Aalborg Universitet. Published in: Proceedings of the 2018 American Control Conference (ACC)

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1 Aalborg Universitet Plug-and-Play oltage/urrent Stabilizaon D Microgrid lusters with Grid- Forming/Feeding onverters Han, Renke; Tucci, Michele; Marnelli, Andrea; Guerrero, Josep M.; Ferrari-Trecate, Giancarlo Published in: Proceedings of the 28 American ontrol onference (A) DOI (link to publicaon from Publisher):.2399/A Publicaon date: 28 Document ersion Publisher's PDF, also known as ersion of record Link to publicaon from Aalborg University itaon for published version (APA): Han, R., Tucci, M., Marnelli, A., Guerrero, J. M., & Ferrari-Trecate, G. (28). Plug-and-Play oltage/urrent Stabilizaon D Microgrid lusters with Grid-Forming/Feeding onverters. In Proceedings of the 28 American ontrol onference (A) (pp ). IEEE Press. General rights opyright and moral rights for the publicaons made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condion of accessing publicaons that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publicaon from the public portal for the purpose of private study or research.? You may not further distribute the material or use it for any profit-making acvity or commercial gain? You may freely distribute the URL idenfying the publicaon in the public portal? Take down policy If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to the work immediately and invesgate your claim. Downloaded from vbn.aau.dk on: oktober 29, 28

2 28 Annual American ontrol onference (A) June 27 29, 28. Wisconsin enter, Milwaukee, USA Plug-and-Play oltage/urrent Stabilizaon D Microgrid lusters with Grid-Forming/Feeding onverters Renke Han, Student Member, IEEE, Michele Tucci 2, Student Member, IEEE, Andrea Marnelli 3, Josep M. Guerrero, Fellow, IEEE, Giancarlo Ferrari-Trecate 4, Senior Member, IEEE Abstract In this paper, we propose a new decentralized control scheme for Microgrid (MG) clusters, given by the interconnecon of atomic dc MGs, each composed by grid-forming and grid-feeding converters. In parcular, we develop a new Plug-and-Play (PnP) voltage/current controller for each MG in order to achieve simultaneous voltage support and current feeding funcon with local references. The coefficients of each stabilizing controller are characterized by explicit inequalies, which are related only to local electrical parameters of the MG. With the proposed controller, each MG can plug-in/out of the clusters seamlessly irrespecvely of the power line parameters and models of other MGs. A profound proof of closed-loop stability of MG clusters is provided. Moreover, theorecal results are validated by hardware-in-loop (HiL) tests. I. INTRODUTION During the past decades, dc MGs have been recognized more and more attracve compared with ac MG [], as they provide a higher efficiency, more natural interface. Gridforming converters are used as the interface between energy storage system (ESSes) and the system to provide voltage support in dc MGs. To achieve simultaneous voltage support and communicaon-less current sharing among ESSes, voltage-current (-I) or I- droop control [2], [3] is widely adopted by imposing virtual impedance for output voltages. However, voltage deviaons and current sharing errors cannot be eliminated due to different line impedances. Another key challenge is that the stability of connected ESSes is sensive to the chosen virtual impedances which should be designed taking into account the specific MG topologies [4], [5], [6]. Recently, an alternave class of decentralized primary controllers, called PnP controller according to the terminology used in [7], has been proposed in [8], [9], []. In [], to achieve PnP robust voltage control, informaon