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On the Robustness of a Passivity–based Controller for Microgrids*
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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the Robustness of a Passivity–based the Robustness of a theController Robustness ofMicrogrids a Passivity–based Passivity–based for Controller for Microgrids Controller for Microgrids ∗ ∗
Sof´ıa Avila-Becerril ∗ Diego Silva-Mart´ınez ∗ ∗ Diego Silva-Mart´ ∗ Sof´ ıa Avila-Becerril ınez Sof´ ınez ∗ Diego Silva-Mart´ Gerardo Espinosa-P´ erez ∗∗ Sof´ıa ıa Avila-Becerril Avila-Becerril Diego Silva-Mart´ ınez ∗ ∗ Gerardo Espinosa-P´ e rez Gerardo e Gerardo Espinosa-P´ Espinosa-P´ erez rez ∗ ∗ Facultad de Ingenier´ ıa, Universidad Nacional Aut´ onoma de M´exico, ∗ ∗ de Ingenier´ ıa, Universidad Nacional Aut´ o noma de M´ exico, Facultad de Ingenier´ ıa, Universidad Nacional Aut´ o noma ∗ Facultad Ciudad Universitaria, Ciudad de M´ e xico Facultad Ciudad de Ingenier´ ıa, Universidad Nacional Aut´ o(e-mail: noma de de M´ M´eexico, xico, Universitaria, Ciudad de M´ e xico (e-mail: Ciudad Universitaria, Ciudad de M´ e xico (e-mail: [email protected]; [email protected]; Ciudad Universitaria, Ciudad de M´exico (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). [email protected]; [email protected]; [email protected]). [email protected]). [email protected]). Abstract: Motivated by the growing interest on microgrids, in this paper, it is presented a Abstract: Motivated the growing interest on microgrids, in this reported paper, itpassivity-based is presented aa Abstract: Motivated by the interest on in paper, is numerical evaluation ofby some robustness properties of a previously Abstract: Motivatedof bysome the growing growing interest on microgrids, microgrids, in this this reported paper, it itpassivity-based is presented presented a numerical evaluation robustness properties of a previously numerical some robustness properties of aa previously reported passivity-based control law.evaluation The mainof feature of the approached controller is that, unlike commonly reported numerical evaluation of some robustness properties of previously reported passivity-based control law. The The dynamics main feature feature of the the approached controller is that, that, unlike commonly reported control law. main of approached is unlike commonly in the literature, of the power converterscontroller are included. Hence, a local controlreported for the control law. The main feature of the approached controller is that, unlike commonly reported in the literature, of the converters are included. Hence, aa local control for the in the literature, dynamics of the power converters are included. Hence, local control for the power converters dynamics is considered andpower complemented with a secondary control of the well–known in the literature, dynamics of the power converters are included. Hence, a local control for the power converters is considered and complemented with a secondary control of the well–known power converters is considered and complemented with a secondary control of the well–known droop type, where instead of measuring all the buses powers, as indicated in the original version power converters is considered and complemented with a secondary control of the well–known droop where instead ofameasuring all the buses powers, as indicated in the the original version droop type, where instead all powers, as in original version of the type, microgrid controller, state–estimation scheme is included to generate information droop type, wherecontroller, instead of ofameasuring measuring all the the buses buses powers, as indicated indicated in the the the original version of the microgrid state–estimation scheme is included to generate information of the microgrid controller, a state–estimation scheme is included to generate the information required by the former. Thea estimator allowed scheme to incorporate measurement noise and showed of the microgrid controller, state–estimation is included to generate the information required by the former. The estimator allowed to incorporate measurement noise and showed required the estimator allowed incorporate measurement the good by performance ofThe the controller in a moreto realistic scenario. required by the former. former.of The estimator in allowed torealistic incorporate measurement noise noise and and showed showed the good performance the controller aa more scenario. the good performance of the controller in more realistic scenario. the good performance of the controller in a more realistic scenario. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Microgrids, Passivity Based Control, Control Droop. Keywords: Keywords: Microgrids, Microgrids, Passivity Passivity Based Based Control, Control, Control Control Droop. Droop. Keywords: Microgrids, Passivity Based Control, Control Droop. 1. INTRODUCTION In spite of the remarkable advances achieved in under1. In spite of remarkable advances achieved in 1. INTRODUCTION INTRODUCTION In spite the remarkable advances achieved understanding thethe stability properties of Microgrids, theundermain 1. INTRODUCTION In spite of of the remarkable advances achieved in in understanding the stability properties of Microgrids, the standing the stability properties of Microgrids, the main drawback of the reported results lies in the fact that the Microgrids are part of a new generation of electrical net- standing the stability properties of Microgrids, the main main drawback of the reported results lies in the fact that the Microgrids are part of aa new generation of electrical netdrawback of the reported results lies in the fact that the Microgrids are part of new generation of electrical netdynamics of the power converters are over simplified; this works that allow the integration of renewable energies and drawback of the the power reported results lies in thesimplified; fact that this the Microgrids are part of a new generation of electrical netdynamics of converters are over works that allow the integration of renewable energies and dynamics of the power converters are over simplified; this works that allow the integration of renewable energies and assumption makes possible the application of the droop that have changed the studies of Electrical Power Systems dynamics of the power converters are over simplified; this works that allow the integration of renewable energies and makes possible application of the droop that have changed the studies Systems assumption makes possible the application of droop that have changed to theFang studies of Electrical Power Systems scheme. In this sense, even the if the use of time scales to (EPS). According et of al.Electrical (2012), a Power microgrid is a assumption makes possible the application of the the droop that have changed the studies of Electrical Systems scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), aa Power microgrid is aa assumption scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), microgrid is a certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), a microgrid is a a certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such aa certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such converters, this situation does not seem natural since this as solar panels and small wind turbines) interconnected via certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such converters, this situation doesofnot not seem natural natural since this this as solar panels small turbines) via this does seem since as solarconverters. panels and and These small wind wind turbines) interconnected via converters, model limits thesituation possibility including phenomena expower microgrids caninterconnected operate isolated converters, this situation doesofnot seem natural since this as solar panels and small wind turbines) interconnected via model limits the possibility including phenomena expower converters. These microgrids can operate isolated model limits the possibility of including phenomena expower converters. These microgrids can operate isolated hibited by the microgrids related to the power quality and from or interconnected to the main network, addressing model limits the possibility of including phenomena expower converters. Thesetomicrogrids can operate isolated hibited by the microgrids related to the power quality and from or interconnected the main network, addressing hibited by the microgrids related to the power quality and from or interconnected to the main network, addressing disturbances during the operation of the equipment based stability problems while sharing the power demand behibited by the microgrids related to the power quality and from or interconnected to the main network, addressing during the operation of the equipment based stability while sharing the the power demand demand be- disturbances disturbances during stability problems while sharing power electronics. tween theproblems generation units. disturbances during the the operation operation of of the the equipment equipment based based stability problems while sharing the power power demand bebe- on on power electronics. tween the generation units. on power electronics. tween the generation units. on power electronics. tween units. based structure, control of On the other hand, exploiting the structural properties of Due tothe itsgeneration power electronics the other hand, exploiting the structural Due to power electronics structure, control of On the hand, exploiting the properties of Due to its its is power electronics based structure, control of On electrical networks in Avila-Becerril et properties al. (2016),of a microgrids accomplished in based two levels, namely, a local On the other other hand,stated exploiting the structural structural properties of Due to its power electronics based structure, control of electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in two levels, namely, a local electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in two levels, namely, a local control scheme for Microgrids has been reported in Avilacontrol is required to command the voltage supplied by electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in twothe levels, namely, a local control scheme for Microgrids has been reported in Avilacontrol is required to command voltage supplied by control scheme for