer without any significant error. For instance, for the 6th orer system the Hankel singular value is about. If the orer of the system is further ecrease, the accuracy will also ecrease. The analysis of the Hankel singular values also allows to select the orer of the reuce system without performing simulations, i.e., to estimate the error an the orer a priori. After checking the Hankel singular values, time-omain simulation is performe in orer to further investigate the results of the reuce moel. Therefore, a istant three-phase fault in the gri is represente by specifying the voltage of the.6 The linearization is one using numerical perturbation in the Control Design toolbox of MATLAB/Simulink to get the state space of the external area. In the next step, the linearize state space equations in incremental form are: x(t) = A x(t) B u(t) t (45) y(t) = C x(t) D u(t) (46) Value 2 with x = x(t) x, u = u(t) u an y = y(t) y. Accuracy of linearization is checke by comparing the timeomain response of the original system with the linearize system. The results of this comparison are shown in Fig Inex of singular value Fig. 9. Hankel singular values of the system.

7 .8 Original Reuce.5.8 p.u. voltage ip.6 p.u. voltage ip Voltage[p.u.] Fig.. Comparison of the original full orer (3 states) nonlinear moel an the reuce orer (6 states) linearize moel. Response of the photovoltaic system current of phase a at the bounary bus (bus 2) for a.8 p.u. voltage ip of. s..8.6 Original Reuce Fig. 2. Different voltage ips with the uration of. s that are applie for the reuce orer (6 states) linearize system p.u. voltage ip (original).8 p.u. voltage ip (reuce).6 p.u. voltage ip (original).6 p.u. voltage ip (reuce) Fig.. Comparison of the original full orer (3 states) nonlinear moel an the reuce orer (5 states) linearize moel. Response of the photovoltaic system current of phase a at the bounary bus (bus 2) for a.8 p.u. voltage ip of. s. infinite bus to.8 pu for. s. The fault occurs at t =. s an is cleare at t =. s. In Fig. the comparison of the original full orer (3 states) nonlinear moel an the reuce orer (6 states) moel is epicte. By comparing the reuce an the full system, the results show that the system can be reuce to the 6th orer with a reasonably goo accuracy. In Fig. the comparison of the original full orer (3 states) nonlinear moel an the reuce orer (5 states) moel is shown. By comparing the reuce an the full system, the results show that the system shoul not be reuce to the 5th orer ue to the relatively poor accuracy. In relation to Fig., it shoul be note that the accuracy is consierably ecrease. Accoring to the Hankel singular values of the system shown in Fig. 9, this ecrease can be also ientifie in the ifference of the singular values between the 5th an the 6th orer. This means further reuction causes the ecrease in accuracy an results in a ba agreement with the original system. Another comparison is carrie out investigating the response of the reuce orer (6 states) linearize moel for ifferent voltage ips, as shown in Fig. 2. Because Hankel norm approximation is applicable only to linear (or linearize) systems, this stuy is carrie out in orer to confirm the accuracy Fig. 3. Comparison of the reuce orer (6 states) linearize system applie to ifferent voltage ips with the uration of. s. Response of the photovoltaic system current of phase a at the bounary bus (bus 2) p.u. voltage ip (original).8 p.u. voltage ip (reuce).6 p.u. voltage ip (original).6 p.u. voltage ip (reuce) Fig. 4. Comparison of the reuce orer (6 states) linearize system applie to ifferent voltage ips with the uration of. s. Response of the photovoltaic system current of phase a at the bounary bus (bus 2). Zoom of Fig. 3. for ifferent eviations from the steay-state operating point. The results of the responses of the PV system currents at the bounary bus (bus 2) are presente in Fig. 3. The zoom of Fig. 3 is shown in Fig. 4. From the results it is obvious that the bigger the voltage ip, the worse is the accuracy between the original full orer nonlinear moel an the reuce orer linearize moel. For voltage ips own to V =.8 p.u. the accuracy is still goo that can be seen by the black lines in

8 Fig. 4. For a voltage ip own to V =.6 p.u., inicate by the grey line, it can be seen that the accuracy between the original full orer nonlinear moel an the reuce orer linearize moel is very poor. It can be conclue that Hankel norm approximation can only be use for small isturbances in the power system. One of the most important avantages of applying power system moel reuction methos is the improvement of the computational time. Hence, the simulation time of ifferent system orers is also investigate. The simulations are carrie out in MATLAB/Simulink R23a an execute on a 64-bit Winows 8 operating system running on Intel core i5.7 GHz processor an 4 GB of RAM. The results of the performance comparison, consiering ifferent system orers, are shown in Table II. For performing a simulation with the original full orer nonlinear moel takes about s, while consiering the reuce orer linearize moel the simulation time ecreases rastically to about 5 s. Therefore, for the simple test system use in this stuy, a 6th orer Hankel norm approximation moel appears to be optimal in the sense of accuracy versus performance. V. CONCLUSIONS The new contribution of this paper is to perform ynamic moel reuction of a power system using Hankel norm approximation with a new technology, namely gri-connecte PV systems. The overall methoology of the moel reuction algorithm an the test system is presente. Furthermore, Hankel norm approximation is applie for ifferent system orers as well as various isturbances. The results inicate that for small eviations from the operating point, Hankel norm approximation can significantly reuce the orer an the simulation time of the system while retaining reasonably goo accuracy with respect to the original system. Base on the results, Hankel norm approximation can be applie in the future to istribution networks with high penetration levels of gri-connecte PV systems for ynamic stuies. The obtaine ynamic equivalents can be use on the high voltage transmission an the meium voltage istribution level to perform ifferent bulk power system stuies, such as transient an small-signal stability as well as voltage an frequency stability stuies base on root mean square simulations. Orer of the system TABLE II SIMULATION TIME Simulation time [s] 3 states (original system) states (reuce system) states (reuce system) 5. ACKNOWLEDGMENT The authors gratefully acknowlege the fruitful an insightful iscussions uring this research work with Tina Pascheag (University of Kassel, Kassel, Germany), Salman Zaii (University of Kassel, Kassel, Germany) an Daniel Duckwitz (Fraunhofer IWES, Kassel, Germany). Furthermore, the authors thank Thomas Stetz (University of Applie Sciences Mittelhessen, Gießen, Germany) for proviing the PV system moel in MATLAB/Simulink. 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