about line impedances need be known to form the upper and lower boundary for system model. On the other hand, PnP controllers in [8], [9] form a decentralized control architecture each regulator can be synthesized using informaon about the corresponding ESSes only [9] or, at most, specific parameters of the power lines connected to the R. Han and J. M. Guerrero is with Department of Energy Technology, Aalborg University, Aalborg, Denmark {rha,joz}@et.aau.dk 2 M. Tucci is with Diparmento di Ingegneria Industriale e dell Informazione, Università degli Studi di Pavia, Pavia, Italy michele.tucci2@universitadipavia.it 3 A. Marnelli is with Diparmento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy andrea5.marnelli@mail.polimi.it 4 G. Ferrari-Trecate is with Automac ontrol Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland giancarlo.ferraritrecate@epfl.ch ESSes [8]. The latter pieces of informaon are not required in the design procedure of [9], whics therefore termed line-independent method. However, for the PnP methods menoned above, the synthesis of a PnP controller requires to solve a convex opmizaon problem, and if it is unfeasible, the plug-in/out of corresponding ESS must be denied. Moreover, they are only suited for grid-forming converters, as they provide only voltage support in the system. However, one complete MG must be composed of renewable energy source(reses), ESSes and loads to achieve the power generaon, storage and consumpon. When RESes such as P sources are added in dc MGs, grid-feeding converters should be used as the interface to achieve current feeding for the system according to the reference given by e.g. maximum power point tracking algorithm []. Furthermore the current stabilizaon should be guaranteed. In [2], a current-based PI primary droop control is proposed considering the constant current load. Then, a current-based PnP controller is proposed for grid-feeding converters [3]. The joint ulizaon of grid-feeding and gridforming converters has been considered for studying energy management problems [4], [5], [6]. However, from the stability point of view, if the grid-forming and grid-feeding converters are considered simultaneously, the controller and its stability should be redesigned. To authors knowledge, the stabilizaon of MG clusters including both types of converters has never been considered before, independently of the number of MGs and the topologies. In this paper, considering a MG cluster, a PnP voltage/current controller is proposed to achieve voltage support and current feeding funcons simultaneously. The set of coefficients of each local controller is explicitly characterized through a set of inequalies which only depends on the local parameters. In parcular, we show that controller design is always feasible and does not require to solve an opmizaon problem. Global closed-loop stability is formally proven. The paper is structured as follows. In Secon II and III-A, the model of a MG cluster and the proposed controller are introduced. In Secon III-B, the global closed-loop stability is proven. Finally, the HiL tests are shown in Secon I. II. D MGS WITH GRID-FORMING/FEEDING A. Electrical model of MGs ONERTERS A MG composed of one grid-forming converter and one grid-feeding converter connected to a single point of common /$3. 28 AA 5362