Microgrids has been reported in Avilacontrol is required to command the voltage supplied by Becerril and Espinosa-P´ e rez (2016) that solves a voltage these devices whiletothe power demanded bysupplied the loads control scheme for Microgrids has been reported in Avilacontrol is required command the voltage by and Espinosa-P´ eerez (2016) that a voltage these devices while the power by loads Becerril and (2016) that solves a these devices while by theimplementing power demanded demanded by the the droop loads Becerril stabilization problem. In this paper, it solves is showed that is usually achieved the so-called Becerril and Espinosa-P´ Espinosa-P´ erez rez (2016) that solves a voltage voltage these devices while the power demanded by the loads stabilization problem. In this paper, it is showed that is usually achieved by implementing the so-called droop stabilization problem. In this paper, it is showed is usually achieved by implementing the so-called droop the microgrid can be decomposed as the interconnection control (see Barklund et al. (2008); Pedrasa and Spooner problem. In this paper, it is showed that that is usually achieved byetimplementing the so-called droop stabilization the microgrid can be decomposed as the interconnection control (see Barklund al. (2008); Pedrasa and Spooner the microgrid can be decomposed as the interconnection control (seereferences Barklund therein). et al. al. (2008); (2008); Pedrasa Pedrasa and and Spooner Spooner the of amicrogrid main Port-Controlled Hamiltonian (PCH) system (2006) and can be decomposed as the interconnection control (see Barklund et of a main Port-Controlled Hamiltonian (PCH) system (2006) of aa main Port-Controlled Hamiltonian system (2006) and and references references therein). therein). equipped sources and loads ports to (PCH) which another mainwith Port-Controlled Hamiltonian (PCH) system (2006) and references therein). Although the effectiveness of the aforementioned control of equipped with sources and loads ports to which another equipped with sources and loads ports to which another systems, of the same kind, are attached. The model exAlthough the effectiveness of the aforementioned control equipped with sources and loads ports to which another Although the effectiveness of the the aforementioned controla systems, strategy isthe widely recognized in practice, there exists of the same kind, are attached. The model exsystems, of the same kind, are attached. The model exAlthough effectiveness of aforementioned control plicitly includes the dynamic of the power converters and strategy is widely practice, there aa systems, of the same kind, are attached. The model and exstrategy is understanding widely recognized recognized intheoretical practice, stability there exists exists lack in the of itsin propplicitly includes the dynamic of the power converters plicitly includes the dynamic of the power converters and strategy is widely recognized in practice, there exists a a Passivity–based Control (PBC) is presented to guarantee lack in the understanding of its theoretical stability propplicitly includes the dynamic of the power converters and lack in the understanding of its theoretical stability properties. Some prominent results explaining these feature are a Passivity–based Control (PBC) is presented to guarantee a Passivity–based Control (PBC) is guarantee lack inSome the understanding of its theoretical stability prop- that a prescribed output for theto conerties. prominent these feature Passivity–based Controlvoltage (PBC) profile is presented presented topower guarantee erties. Some prominent results results explaining these feature are given by Simpson-Porco et al.explaining (2013) and Schiffer et are al. athat aa prescribed output voltage profile for the power conthat prescribed output voltage profile for the power erties. Some prominent results explaining these feature are verters is achieved. This reference voltage, corresponding given by Simpson-Porco et al. (2013) and Schiffer et al. that a prescribed output voltage profile for the power concongiven by Simpson-Porco et al. (2013) and Schiffer et al. (2014). In the former, the authors show that, under some verters is achieved. This reference voltage, corresponding verters is achieved. This reference corresponding given byInSimpson-Porco etauthors al. (2013) and Schiffer et al. to a desired steady–state behavior,voltage, is computed based on (2014). the former, the show that, under some verters is achieved. This reference voltage, corresponding (2014). In the former, the authors show that, under some assumptions, the microgrid equipped withthat, a droop control, aa desired behavior, is computed based on to steady–state behavior, is based on (2014). In thethe former, the authors show under some to a given loadsteady–state power demand, by solving the power assumptions, microgrid with aa droop a desired desired steady–state behavior, is computed computed basedflow on assumptions, the as microgrid equipped with droop control, control, can be depicted a modelequipped of coupled oscillators stating to aequations given load power demand, by solving the power flow a given load power demand, by solving the power flow assumptions, the microgrid equipped with a droop control, of the network; hence, measurement of all the can be depicted as a model of coupled oscillators stating a given load power demand, by solving the power flow can bestability depicted as aa model model of coupled oscillators stating somebe properties. In of thecoupled latter, it is assumed that equations of the(voltage, network; hence, of all the equations of network; hence, measurement of can depicted as oscillators stating buses variables andmeasurement powers) is required. some stability properties. In the latter, it is assumed that equations of the the(voltage, network;phase hence, measurement of all all the the some stability properties. In the latter, it is assumed that each bus in the microgrid has a power converter connected buses variables phase and powers) is required. buses variables (voltage, phase and powers) is required. some stability properties. In the latter, it is assumed that each bus in the microgrid has aa power converter connected buses variables (voltage, phase and powers) isevaluate required.the each bus in the microgrid has power converter connected The purpose of this paper is to numerically whose frequency and voltage are controlled independently each bus in the microgrid has are a power converter connected The purpose of this paper is to numerically evaluate the whose and independently The purpose of is to evaluate the whose frequency and voltage voltage are controlled controlled independently properties of PBC of Avila-Becerril of eachfrequency other by means of a droop control, hence a stability robustness The purpose of this this paper paper is scheme to numerically numerically evaluate and the whose frequency and voltage are controlled independently robustness properties ofinPBC PBC scheme of Avila-Becerril Avila-Becerril and of each other by means of aa droop control, hence aa stability robustness properties of scheme of and of each other by means of droop control, hence stability Espinosa-P´ e rez (2016) two directions, namely: analysis in terms of consensus is developed. robustness properties of PBC scheme of Avila-Becerril and of each other by means of a droop control, hence a stability Espinosa-P´ e rez (2016) in two directions, namely: analysis Espinosa-P´ analysis in in terms terms of of consensus consensus is is developed. developed. Espinosa-P´eerez rez (2016) (2016) in in two two directions, directions, namely: namely: analysis of consensus is developed. Part of in thisterms work was supported by DGAPA-UNAM under grant
of Part IN116516. of this this work work was was supported supported by by DGAPA-UNAM DGAPA-UNAM under under grant grant Part Part of this work was supported by DGAPA-UNAM under grant IN116516. IN116516. IN116516. Copyright 6847Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 6847 Copyright © under 2017 IFAC IFAC 6847Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 6847 10.1016/j.ifacol.2017.08.1531
Proceedings of the 20th IFAC World Congress Sofía Avila-Becerril et al. / IFAC PapersOnLine 50-1 (2017) 6648–6653 Toulouse, France, July 9-14, 2017
• The computation of the output voltage of the power converters is obtained by implementing a droop control instead of solving the power flow equations of the network. • It is assumed that only part of the buses variables are available for measurement and that these measurements are contaminated with noise. To recover the real variables, a static state estimation is included in the control scheme. This algorithm is based in the well–known Weighted Least Squares technique. The robustness of the control scheme is evaluated through numerical simulations showing that converge to the desired behavior is achieved in spite of the existence of noise measurement, network model inaccuracies, as well as errors introduced in various stages of the communication path Abur (2015). The rest of the paper is organized as follows: For clarity, in Section 2 it is recovered the microgrid structure considered by Avila-Becerril and Espinosa-P´erez (2016) together with the controller. Section 3 is devoted to explain the implementation of the control law with the state estimation scheme while in Section 4 the numerical evaluation is carried out. Finally, some concluding remarks are included in Section 5. 2. MICROGRID STRUCTURE In this paper it is considered a generic balanced, lossless microgrid with a meshed topology, operating under normal conditions, modeled by a single-phase equivalent circuit. On the one hand, each of the n1 generation units are composed by a constant voltage source Vi ∈ R > 0 modulated by a switching array via ui ∈ R to later on fed a second order LC filter, such that vCi ∈ R is the output capacitor voltage while ILi ∈ R is the port current delivered to the network. The set of n1 power converters can be represented in port Hamiltonian (pH) form with the definition of the stored energy function Hc : Rn1 × Rn1 → R≥0 given by 1 1 (1) Hc (x1 , x2 ) = xT1 L−1 x1 + xT2 C −1 x2 , 2 2 with the collection of the linkage inductor fluxes x1 ∈ Rn1 and x2 ∈ Rn1 the electrical capacitor charges, where it has been assumed a linear constitutive relationship for both the inductors and the capacitors, with L ∈ Rn1 ×n1 > 0 and C ∈ Rn1 ×n1 > 0 diagonal inductance and capacitance matrices. The dynamic behavior of the n1 power converters can be represented by the compact form 0 x˙ 12 = J12 ∇x12 Hc + G12 u − (2) IL T ∈ Rn1 , the matrices where x12 = xT1 xT2 0 −I V = −JT12 ∈ Rn1 ×n1 ; G12 = , J12 = I 0 0
u = col(ui ) ∈ Rn1 , and the inductor currents vector satisfies iL = ∇x1 Hc ∈ Rn1 , while the capacitor voltages vector is given by vC = ∇x2 Hc ∈ Rn1 . Remark 1. The structure considered for the power converters essentially captures the dynamic behavior of a large class of these devices since common converter topologies
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can be represented under a port Hamiltonian structure (see for example Noriega-Pineda et al. (2010)). On the other hand, the transmission lines are modeled by the Π model such that the network with the loads are viewed as an electrical circuit on a graph (for details see Avila-Becerril et al. (2016)). It is defined the energy function of the network Ha : Rn1 ×n3 → R≥0 as 1 1 (3) Ha (x3 , x4 ) = xT3 Ca−1 x3 + xT4 L−1 a x4 2 2 n1 where x3 ∈ R is the electrical capacitor charges vector and x4 ∈ Rn3 the linkage inductor fluxes vector, with La ∈ Rn3 ×n3 > 0 and Ca ∈ Rn1 ×n1 > 0 the diagonal inductance and capacitance matrices. Now, let the current at the i–th load in parallel with the i-th capacitor be −1 fRci = ψci (eRci ), with the load voltage eRci and ψci (·) a bijective function, then the model of the transmission lines can be represented by the pH system x˙ 34 = J34 ∇x34 Ha (x3 , x4 ) − F34 fRc + G34 e1 (4) T T T (n1 +n3 ) ∈ R , fRc = col (fRci ) ∈ where x34 = x3 x4 Rn1 ∂Ha (x3 , x4 ) I ∇x34 Ha (x3 , x4 ) = , F34 = , 0 ∂x34
0 0 −HCL T = −J ; G = 34 T T 34 H1L HCL 0 subject to the port constraints f1 = H1L fL , eRc = eC = ψc (fRc ), (5) n1 with e1 , f1 ∈ R the voltage and current of the sources, eC the capacitor voltages parallel connected with the loads; while HCL and H1L are submatrices of the fundamental loop matrix (see Bollob´as (1998)) and represents the interconnections between capacitors and inductors, and sources with inductors, respectively. Remark 2. Equation (5) shows the current that the transmission lines demand from each source. In the sequel it is assumed ideal sources in the sense that they can provide any amount of current, so that only steady state stability issues can be approached. J34 =
The Hamiltonian model of the microgrid is obtained from (2) and (4) under the ports choice given by ∂Ha (x34 ) , e1 = C −1 x2 , (6) IL = H1L ∂x4 while the currents at the loads as can be written ∂Ha (x34 ) −1 fRc = ψc (7) ∂x3 The previous models (2), (4), together with (6) and (7), leads to the model of the complete Microgrid x˙ = JT ∇x HT (x) − gRT Ψ(x34 ) + GT u (8) T T T T T (3n1 +n3 ) ∈R , where the with state x = x1 x2 x3 x4 total stored energy function are (9) HT (x) = Hc (x1 , x2 ) + Ha (x3 , x4 ) and the matrices of appropriate dimensions
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0 −I 0 0 0 0 0 −H1L I 0 0 0 T JT = 0 0 0 −HCL = −JT ; gRT = 0 I ; T T I 0 0 H1L HCL 0 Vu 0 0 ∂Ha (x34 ) . Ψ(x34 ) = GT u = −1 0 ψc ∂x3 0 Likewise, the admissible trajectories are the solutions of x˙ = JT ∇x HT (x ) − gRT Ψ(x34 ) + GT u , (10) with u the control input that generates x . Let the error variable be x ˜ = x − x ∈ R3n1 +n3 and assume that:
A.1 The state are available for measurement. A.2 The parameters C, L, Ca and La are known and positive. A.3 The voltage source V vector is strictly positive. A.4 The desired steady-state behavior x2 is a known bounded and differentiable function with bounded first derivative. A.5 The port–variables of each load satisfies the passivity condition T (eC − eC ) ψc−1 (eC ) − ψc−1 (eC ) > 0 for a given admissible trajectory eRc (t) ∈ Rn1 .
Under the assumptions above, the stability properties of model (8), in closed loop with the control law
u = V −1 [x˙ 1 + C −1 x2 − K1 L−1 x ˜1 ] n1 ×n1 and admissible with K1 = diag{K1i } ∈ R satisfying the constraints x˙ 2 = L−1 x1 − H1L L−1 a x4 , x˙ 3 = −HCL ∇x 4 HT (x ) − ϕ−1 c (eC )
(11) state (12a) (12b)
T T x˙ 4 = H1L ∇x 3 HT (x ) (12c) ∇x 2 HT (x ) + HCL have been studied in Avila-Becerril and Espinosa-P´erez (2016) and guarantees that lim x ˜ = 0. t→∞
The main problem when implementing the proposed controller is the definition of an admissible trajectory x ∈ R3n1 +n3 . In this sense, the achievement of a desired steady state behavior depends on (10). 3. SET POINT DESIGN In this section it is presented the main contribution of the paper, which refers to an improvement in the implementation of the controller above. In particular, it is proposed to incorporate an external control loop that allows to modify the desired trajectories in terms of the demanded power. Since this modification is not contemplated in the stability proof, it will be a good measure of the controller robustness, which will be evaluated numerically in Section 4. First, let us consider that the microgrid is formed by n = n1 + n3 nodes and that the i−th node has associated a voltage amplitude Vi and a phase angle δi . In turn, two nodes i and k of the microgrid are connected by means of a complex admittance denoted as Yik := Gik + jBik ∈ C, with conductance Gik ∈ R and susceptance Bik ∈ R. In addition, assume that:
A.6 The desired steady-state is sinusoidal. Under the last condition and since the network is lossless (Schiffer et al. (2014)), the total steady-state power flows (injected or consumed) Pi , Qi in the i–th node can be written as Pi = |Bik | Vi Vk sin(δik ), k∈Ni
Qi = |Bii | (Vi )2
k∈Ni
|Bik | Vi Vk cos(δik ).
(13)
where Ni is the set of neighbors of the i − th ˆii + Bik , with B ˆii ∈ node for which Yik = 0 and Bii B δik
δi − δk ,
k∈Ni
R the shunt susceptances in the node i. This power flow equations are static, model the network when the net sum of consumption, injections and dissipated power is zero, and determine the steady state operation of the network. It should be noted that this set of algebraic equations are non linear both in the voltage and in the angle, therefore their solution involves the use of a numerical method. In Avila-Becerril and Espinosa-P´erez (2016), the desired behavior is stated in terms of the power demanded by the loads. Hence, once this power is fixed, the corresponding voltage magnitud and phase angle of the nodes are found, to later on compute the entire value of x . In this sense, it is defined for each power converter the output voltage Ci−1 x2i = Vi sin(ωs t + δi ), (14) where ωs ∈ R takes the same value for all the power converters, while the magnitude Vi and the phase δi must be computed off–line to get an adequate power flow (13).
3.1 Droop Control The previous proposal has as a disadvantage namely, that the implementation of the controller lies in the necessity to compute off–line the admissible trajectories x . With the aim of solving this problems it is incorporated a secondary controller that computes the desired output voltage of the power converters by implementing a droop control instead of solving (13). Basically, the droop control (Kundur et al. (1994)) is a proportional control that, for inductive lines, balances the active and reactive power demands in the network by instantaneously changing the frequency ωi and the magnitud Vi of the voltage signal at the i–th power converter, so it allows to establish the power in a desired steady state value. In the microgrids literature, usually this control is implemented as ui1 = ωd − kpi (Pi − Pid ) , (15) ui2 = Vid − kqi (Qi − Qid ) , with ωd ∈ R and Vid ∈ R>0 the desired nominal frequency and the nominal voltage amplitude, kpi and kqi ∈ R>0 the controller gains and Pid , Qid the reference powers. The variables ui1 and ui2 are the control inputs of the i– th power converter so that to directly include the control law (15) in the model of the microgrid it is necessary to assume that the voltage and the phase are decoupled (see for example Schiffer et al. (2014); Simpson-Porco et al. (2013)).
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Nevertheless, since in this work the microgrid model (8) takes into account the dynamics of the converter, implementing the droop merely as in (15) is not possible. Thus, it is proposed to define the desired state x (t) to take into account power variations. That is, the voltage output of the converter has been chosen as Ci−1 x 2i = Vi sin(δ˙i t), (16) where δ˙i = ωd − kpi (Pi − Pid ) Vi = Vid − kqi (Qi − Qid ) with ωd the nominal frecuency. Note that if the power of the node is the required, then there will be no phase change. Once x 2 is calculated, the rest of the admissible trajectories x are obtained from equation (12). For this, it should be noted that equations (12b) and (12c) are of the form T x˙ 34 = f (x34 ) + H1L C −1 x 2 , so it is necessary to solve the differential equation for x 34 and then the variable x 1 can be algebraically obtained from equation (12a). Remark 3. It is important to note that the dynamic obtained for the rest of the desired trajectories imposes a new challenge in the stability analysis because now it is necessary to show that the admissible trajectories are bounded.
∂ J(xm ) ∂xm = H T (xm )W [z − h(xm )] = 0
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G(x) =
(17)
∂ ∂xm h(xm ).
In this paper, solution of (17) where H(xm ) = is obtained by the Newton-Raphson method. Remark 4. The values used for the weight matrix W in (17) associated with each measurement are given in the literature (see for example G´omez-Exp´osito et al. (2016); Crow (2009); Abur and Exposito (2004)); however, this values can also be found as reported by Allemong (1982). Proposition 1. Consider the microgrid model (8) under A1–A6. Additionally suppose that: B.3 The measurement of power in some nodes is known and that the measurement noise has Gaussian behavior. B.4 The voltage and phase in one bus is known, and are taken as reference. B.5 The transient of the closed loop system is fast enough to assume that the system is in steady state. Then, the control law (11) with the desired trajectories (16) and δ˙i = ωd − kpi Pˆi − Pid , (18) ˆ i − Qid . Vi = Vid − kqi Q
A disadvantage of the proposed set point design lies in the necessity to measure powers in all nodes. Under this scenario, the performance of the controller may be affected by noise measurements.