3 coupling (P) bus is considered achieving both voltage support and current feeding. A MG cluster is obtained by interconnecng N MGs, induced by the set D = {,..., N}. Two MGs are neighbors if there is a power line connecng them and denote with N i D the subset of neighbors of MG i. The neighboring relaon is symmetric which means j N i implies i N j. Furthermore, let E = {(i, j) : i D, j N i } collect unordered pairs of indices associated to lines. The topology of the MG cluster is then described by the undirected graph G el with nodes D and edges E. The electrical scheme of the i-th MG is illustrated in left block of Fig.. MG i : d i dt = I + I + di dt di dt (I Li + i ) R Li j N i = L i R L I + L = L i R L I + L ( j R ij i R ij variables i, I, I are the i-th P voltage, filter current from RES and filter current from ESS, respecvely. For the grid-feeding converter, represents the command and R, L are the electrical parameters. For the gridforming converter, represents the command and R, L are the electrical parameters; is the capacitor at the P bus. Moreover, j is the voltage at the P of each neighboring MGs j N i and R ij and L ij are the resistance and inductance of the dc power line connecng MGs i and j. In general, the RL parameters are different for grid-feeding and grid-forming converters. Microgrid i ESS ESS RES Microgrid i with Grid-Forming/Feeding onverters RES I I R R L L i P Bus RLi Iij ILi k k k, k, 2, 3, k k3, i i i i i 2, i Rij Line ij i I s I s Lij oltage/urrent-based PnP ontroller i Fig. : Electrical Scheme of MG i with PnP oltage/urrent ontroller. B. State-space model of MG lusters Dynamics () provides the state-space model: ẋ (t) = A iix (t) + B iu (t) + M id (t) Σ MG : + ξ (t) + A load,i x (t) z (t) = H ix (t) x = [ i, I, I ]T is the state of the system, u = [, ] is the control input, d = I Li is the exogenous input and z = [Ii, i] is the controlled variable. The term ξ = j N i A ij (x [j] x ) accounts for the coupling with each MG j N i. The matrices of Σ MG are obtained from (). The details can be found in [7]. ) Icap,i j Pri, pu I ref, i Pri ref, i () I i The overall model for a MG cluster is given by ẋ(t) = Ax(t) + Bu(t) + Md(t) z(t) = Hx(t) x = (x [],..., x [N] ) R 3N, u = (u [],..., u [N] ) R 2N, d = (d [],..., d [N] ) R N, z = (z [],..., z [N] ) R 2N. Matrices A, B, M and H are reported in Appendix A in [7]. III. DESIGN OF STABILIZING OLTAGE/URRENT ONTROLLERS A. Structure of PnP oltage/urrent controllers In order to track constant references z ref (t), when d(t) is constant as well, the MG model is augmented with integrators [8]. A necessary condion for making error e(t) = z ref (t) z(t) equal to zero as t, is that, for arbitrary d and z ref, there are equilibrium states and inputs x and ū verifying (2). The existence of these equilibrium points can be shown as in the proof of Proposion in [8]. Let I cap,i > define the maximal output current capability provided by MG i. According to the block on the bottom right of Fig., the dynamics of integrators are given by v (t) = e (t) = z P ri, ref (t) I (t) (3a) v (t) = e (t) = z P ri, ref (t) i(t) z P ri, ref = I P ri,pu ref,i I cap,i, z P ri, ref Hence, the augmented MG model is ˆΣ MG : = P ri ref,i, ˆx (t) = Âiiˆx (t) + ˆB iu (t) + ˆM i ˆd (t) + ˆξ (t) + Âload,iˆx (t) z (t) = Ĥiˆx (t) (2) (3b) ˆx = [ i, I, v i, I, v i ]T R 5 is the state, ˆd = P ri, [d, z ref (4), z ref P ri, ] T R 3 collects the exogenous signals and ˆξ = j N i  ij (ˆx [j] ˆx ). Matrices in (4) can be established by combining () with (3). Based on Proposion 2 in [8], it can be proven that the pair (Âii, ˆB i ) is controllable. Hence, system (4) can be stabilized. The overall augmented system is obtained from (4) as { ˆx(t) = ˆx(t) + ˆBu(t) + ˆMˆd(t) (5) z(t) = Ĥˆx(t) ˆx and ˆd collect variables ˆx and ˆd respecvely, and matrices Â, ˆB, ˆM and Ĥ are obtained from systems (4). MG Each MG ˆΣ is with the following state-feedback controller MG : u (t) = K iˆx (t) (6) [ ] k K i =,i k2,i k3,i k,i k2,i k3,i R 2 5. Nong that the control variables and are coupled through the coefficients k,i and k,i appearing in the first column of K i. In parcular, the overall control architecture is decentralized since the computaon of u requires the state of only. ˆΣ MG 5363