ˆ i the estimated power, guarantees that the with Pˆi and Q microgrid state converges to the admissible trajectories, lim x ˜=0
The introduction of the modifications mentioned above has led to the proposal of a state estimator that is presented in the next section.
3.2 Static State Estimation In this section it is incorporated a static state estimator (G´ omez-Exp´ osito et al. (2016)) which calculates the statistically optimal state of the network. The static state of the network xm is defined by the vector with voltage magnitude and phase of all buses, while the estimator (Abur (2015)) processes the measurements of the network model and converts them into the best state estimate, despite measurements are corrupted by noise, may be missing or grossly inaccurate. Thereby, from the power flow values of some nodes, with their respective uncertainty associated with the measurement, the voltage magnitude and phase are calculated using a classical Weighted Minimum Square error. Consider the nonlinear set of measurements z = h(xm ) + e where xm is the state vector, z is the measurement vector, h(xm ) is a nonlinear measurement function, that for this case corresponds to the power flow equations (13), and e is the measurement error. To calculate the estimates x ˆm it is minimized the objective function J(xm ) = [z − h(xm )]T W [z − h(xm )] with W = diag 1/σi2 and σi2 the variance error of the i-th instrument. So, by minimizing J(xm ), the estimates x ˆ are those values which satisfy
t→∞
Proposition above considers a more realistic scenario. On the other hand, assumption B.5 is related to the fact that the estimator is build on the assumption that the state variable is in steady state or quasi steady state. The next section presents its numerical evaluation. 4. NUMERICAL EVALUATION A scheme of the control implementation is shown in Figure 1, where, for simulation purposes, the power measurement were done by decoupling the magnitude and the phase of voltage vc = C −1 x2 at the converters. The estimators output is the voltage magnitude and phase of all nodes that the droop control uses to calculate the desired converters voltage. The proposed controller was evaluated in the network presented by Schiffer et al. (2014) and is the one shown in Figure 2, where it have been neglected the non-controllable sources (solar and wind) with a series impedance reduction. So, it is assumed that the batteries in nodes 1 and 5 are operated in charging mode and that the maximum power capacity for the i-th source is SiN . The base units in the calculations per unit are SB = 4.75M V A, VB = 20kV and ZB = 84.21Ω, while the generation units are modeled as sources of 1.0 pu as constant magnitude. For this simulation, the design parameters are those shown in the Table 1. Consider that each inverter is represented by (2) and its capacitance and inductance values have been set in 8 [µF ] y 1.5 [µH], respectively, while Vi = 1 pu (Pogaku et al. (2007)). In addition, the droop gains have
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V vc(t)
Power Converters
i(t)
Pm
Static State Estimation
Sensor Qm
|vc|
u
Power Flow
Controller Pd
Qd
Vd
d
|vcd|
P, Q
Droop Control
vcd= |vcd| sin ( dt + ) d
Fig. 1. Controller Implementation
0
V
Cd
[pu]
1
−1 0
0.1
0.2
0.3
Fuel cell
Batery
voltage error
0.06
Power converter
Fig. 2. Illustrative microgrid SBase = 4.75M V A, VBase = 20kV (0.505, 0.028, 0.261, 0.179, 0.168, 0.012)[pu] (-0.0173, 0.014, 0.131, 0.089, -0.067, 0.006)[pu] (-0.007, 0.00, 0.003, 0.002, 0.002, 0.00 [pu]) Table 1. Parameters
The second scenario evaluate the controller (11) when the state estimator has been incorporated under the specifi-
0.7
0.8
0
0.85
0.9
0.95
0.3
0.4
0.5 t [s]
0 0.1
0.2
0.6
0.7
0.8
0.9
1
Fig. 3. (a)Converters reference voltage in pu. (b) Tracking error 1 0.5
δ [rad]
0 0 0.04 0.02 0 −0.02 60.10
f [Hz]
The evaluation has been divided into two scenarios: the first, besides evaluating the performance of the proposed controller, aims to test its robustness against a change of 50% in the loads of nodes 1 and 5 at t = 0.7[s]. Accordingly, the first part of Figure 3 shows the reference value of voltage converters C −1 x2 ∈ R2 in per unit. In the second part of this figure, we show the tracking error between the actual voltage vC = C −1 x2 and the desired vCd = C −1 x2 , where it can be seen that the convergence is almost instantaneous, giving a solution to the tracking problem. Figure 4, presents the obtained voltage values at the six nodes, the frequency and the phase, with the aim of showing that they reaches synchronization in the first two. Figure 5 exhibits the active and reactive powers, it can be notice that, before the change of load, the power values coincide with those specified in Table 1 and that in t = 7s the loads have been increased 50% which causes an adjustment in the values of the powers of the rest of the nodes.
0.6
−2
0.02
−0.02 0
been fixed in Kp = diag{0.12, 0.1, 0.1, 0.6, 0.3, 0.5} and Kq = diag{0.15, 0.1, 1.3, 0.2, 1.5, 0.6}, for the six converters, while the desired frequency value ωrd is 60 [Hz]. The numerical evaluation of the controller was performed in SIMULINK of MATLAB, with variable integration step and Runge-Kutta numerical method. The gains of the controller (11) were defined as k1i = 4.5 and k2i = 500, and all initial conditions were set to zero.
0.5
2
0.04
V [pu]
Values SiN Pid Qdi
0.4
−5
x 10
60.05 60 59.95 0
1.002 1
1
0.5 0
0.01
0.03 0.02 0.01 0 −0.01 0
0.998 0.6
0.02
0.2
0.4
0.6
0.7
0.8
0.8
1
1.2
−4
x 10
0
0.01
0.02 −5 0.6
0.5
0.7
0.8
1
1.5
60.05 60
60 59.95 0
0.05
59.99
0.1
59.98 0.6
0.5
t [s]
0.7
1
0.8
1.5
Fig. 4. Voltage magnitud, phase and frequency cations given in Section 3, with 10% of active and reactive power measurement noise. The Newton Raphson algorithm converges in the fourth iteration with the vector xm initialized in xm = x0 , with the values of Vi = 1pu and θi = 0. In Figure 4 we present the active and reactive power estimated, which shows that the estimator converges to the actual values, that are those in Table 1, which guarantees the correct operation of the system despite of the noise. 5. CONCLUDING REMARKS In this paper we present some numerical robustness properties of a passivity based control for microgrids where, by exploiting the properties of the droop controller, it
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P [pu]
REFERENCES
P1
0.2
P
0.1
2
P
3
0
P4
−0.1
P5
−0.2
P
6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 Q
1
Q [pu]
0.05
Q2 Q3 Q
0
4
Q
5
Q6
−0.05 0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
1
Fig. 5. Active and reactive power at the nodes.
Fig. 6. (a)Actual active power (b) Actual reactive power 0.2
P
P [pu]
1
0.1
P2
0
P4
P
3
P5 P
−0.1
Q [pu]
−0.2 0 0.1
6
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1
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2
Q
5 6
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
0.8
0.9
6653
1
Fig. 7. Droop Output (Actual Active and Reactive Power) using the estimator
Abur, A. (2015). Observability and dynamic state estimation. In 2015 IEEE Power & Energy Society General Meeting, 1–5. IEEE. Abur, A. and Exposito, A.G. (2004). Power system state estimation: theory and implementation. CRC press. Allemong, J.J.; Radu, L.S.A. (1982). A fast and reliable state estimation algorithm for aep’s new control center. IEEE Transactions on Power Apparatus and Systems, PAS-101. doi:10.1109/tpas.1982.317159. Avila-Becerril, S. and Espinosa-P´erez (2016). A hamiltonian approach for stabilization of microgrids including power converters dynamic. Congreso Latinoamericano de Control Autom´ atico CLCA, 2016, 246–251. Avila-Becerril, S., Espinosa-P´erez, G., and Fernandez, P. (2016). Dynamic characterization of typical electrical circuits via structural properties. Mathematical Problems in Engineering, 2016. Barklund, E., Pogaku, N., Prodanovi´c, M., HernandezAramburo, C., and Green, T.C. (2008). Energy management in autonomous microgrid using stabilityconstrained droop control of inverters. Transactions on Power Electronics, IEEE, 23(5), 2346–2352. Bollob´as, B. (1998). Modern graph theory, volume 184. Springer Science & Business Media. Crow, M. (2009). Computational methods for electric power systems. Fang, X., Misra, S., Xue, G., and Yang, D. (2012). Smart grid—the new and improved power grid: A survey. Communications Surveys & Tutorials, IEEE, 14(4), 944–980. G´omez-Exp´osito, A., Conejo, A.J., and Ca˜ nizares, C. (2016). Electric energy systems: analysis and operation. CRC Press. Kundur, P., Balu, N., and Lauby, M. (1994). Power system stability and control, volume 7. McGraw-hill New York. Noriega-Pineda, D., Espinosa-P´erez, G., Cardenas, V., and Alvarez-Ram´ırez, J. (2010). Passivity-based control of multilevel cascade inverters: High-performance with reduced switching frequency. International Journal of Robust and Nonlinear Control, 20(9), 961–974. Pedrasa, M.A. and Spooner, T. (2006). A survey of techniques used to control microgrid generation and storage during island operation. In Proceedings of the 2006 Australasian Universities Power Engineering Conference (AUPEC’06), 1–6. Pogaku, N., Prodanovi´c, M., and Green, T.C. (2007). Modeling, analysis and testing of autonomous operation of an inverter-based microgrid. Power Electronics, IEEE Transactions on, 22(2), 613–625. Schiffer, J., Ortega, R., Astolfi, A., Raisch, J., and Sezi, T. (2014). Conditions for stability of droop-controlled inverter-based microgrids. Automatica, 50(10), 2457– 2469. Simpson-Porco, J.W., D¨ orfler, F., and Bullo, F. (2013). Synchronization and power sharing for droop-controlled inverters in islanded microgrids. Automatica, 49(9), 2603–2611.