4 B. ondions for stability of the closed-loop MG luster For showing stability, we will use local Lyapunov funcons i (ˆx ) = [ˆx ] T P iˆx. (7) Assumpon. The posive definite matrix P i R 5 5 in (7) fulfills η i 2 2 P i = 2 P22,i 2 2, (8) P44,i [ P22,i p = 22,i p 23,i p 23,i p 33,i ] [, P44,i p = 44,i p 45,i p 54,i p 55,i ]. (9) And η i > is a local parameter sasfying η i = σ, i D σ > is a constant parameter, common to all MGs. In absence of coupling terms ˆξ (t), and load terms  load,iˆx (t), we would like to guarantee asymptoc stability of the nominal closed-loop MG ˆx (t) = (Âii + ˆB i K i ) }{{} ˆx (t) + ˆM i ˆd (t). () F i By direct calculaon, one can show that F i has the following structure f 2,i f 4,i f 2,i f 22,i f 23,i F i = f 32,i f 4,i f 44,i f 45,i f 5,i (k,i ) (k2,i R ) k 3,i L L L = () (k,i ) (k2,i R ) k3,i L L L F2,i F4,i = F2,i F22,i. F4,i F44,i From Lyapunov theory, asymptoc stability of () can be cerfied by the existence of a Lyapunov funcon i (ˆx ) = [ˆx ] T P iˆx P i = Pi T > and Q i = F T i P i + P i F i (2) is negave definite. In presence of nonzero coupling terms, we will show that asymptoc stability can be achieved under Assumpon. Based on (8) and (), (2) can be rewritten as (3) shown in the upper part of the next page. Next, we provide two lemmas and two proposions by which we can analyze the closed-loop system and parametrize explicitly the set of stabilizing controllers. Lemma. Under Assumpon, if Q i, Q i R 5 5 has the following structure 2 2 Q i = 2 Q 22,i 2 2 (4) Q 44,i Furthermore, the blocks on the diagonal must verify { Q 22,i Q 44,i Proof. See the proof of Lemma 2 in [7]. (5a) (5b) Remark. Since the blocks Q 22,i and Q 44,i belong to R 2 2, from (5), the determinants of Q 22,i and Q 44,i are nonnegave. Proposion. [9] If Q = Q T and an element q ii on the diagonal are verified q ii =, then (i) The matrix Q cannot be negave definite. (ii) The i-th row and column of Q have zero entries. Proposion 2. Under Assumpon, matrices P i and Q i have the structure given in (6) (displayed in the upper part of next page), = L k 3,i (k,i )(k 2,i R ). Moreover, if P i >, Q i and Q i, one has k,i < k,i < k2,i < R, k3,i > k 2,i < R < k3,i < (k L,i )(k2,i R) (7) Proof. See proof of Proposion 4 in [7]. Lemma 2. Let Assumpons and Proposion 2 hold, define (v i ) = v T i Q 44,i v i, with v i R 2. If Q i, and Q i, then ( v i ) = v i Ker(F 44,i). Proof. See the proof of Proposion 3 in [9]. Proposion 3. Let g i (w i ) = w T i Q iw i, i D with w i R 5. Under Assumpon, Proposion 2, and Lemma 2, only vectors w i in the form w i = [ α i γ i β i δ i β i ] T with α i, γ i, β i R, and δ i = k 2,i R, fulfill k3,i g i ( w i ) = w T i Q i w i =. (8) Proof. See proof of Proposion 5 in [7]. onsider the overall closed-loop MG cluster model { ˆx(t) = ( + ˆBK)ˆx(t) + ˆMˆd(t) z(t) = Ĥˆx(t) (9) 5364

5 Q i = [F 2,i ]T P 22,i + η if 2,i [F 4,i ]T P 44,i + η if 4,i [F 2,i ]T η i + P 22,i F 2,i [F 22,i ]T P 22,i + P 22,i F 22,i 2 2 = Q 2,i Q 4,i [Q 2,i ]T Q 22,i 2 2 (3) [F 4,i ]T η i + P 44,i F 4,i 2 2 [F 44,i ]T P 44,i + P 44,i F 44,i [Q 4,i ]T 2 2 Q 44,i P i = η i p 22,i k 3,i L p 22,i L (k 2,i R ) L k 3,i L k 3,i k 3,i (k,i ), Q i = 2 (k 2,i R ) L p 22,i 2 (k 2,i R )2 2 (k 2,i R )k 3,i 2 (k 2,i R )k 3,i 2 (k 3,i )2 (6) obtained by combining (5) and (6), with K = diag(k,..., K N ). onsidering also the collecve Lyapunov funcon N (ˆx) = i (ˆx ) = ˆx T Pˆx (2) i= P = diag(p,..., P N ). One has (ˆx) = ˆx T Qˆx Q = ( + ˆBK) T P + P( + ˆBK). (2) A consequence of Proposion is that, under Assumpon, the matrix Q cannot be negave definite. At most, one has Q. (22) Moreover, even if Q i holds for all i D, the inequality (22) might be violated because of the nonzero coupling terms  ij and load terms Âload,i in matrix Â. The next result shows that this cannot happen. In order to derive that Q i, i D implies (22), the following decomposion of matrix  is considered  =  D + Â Ξ +  L + Â, (23)  D = diag(âii,..., ÂNN ) collects the local dynamics only,  collects the coupling dynamic represenng the off-diagonal items of matrix Â. Moreover, matrices Â Ξ = diag(âξ,..., ÂξN) and  L = diag(âload,,..., Âload,N),  ξi = R ij 4 j N i, Âload,i = R Li 4, take into account the dependence of each local state on the neighboring MGs and the local resisve load respecvely. According to the decomposion (23), the inequality (22) is equivalent to ( D + ˆBK) T P + P( D + ˆBK) + 2(Â Ξ +  L )P +  T }{{}}{{} P + P }{{} (a) (b) (c) (24) Proposion 4. If gains K i are chosen according to (7), then Q i for all i D, finally (22) holds. Proof. See proof of Proposion 6 in [7]. Theorem. If Assumpons is fulfilled, the graph G el is connected, control coefficients are chosen according to (7), then the origin of (5) is asymptocally stable. Proof. See proof of Theorem 2 in [7]. Remark 2. The design of stabilizing controller for each MG can be conducted according to Proposion 2. In parcular, differently from the approacn [9], no opmizaon problem has to be solved for compung a local controller. Indeed, it is enough to choose control coefficient k,i, k 2,i, k 3,i and k,i, k2,i, k 3,i fulfilling the inequalies in (7). Note that these inequalies are always feasible, implying that a stabilizing controller always exists. Moreover, the inequalies depend only on the parameters R and R of the MG i. Therefore, the control synthesis is independent of parameters of MGs and power lines, which means that controller design can be executed only once for each converter in a plug-and play fashion. I. HARDWARE-IN-LOOP TEST In order to verify the correctness of theorecal results, real-me HiL tests have been carried out using dspae 6 platform. The real-me test model comprises four MGs with meshed electrical topology shown in Fig. 2. Meanwhile, the electrical setup informaon are shown in TABLE I and the transmission lines parameters are shown in TABLE II and the control coefficients are shown in TABLE III A. ase : PnP Test In this subsecon, the effecveness of the proposed primary PnP controller is verified. Each MG is started separately. At the beginning, we set different voltage and current references for different MGs. The results are shown in Fig. 3. At t = T, MGs 3 are connected together without 5365

6 TABLE I: Electrical setup parameters Parameter Symbol alue D source voltage - Nominal voltage ref,i P ri Output capacitance t 2.2 mf Inductance for DGU L t.8 H Inductor + switch loss resistance for DGU Rt.2 Ω Inductance for DGU L t.8 H Inductor + switch loss resistance for DGU Rt. Ω Switching frequency f sw khz TABLE II: Transmission lines parameters onnected MGs (i, j) Resistance R i,j (Ω) Inductance L i,j (mh) (, 2).3.8 (2, 3) (3, 4) (4, ) A to 3.5A. At t = T 4, the current reference for MG 3 is changed from 3A to.5a. At t = T 5, the current reference for MG 4 is changed from 4A to 5.5A. As shown in Fig. 4b, whether the current references are increased or decreased, the output currents can track the changed reference. In addion, as shown in Fig. 4a, when the current references are changed, the output voltages are only affected by little oscillaons approximately.5. Output oltage () MG MG 2 T T2 T3 T4 T5 Output urrent from RES (A) MG MG 2 T T2 T3 T4 T5.6 RES RES L,2 R,2 MG MG 2 R4, L2, (a) oltage Tracking Performance (b) urrent Tracking Performance. ESS ESS L4, R2,3 MG 4 MG 3 Fig. 4: oltage and urrent Tracking Performance of PnP decentralized controllers Fig. 2: System onfiguraon of Hardware-in-Loop Test. changing