allows, in addition to the stabilization of the microgrid, a dispatch of power flows. The controller is evaluated numerically for load disturbances. Also, a state estimator that allows inclusion of an error in the measurement, has been incorporated, making the scenario more realistic. 6852
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the Robustness of a Passivity–based the Robustness of a theController Robustness ofMicrogrids a Passivity–based Passivity–based for Controller for Microgrids Controller for Microgrids ∗ ∗
Sof´ıa Avila-Becerril ∗ Diego Silva-Mart´ınez ∗ ∗ Diego Silva-Mart´ ∗ Sof´ ıa Avila-Becerril ınez Sof´ ınez ∗ Diego Silva-Mart´ Gerardo Espinosa-P´ erez ∗∗ Sof´ıa ıa Avila-Becerril Avila-Becerril Diego Silva-Mart´ ınez ∗ ∗ Gerardo Espinosa-P´ e rez Gerardo e Gerardo Espinosa-P´ Espinosa-P´ erez rez ∗ ∗ Facultad de Ingenier´ ıa, Universidad Nacional Aut´ onoma de M´exico, ∗ ∗ de Ingenier´ ıa, Universidad Nacional Aut´ o noma de M´ exico, Facultad de Ingenier´ ıa, Universidad Nacional Aut´ o noma ∗ Facultad Ciudad Universitaria, Ciudad de M´ e xico Facultad Ciudad de Ingenier´ ıa, Universidad Nacional Aut´ o(e-mail: noma de de M´ M´eexico, xico, Universitaria, Ciudad de M´ e xico (e-mail: Ciudad Universitaria, Ciudad de M´ e xico (e-mail: [email protected]; [email protected]; Ciudad Universitaria, Ciudad de M´exico (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). [email protected]; [email protected]; [email protected]). [email protected]). [email protected]). Abstract: Motivated by the growing interest on microgrids, in this paper, it is presented a Abstract: Motivated the growing interest on microgrids, in this reported paper, itpassivity-based is presented aa Abstract: Motivated by the interest on in paper, is numerical evaluation ofby some robustness properties of a previously Abstract: Motivatedof bysome the growing growing interest on microgrids, microgrids, in this this reported paper, it itpassivity-based is presented presented a numerical evaluation robustness properties of a previously numerical some robustness properties of aa previously reported passivity-based control law.evaluation The mainof feature of the approached controller is that, unlike commonly reported numerical evaluation of some robustness properties of previously reported passivity-based control law. The The dynamics main feature feature of the the approached controller is that, that, unlike commonly reported control law. main of approached is unlike commonly in the literature, of the power converterscontroller are included. Hence, a local controlreported for the control law. The main feature of the approached controller is that, unlike commonly reported in the literature, of the converters are included. Hence, aa local control for the in the literature, dynamics of the power converters are included. Hence, local control for the power converters dynamics is considered andpower complemented with a secondary control of the well–known in the literature, dynamics of the power converters are included. Hence, a local control for the power converters is considered and complemented with a secondary control of the well–known power converters is considered and complemented with a secondary control of the well–known droop type, where instead of measuring all the buses powers, as indicated in the original version power converters is considered and complemented with a secondary control of the well–known droop where instead ofameasuring all the buses powers, as indicated in the the original version droop type, where instead all powers, as in original version of the type, microgrid controller, state–estimation scheme is included to generate information droop type, wherecontroller, instead of ofameasuring measuring all the the buses buses powers, as indicated indicated in the the the original version of the microgrid state–estimation scheme is included to generate information of the microgrid controller, a state–estimation scheme is included to generate the information required by the former. Thea estimator allowed scheme to incorporate measurement noise and showed of the microgrid controller, state–estimation is included to generate the information required by the former. The estimator allowed to incorporate measurement noise and showed required the estimator allowed incorporate measurement the good by performance ofThe the controller in a moreto realistic scenario. required by the former. former.of The estimator in allowed torealistic incorporate measurement noise noise and and showed showed the good performance the controller aa more scenario. the good performance of the controller in more realistic scenario. the good performance of the controller in a more realistic scenario. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Microgrids, Passivity Based Control, Control Droop. Keywords: Keywords: Microgrids, Microgrids, Passivity Passivity Based Based Control, Control, Control Control Droop. Droop. Keywords: Microgrids, Passivity Based Control, Control Droop. 1. INTRODUCTION In spite of the remarkable advances achieved in under1. In spite of remarkable advances achieved in 1. INTRODUCTION INTRODUCTION In spite the remarkable advances achieved understanding thethe stability properties of Microgrids, theundermain 1. INTRODUCTION In spite of of the remarkable advances achieved in in understanding the stability properties of Microgrids, the standing the stability properties of Microgrids, the main drawback of the reported results lies in the fact that the Microgrids are part of a new generation of electrical net- standing the stability properties of Microgrids, the main main drawback of the reported results lies in the fact that the Microgrids are part of aa new generation of electrical netdrawback of the reported results lies in the fact that the Microgrids are part of new generation of electrical netdynamics of the power converters are over simplified; this works that allow the integration of renewable energies and drawback of the the power reported results lies in thesimplified; fact that this the Microgrids are part of a new generation of electrical netdynamics of converters are over works that allow the integration of renewable energies and dynamics of the power converters are over simplified; this works that allow the integration of renewable energies and assumption makes possible the application of the droop that have changed the studies of Electrical Power Systems dynamics of the power converters are over simplified; this works that allow the integration of renewable energies and makes possible application of the droop that have changed the studies Systems assumption makes possible the application of droop that have changed to theFang studies of Electrical Power Systems scheme. In this sense, even the if the use of time scales to (EPS). According et of al.Electrical (2012), a Power microgrid is a assumption makes possible the application of the the droop that have changed the studies of Electrical Systems scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), aa Power microgrid is aa assumption scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), microgrid is a certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such scheme. In this sense, even if the use of time scales to (EPS). According to Fang et al. (2012), a microgrid is a a certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such aa certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such converters, this situation does not seem natural since this as solar panels and small wind turbines) interconnected via certain extent justifies neglecting the dynamics of the group of loads, lines and distributed generation units (such converters, this situation doesofnot not seem natural natural since this this as solar panels small turbines) via this does seem since as solarconverters. panels and and These small wind wind turbines) interconnected via converters, model limits thesituation possibility including phenomena expower microgrids caninterconnected operate isolated converters, this situation doesofnot seem natural since this as solar panels and small wind turbines) interconnected via model limits the possibility including phenomena expower converters. These microgrids can operate isolated model limits the possibility of including phenomena expower converters. These microgrids can operate isolated hibited by the microgrids related to the power quality and from or interconnected to the main network, addressing model limits the possibility of including phenomena expower converters. Thesetomicrogrids can operate isolated hibited by the microgrids related to the power quality and from or interconnected the main network, addressing hibited by the microgrids related to the power quality and from or interconnected to the main network, addressing disturbances during the operation of the equipment based stability problems while sharing the power demand behibited by the microgrids related to the power quality and from or interconnected to the main network, addressing during the operation of the equipment based stability while sharing the the power demand demand be- disturbances disturbances during stability problems while sharing power electronics. tween theproblems generation units. disturbances during the the operation operation of of the the equipment equipment based based stability problems while sharing the power power demand bebe- on on power electronics. tween the generation units. on power electronics. tween the generation units. on power electronics. tween units. based structure, control of On the other hand, exploiting the structural properties of Due tothe itsgeneration power electronics the other hand, exploiting the structural Due to power electronics structure, control of On the hand, exploiting the properties of Due to its its is power electronics based structure, control of On electrical networks in Avila-Becerril et properties al. (2016),of a microgrids accomplished in based two levels, namely, a local On the other other hand,stated exploiting the structural structural properties of Due to its power electronics based structure, control of electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in two levels, namely, a local electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in two levels, namely, a local control scheme for Microgrids has been reported in Avilacontrol is required to command the voltage supplied by electrical networks stated in Avila-Becerril et al. (2016), a microgrids is accomplished in twothe levels, namely, a local control scheme for Microgrids has been reported in Avilacontrol is required to command voltage supplied by control scheme for Microgrids has been reported in Avilacontrol is required to command the voltage supplied by Becerril and Espinosa-P´ e rez (2016) that solves a voltage these devices whiletothe power demanded bysupplied the loads control scheme for Microgrids has been reported in Avilacontrol is required command the voltage by and Espinosa-P´ eerez (2016) that a voltage these devices while the power by loads Becerril and (2016) that solves a these devices while by theimplementing power demanded demanded by the the droop loads Becerril stabilization problem. In this paper, it solves is showed that is usually achieved the so-called Becerril and Espinosa-P´ Espinosa-P´ erez rez (2016) that solves a voltage voltage these devices while the power demanded by the loads stabilization problem. In this paper, it is showed that is usually achieved by implementing the so-called droop stabilization problem. In this paper, it is showed is usually achieved by implementing the so-called droop the microgrid can be decomposed as the interconnection control (see Barklund et al. (2008); Pedrasa and Spooner problem. In this paper, it is showed that that is usually achieved byetimplementing the so-called droop stabilization the microgrid can be decomposed as the interconnection control (see Barklund al. (2008); Pedrasa and Spooner the microgrid can be decomposed as the interconnection control (seereferences Barklund therein). et al. al. (2008); (2008); Pedrasa Pedrasa and and Spooner Spooner the of amicrogrid main Port-Controlled Hamiltonian (PCH) system (2006) and can be decomposed as the interconnection control (see Barklund et of a main Port-Controlled Hamiltonian (PCH) system (2006) of aa main Port-Controlled Hamiltonian system (2006) and and references references therein). therein). equipped sources and loads ports to (PCH) which another mainwith Port-Controlled Hamiltonian (PCH) system (2006) and references therein). Although the effectiveness of the aforementioned control of equipped with sources and loads ports to which another equipped with sources and loads ports to which another systems, of the same kind, are attached. The model exAlthough the effectiveness of the aforementioned control equipped with sources and loads ports to which another Although the effectiveness of the the aforementioned controla systems, strategy isthe widely recognized in practice, there exists of the same kind, are attached. The model exsystems, of the same kind, are attached. The model exAlthough effectiveness of aforementioned control plicitly includes the dynamic of the power converters and strategy is widely practice, there aa systems, of the same kind, are attached. The model and exstrategy is understanding widely recognized recognized intheoretical practice, stability there exists exists lack in the of itsin propplicitly includes the dynamic of the power converters plicitly includes the dynamic of the power converters and strategy is widely recognized in practice, there exists a a Passivity–based Control (PBC) is presented to guarantee lack in the understanding of its theoretical stability propplicitly includes the dynamic of the power converters and lack in the understanding of its theoretical stability properties. Some prominent results explaining these feature are a Passivity–based Control (PBC) is presented to guarantee a Passivity–based Control (PBC) is guarantee lack inSome the understanding of its theoretical stability prop- that a prescribed output for theto conerties. prominent these feature Passivity–based Controlvoltage (PBC) profile is presented presented topower guarantee erties. Some prominent results results explaining these feature are given by Simpson-Porco et al.explaining (2013) and Schiffer et are al. athat aa prescribed output voltage profile for the power conthat prescribed output voltage profile for the power erties. Some prominent results explaining these feature are verters is achieved. This reference voltage, corresponding given by Simpson-Porco et al. (2013) and Schiffer et al. that a prescribed output voltage profile for the power concongiven by Simpson-Porco et al. (2013) and Schiffer et al. (2014). In the former, the authors show that, under some verters is achieved. This reference voltage, corresponding verters is achieved. This reference corresponding given byInSimpson-Porco etauthors al. (2013) and Schiffer et al. to a desired steady–state behavior,voltage, is computed based on (2014). the former, the show that, under some verters is achieved. This reference voltage, corresponding (2014). In the former, the authors show that, under some assumptions, the microgrid equipped withthat, a droop control, aa desired behavior, is computed based on to steady–state behavior, is based on (2014). In thethe former, the authors show under some to a given loadsteady–state power demand, by solving the power assumptions, microgrid with aa droop a desired desired steady–state behavior, is computed computed basedflow on assumptions, the as microgrid equipped with droop control, control, can be depicted a modelequipped of coupled oscillators stating to aequations given load power demand, by solving the power flow a given load power demand, by solving the power flow assumptions, the microgrid equipped with a droop control, of the network; hence, measurement of all the can be depicted as a model of coupled oscillators stating a given load power demand, by solving the power flow can bestability depicted as aa model model of coupled oscillators stating somebe properties. In of thecoupled latter, it is assumed that equations of the(voltage, network; hence, of all the equations of network; hence, measurement of can depicted as oscillators stating buses variables andmeasurement powers) is required. some stability properties. In the latter, it is assumed that equations of the the(voltage, network;phase hence, measurement of all all the the some stability properties. In the latter, it is assumed that each bus in the microgrid has a power converter connected buses variables phase and powers) is required. buses variables (voltage, phase and powers) is required. some stability properties. In the latter, it is assumed that each bus in the microgrid has aa power converter connected buses variables (voltage, phase and powers) isevaluate required.the each bus in the microgrid has power converter connected The purpose of this paper is to numerically whose frequency and voltage are controlled independently each bus in the microgrid has are a power converter connected The purpose of this paper is to numerically evaluate the whose and independently The purpose of is to evaluate the whose frequency and voltage voltage are controlled controlled independently properties of PBC of Avila-Becerril of eachfrequency other by means of a droop control, hence a stability robustness The purpose of this this paper paper is scheme to numerically numerically evaluate and the whose frequency and voltage are controlled independently robustness properties ofinPBC PBC scheme of Avila-Becerril Avila-Becerril and of each other by means of aa droop control, hence aa stability robustness properties of scheme of and of each other by means of droop control, hence stability Espinosa-P´ e rez (2016) two directions, namely: analysis in terms of consensus is developed. robustness properties of PBC scheme of Avila-Becerril and of each other by means of a droop control, hence a stability Espinosa-P´ e rez (2016) in two directions, namely: analysis Espinosa-P´ analysis in in terms terms of of consensus consensus is is developed. developed. Espinosa-P´eerez rez (2016) (2016) in in two two directions, directions, namely: namely: analysis of consensus is developed. Part of in thisterms work was supported by DGAPA-UNAM under grant
of Part IN116516. of this this work work was was supported supported by by DGAPA-UNAM DGAPA-UNAM under under grant grant Part Part of this work was supported by DGAPA-UNAM under grant IN116516. IN116516. IN116516. Copyright 6847Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 6847 Copyright © under 2017 IFAC IFAC 6847Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 6847 10.1016/j.ifacol.2017.08.1531
Proceedings of the 20th IFAC World Congress Sofía Avila-Becerril et al. / IFAC PapersOnLine 50-1 (2017) 6648–6653 Toulouse, France, July 9-14, 2017