the control coefficients. As shown in Fig. 3a, after the connecon of MGs 3, only small disturbances appear in the voltage waveform. Moreover, there is no major variaon affecng the output currents as shown in Fig. 3b. Then at t = T 2, MG 4 is connected to the system. Similarly, as shown in Fig. 3, after small disturbance, both the output voltage and current track the respecve reference values. Fig. R3,4 L3,4 B. ase 2: Instability Test The correctness and accuracy of coefficient sets in (7) derived from Proposion 2 is verified. According to the system parameters shown in TABLE I, for each MG, the controller coefficients must verify k,i < k,i < k2,i <.2, k2,i <. k3,i > < k3,i < (k L,i )(k2,i R) (25) MG MG 2 T T2 T MG MG 2 T T2 T3 Based on the (25), the value of control coefficients are chosen as shown in Table III. At the beginning, MGs are Output oltage () (a) oltage Performance for PnP Test. Output urrent from RES (A) (b) urrent Performance for PnP Test. Fig. 3: The Plug-in/-out Performance of primary PnP controllers. 4 illustrates the current tracking performance by changing the current references for different MGs. At t = T, four MGs are connected together simultaneously. At t = T 2, the current reference for MG is changed from A to 2.5A. At t = T 3, the current reference for MG 2 is changed from TABLE III: ontrol oefficients for MG i =, 2, 3, 4 ontrol oefficients Symbol alue For grid-feeding converters For grid-forming converters k,i -. k2,i k3,i 4.48 k,i -. k2,i -.8 k3,i operated separately and at t =.5s MGs are connected together to form the MG cluster. Then at t = s, the control coefficients are changed in order to violate (25). Fig. 5 including six sub-figures illustrates the system performance when each control coefficient is changed from the stable region to the instable region. Fig. 5a to 5c show that when control coefficients for the grid-forming converter in MG 5366

7 2 go sightly out of the stable region, the system becomes unstable. Fig. 5d to 5f show that the same happens when control coefficients for the grid-feeding converter in MG 2 slightly violate the inequality. Output oltage () MG MG (a) k,i changed from. to.. Output oltage () MG MG (c) k 3,i changed from 3.67 to 8. Output oltage () MG MG (e) k 2,i changed from 2.7 to.3. Output oltage () MG MG (b) k 2,i changed from.8 to.. Output oltage () MG MG (d) k,i changed from. to.. Output oltage () MG MG (f) k3,i changed from 4.4 to. Fig. 5: oltage Profile of Instability Test. ONLUSION In this paper, a PnP oltage/urrent controller for D microgrid clusters is proposed. By choosing the control coefficients according to inequalies which are only related to local parameters, the closed-loop stability can be guaranteed for the MG cluster. Under the proposed control structure, each MG can plug in and out without changing the control coefficients and without knowing the electrical topology of the MG cluster. As in [9], the proofs of closed-loop asymptoc stability exploit structured Lyapunov funcons, the LaSalle invariance theorem and properes of graph Laplacians. Thus it shows that these tools offer a feasible theorecal framework for analyzing different kinds of MGs equipped with various types PnP decentralized control architectures. Finally, HiL tests prove the effecveness and accuracy of the theorecal analysis results. REFERENES [] R. Han, L. Meng, G. Ferrari-Trecate, E. A. A. oelho, J.. asquez, and J. M. Guerrero, ontainment and consensus-based distributed coordinaon control to achieve bounded voltage and precise reacve power sharing in islanded ac microgrids, IEEE Transacons on Industry Applicaons, vol. 53, no. 6, pp , Nov 27. [2] J. M. Guerrero, J.. asquez, J. Matas, D. icuna, L. 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