• The computation of the output voltage of the power converters is obtained by implementing a droop control instead of solving the power flow equations of the network. • It is assumed that only part of the buses variables are available for measurement and that these measurements are contaminated with noise. To recover the real variables, a static state estimation is included in the control scheme. This algorithm is based in the well–known Weighted Least Squares technique. The robustness of the control scheme is evaluated through numerical simulations showing that converge to the desired behavior is achieved in spite of the existence of noise measurement, network model inaccuracies, as well as errors introduced in various stages of the communication path Abur (2015). The rest of the paper is organized as follows: For clarity, in Section 2 it is recovered the microgrid structure considered by Avila-Becerril and Espinosa-P´erez (2016) together with the controller. Section 3 is devoted to explain the implementation of the control law with the state estimation scheme while in Section 4 the numerical evaluation is carried out. Finally, some concluding remarks are included in Section 5. 2. MICROGRID STRUCTURE In this paper it is considered a generic balanced, lossless microgrid with a meshed topology, operating under normal conditions, modeled by a single-phase equivalent circuit. On the one hand, each of the n1 generation units are composed by a constant voltage source Vi ∈ R > 0 modulated by a switching array via ui ∈ R to later on fed a second order LC filter, such that vCi ∈ R is the output capacitor voltage while ILi ∈ R is the port current delivered to the network. The set of n1 power converters can be represented in port Hamiltonian (pH) form with the definition of the stored energy function Hc : Rn1 × Rn1 → R≥0 given by 1 1 (1) Hc (x1 , x2 ) = xT1 L−1 x1 + xT2 C −1 x2 , 2 2 with the collection of the linkage inductor fluxes x1 ∈ Rn1 and x2 ∈ Rn1 the electrical capacitor charges, where it has been assumed a linear constitutive relationship for both the inductors and the capacitors, with L ∈ Rn1 ×n1 > 0 and C ∈ Rn1 ×n1 > 0 diagonal inductance and capacitance matrices. The dynamic behavior of the n1 power converters can be represented by the compact form 0 x˙ 12 = J12 ∇x12 Hc + G12 u − (2) IL T ∈ Rn1 , the matrices where x12 = xT1 xT2 0 −I V = −JT12 ∈ Rn1 ×n1 ; G12 = , J12 = I 0 0
u = col(ui ) ∈ Rn1 , and the inductor currents vector satisfies iL = ∇x1 Hc ∈ Rn1 , while the capacitor voltages vector is given by vC = ∇x2 Hc ∈ Rn1 . Remark 1. The structure considered for the power converters essentially captures the dynamic behavior of a large class of these devices since common converter topologies
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can be represented under a port Hamiltonian structure (see for example Noriega-Pineda et al. (2010)). On the other hand, the transmission lines are modeled by the Π model such that the network with the loads are viewed as an electrical circuit on a graph (for details see Avila-Becerril et al. (2016)). It is defined the energy function of the network Ha : Rn1 ×n3 → R≥0 as 1 1 (3) Ha (x3 , x4 ) = xT3 Ca−1 x3 + xT4 L−1 a x4 2 2 n1 where x3 ∈ R is the electrical capacitor charges vector and x4 ∈ Rn3 the linkage inductor fluxes vector, with La ∈ Rn3 ×n3 > 0 and Ca ∈ Rn1 ×n1 > 0 the diagonal inductance and capacitance matrices. Now, let the current at the i–th load in parallel with the i-th capacitor be −1 fRci = ψci (eRci ), with the load voltage eRci and ψci (·) a bijective function, then the model of the transmission lines can be represented by the pH system x˙ 34 = J34 ∇x34 Ha (x3 , x4 ) − F34 fRc + G34 e1 (4) T T T (n1 +n3 ) ∈ R , fRc = col (fRci ) ∈ where x34 = x3 x4 Rn1 ∂Ha (x3 , x4 ) I ∇x34 Ha (x3 , x4 ) = , F34 = , 0 ∂x34
0 0 −HCL T = −J ; G = 34 T T 34 H1L HCL 0 subject to the port constraints f1 = H1L fL , eRc = eC = ψc (fRc ), (5) n1 with e1 , f1 ∈ R the voltage and current of the sources, eC the capacitor voltages parallel connected with the loads; while HCL and H1L are submatrices of the fundamental loop matrix (see Bollob´as (1998)) and represents the interconnections between capacitors and inductors, and sources with inductors, respectively. Remark 2. Equation (5) shows the current that the transmission lines demand from each source. In the sequel it is assumed ideal sources in the sense that they can provide any amount of current, so that only steady state stability issues can be approached. J34 =
The Hamiltonian model of the microgrid is obtained from (2) and (4) under the ports choice given by ∂Ha (x34 ) , e1 = C −1 x2 , (6) IL = H1L ∂x4 while the currents at the loads as can be written ∂Ha (x34 ) −1 fRc = ψc (7) ∂x3 The previous models (2), (4), together with (6) and (7), leads to the model of the complete Microgrid x˙ = JT ∇x HT (x) − gRT Ψ(x34 ) + GT u (8) T T T T T (3n1 +n3 ) ∈R , where the with state x = x1 x2 x3 x4 total stored energy function are (9) HT (x) = Hc (x1 , x2 ) + Ha (x3 , x4 ) and the matrices of appropriate dimensions
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0 −I 0 0 0 0 0 −H1L I 0 0 0 T JT = 0 0 0 −HCL = −JT ; gRT = 0 I ; T T I 0 0 H1L HCL 0 Vu 0 0 ∂Ha (x34 ) . Ψ(x34 ) = GT u = −1 0 ψc ∂x3 0 Likewise, the admissible trajectories are the solutions of x˙ = JT ∇x HT (x ) − gRT Ψ(x34 ) + GT u , (10) with u the control input that generates x . Let the error variable be x ˜ = x − x ∈ R3n1 +n3 and assume that:
A.1 The state are available for measurement. A.2 The parameters C, L, Ca and La are known and positive. A.3 The voltage source V vector is strictly positive. A.4 The desired steady-state behavior x2 is a known bounded and differentiable function with bounded first derivative. A.5 The port–variables of each load satisfies the passivity condition T (eC − eC ) ψc−1 (eC ) − ψc−1 (eC ) > 0 for a given admissible trajectory eRc (t) ∈ Rn1 .
Under the assumptions above, the stability properties of model (8), in closed loop with the control law
u = V −1 [x˙ 1 + C −1 x2 − K1 L−1 x ˜1 ] n1 ×n1 and admissible with K1 = diag{K1i } ∈ R satisfying the constraints x˙ 2 = L−1 x1 − H1L L−1 a x4 , x˙ 3 = −HCL ∇x 4 HT (x ) − ϕ−1 c (eC )
(11) state (12a) (12b)
T T x˙ 4 = H1L ∇x 3 HT (x ) (12c) ∇x 2 HT (x ) + HCL have been studied in Avila-Becerril and Espinosa-P´erez (2016) and guarantees that lim x ˜ = 0. t→∞
The main problem when implementing the proposed controller is the definition of an admissible trajectory x ∈ R3n1 +n3 . In this sense, the achievement of a desired steady state behavior depends on (10). 3. SET POINT DESIGN In this section it is presented the main contribution of the paper, which refers to an improvement in the implementation of the controller above. In particular, it is proposed to incorporate an external control loop that allows to modify the desired trajectories in terms of the demanded power. Since this modification is not contemplated in the stability proof, it will be a good measure of the controller robustness, which will be evaluated numerically in Section 4. First, let us consider that the microgrid is formed by n = n1 + n3 nodes and that the i−th node has associated a voltage amplitude Vi and a phase angle δi . In turn, two nodes i and k of the microgrid are connected by means of a complex admittance denoted as Yik := Gik + jBik ∈ C, with conductance Gik ∈ R and susceptance Bik ∈ R. In addition, assume that:
A.6 The desired steady-state is sinusoidal. Under the last condition and since the network is lossless (Schiffer et al. (2014)), the total steady-state power flows (injected or consumed) Pi , Qi in the i–th node can be written as Pi = |Bik | Vi Vk sin(δik ), k∈Ni
Qi = |Bii | (Vi )2
k∈Ni
|Bik | Vi Vk cos(δik ).
(13)
where Ni is the set of neighbors of the i − th ˆii + Bik , with B ˆii ∈ node for which Yik = 0 and Bii B δik
δi − δk ,
k∈Ni
R the shunt susceptances in the node i. This power flow equations are static, model the network when the net sum of consumption, injections and dissipated power is zero, and determine the steady state operation of the network. It should be noted that this set of algebraic equations are non linear both in the voltage and in the angle, therefore their solution involves the use of a numerical method. In Avila-Becerril and Espinosa-P´erez (2016), the desired behavior is stated in terms of the power demanded by the loads. Hence, once this power is fixed, the corresponding voltage magnitud and phase angle of the nodes are found, to later on compute the entire value of x . In this sense, it is defined for each power converter the output voltage Ci−1 x2i = Vi sin(ωs t + δi ), (14) where ωs ∈ R takes the same value for all the power converters, while the magnitude Vi and the phase δi must be computed off–line to get an adequate power flow (13).
3.1 Droop Control The previous proposal has as a disadvantage namely, that the implementation of the controller lies in the necessity to compute off–line the admissible trajectories x . With the aim of solving this problems it is incorporated a secondary controller that computes the desired output voltage of the power converters by implementing a droop control instead of solving (13). Basically, the droop control (Kundur et al. (1994)) is a proportional control that, for inductive lines, balances the active and reactive power demands in the network by instantaneously changing the frequency ωi and the magnitud Vi of the voltage signal at the i–th power converter, so it allows to establish the power in a desired steady state value. In the microgrids literature, usually this control is implemented as ui1 = ωd − kpi (Pi − Pid ) , (15) ui2 = Vid − kqi (Qi − Qid ) , with ωd ∈ R and Vid ∈ R>0 the desired nominal frequency and the nominal voltage amplitude, kpi and kqi ∈ R>0 the controller gains and Pid , Qid the reference powers. The variables ui1 and ui2 are the control inputs of the i– th power converter so that to directly include the control law (15) in the model of the microgrid it is necessary to assume that the voltage and the phase are decoupled (see for example Schiffer et al. (2014); Simpson-Porco et al. (2013)).
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Nevertheless, since in this work the microgrid model (8) takes into account the dynamics of the converter, implementing the droop merely as in (15) is not possible. Thus, it is proposed to define the desired state x (t) to take into account power variations. That is, the voltage output of the converter has been chosen as Ci−1 x 2i = Vi sin(δ˙i t), (16) where δ˙i = ωd − kpi (Pi − Pid ) Vi = Vid − kqi (Qi − Qid ) with ωd the nominal frecuency. Note that if the power of the node is the required, then there will be no phase change. Once x 2 is calculated, the rest of the admissible trajectories x are obtained from equation (12). For this, it should be noted that equations (12b) and (12c) are of the form T x˙ 34 = f (x34 ) + H1L C −1 x 2 , so it is necessary to solve the differential equation for x 34 and then the variable x 1 can be algebraically obtained from equation (12a). Remark 3. It is important to note that the dynamic obtained for the rest of the desired trajectories imposes a new challenge in the stability analysis because now it is necessary to show that the admissible trajectories are bounded.
∂ J(xm ) ∂xm = H T (xm )W [z − h(xm )] = 0
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G(x) =
(17)
∂ ∂xm h(xm ).
In this paper, solution of (17) where H(xm ) = is obtained by the Newton-Raphson method. Remark 4. The values used for the weight matrix W in (17) associated with each measurement are given in the literature (see for example G´omez-Exp´osito et al. (2016); Crow (2009); Abur and Exposito (2004)); however, this values can also be found as reported by Allemong (1982). Proposition 1. Consider the microgrid model (8) under A1–A6. Additionally suppose that: B.3 The measurement of power in some nodes is known and that the measurement noise has Gaussian behavior. B.4 The voltage and phase in one bus is known, and are taken as reference. B.5 The transient of the closed loop system is fast enough to assume that the system is in steady state. Then, the control law (11) with the desired trajectories (16) and δ˙i = ωd − kpi Pˆi − Pid , (18) ˆ i − Qid . Vi = Vid − kqi Q
A disadvantage of the proposed set point design lies in the necessity to measure powers in all nodes. Under this scenario, the performance of the controller may be affected by noise measurements.
ˆ i the estimated power, guarantees that the with Pˆi and Q microgrid state converges to the admissible trajectories, lim x ˜=0
The introduction of the modifications mentioned above has led to the proposal of a state estimator that is presented in the next section.
3.2 Static State Estimation In this section it is incorporated a static state estimator (G´ omez-Exp´ osito et al. (2016)) which calculates the statistically optimal state of the network. The static state of the network xm is defined by the vector with voltage magnitude and phase of all buses, while the estimator (Abur (2015)) processes the measurements of the network model and converts them into the best state estimate, despite measurements are corrupted by noise, may be missing or grossly inaccurate. Thereby, from the power flow values of some nodes, with their respective uncertainty associated with the measurement, the voltage magnitude and phase are calculated using a classical Weighted Minimum Square error. Consider the nonlinear set of measurements z = h(xm ) + e where xm is the state vector, z is the measurement vector, h(xm ) is a nonlinear measurement function, that for this case corresponds to the power flow equations (13), and e is the measurement error. To calculate the estimates x ˆm it is minimized the objective function J(xm ) = [z − h(xm )]T W [z − h(xm )] with W = diag 1/σi2 and σi2 the variance error of the i-th instrument. So, by minimizing J(xm ), the estimates x ˆ are those values which satisfy
t→∞
Proposition above considers a more realistic scenario. On the other hand, assumption B.5 is related to the fact that the estimator is build on the assumption that the state variable is in steady state or quasi steady state. The next section presents its numerical evaluation. 4. NUMERICAL EVALUATION A scheme of the control implementation is shown in Figure 1, where, for simulation purposes, the power measurement were done by decoupling the magnitude and the phase of voltage vc = C −1 x2 at the converters. The estimators output is the voltage magnitude and phase of all nodes that the droop control uses to calculate the desired converters voltage. The proposed controller was evaluated in the network presented by Schiffer et al. (2014) and is the one shown in Figure 2, where it have been neglected the non-controllable sources (solar and wind) with a series impedance reduction. So, it is assumed that the batteries in nodes 1 and 5 are operated in charging mode and that the maximum power capacity for the i-th source is SiN . The base units in the calculations per unit are SB = 4.75M V A, VB = 20kV and ZB = 84.21Ω, while the generation units are modeled as sources of 1.0 pu as constant magnitude. For this simulation, the design parameters are those shown in the Table 1. Consider that each inverter is represented by (2) and its capacitance and inductance values have been set in 8 [µF ] y 1.5 [µH], respectively, while Vi = 1 pu (Pogaku et al. (2007)). In addition, the droop gains have
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V vc(t)
Power Converters
i(t)
Pm
Static State Estimation
Sensor Qm
|vc|
u
Power Flow
Controller Pd
Qd
Vd
d
|vcd|
P, Q
Droop Control
vcd= |vcd| sin ( dt + ) d
Fig. 1. Controller Implementation
0
V
Cd
[pu]
1
−1 0
0.1
0.2
0.3
Fuel cell
Batery
voltage error
0.06
Power converter
Fig. 2. Illustrative microgrid SBase = 4.75M V A, VBase = 20kV (0.505, 0.028, 0.261, 0.179, 0.168, 0.012)[pu] (-0.0173, 0.014, 0.131, 0.089, -0.067, 0.006)[pu] (-0.007, 0.00, 0.003, 0.002, 0.002, 0.00 [pu]) Table 1. Parameters
The second scenario evaluate the controller (11) when the state estimator has been incorporated under the specifi-
0.7
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Fig. 3. (a)Converters reference voltage in pu. (b) Tracking error 1 0.5
δ [rad]
0 0 0.04 0.02 0 −0.02 60.10
f [Hz]
The evaluation has been divided into two scenarios: the first, besides evaluating the performance of the proposed controller, aims to test its robustness against a change of 50% in the loads of nodes 1 and 5 at t = 0.7[s]. Accordingly, the first part of Figure 3 shows the reference value of voltage converters C −1 x2 ∈ R2 in per unit. In the second part of this figure, we show the tracking error between the actual voltage vC = C −1 x2 and the desired vCd = C −1 x2 , where it can be seen that the convergence is almost instantaneous, giving a solution to the tracking problem. Figure 4, presents the obtained voltage values at the six nodes, the frequency and the phase, with the aim of showing that they reaches synchronization in the first two. Figure 5 exhibits the active and reactive powers, it can be notice that, before the change of load, the power values coincide with those specified in Table 1 and that in t = 7s the loads have been increased 50% which causes an adjustment in the values of the powers of the rest of the nodes.
0.6
−2
0.02
−0.02 0
been fixed in Kp = diag{0.12, 0.1, 0.1, 0.6, 0.3, 0.5} and Kq = diag{0.15, 0.1, 1.3, 0.2, 1.5, 0.6}, for the six converters, while the desired frequency value ωrd is 60 [Hz]. The numerical evaluation of the controller was performed in SIMULINK of MATLAB, with variable integration step and Runge-Kutta numerical method. The gains of the controller (11) were defined as k1i = 4.5 and k2i = 500, and all initial conditions were set to zero.
0.5
2
0.04
V [pu]
Values SiN Pid Qdi
0.4
−5
x 10
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1.002 1
1
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0.01
0.03 0.02 0.01 0 −0.01 0
0.998 0.6
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0.2
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−4
x 10
0
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0.02 −5 0.6
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1.5
60.05 60
60 59.95 0
0.05
59.99
0.1
59.98 0.6
0.5
t [s]
0.7
1
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1.5
Fig. 4. Voltage magnitud, phase and frequency cations given in Section 3, with 10% of active and reactive power measurement noise. The Newton Raphson algorithm converges in the fourth iteration with the vector xm initialized in xm = x0 , with the values of Vi = 1pu and θi = 0. In Figure 4 we present the active and reactive power estimated, which shows that the estimator converges to the actual values, that are those in Table 1, which guarantees the correct operation of the system despite of the noise. 5. CONCLUDING REMARKS In this paper we present some numerical robustness properties of a passivity based control for microgrids where, by exploiting the properties of the droop controller, it
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P [pu]
REFERENCES
P1
0.2
P
0.1
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P5
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P
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0
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1 Q
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Q [pu]
0.05
Q2 Q3 Q
0
4
Q
5
Q6
−0.05 0
0.1
0.2
0.3
0.4
0.5 t [s]
0.6
0.7
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0.9
1
Fig. 5. Active and reactive power at the nodes.
Fig. 6. (a)Actual active power (b) Actual reactive power 0.2
P
P [pu]
1
0.1
P2
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P
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Fig. 7. Droop Output (Actual Active and Reactive Power) using the estimator
Abur, A. (2015). Observability and dynamic state estimation. In 2015 IEEE Power & Energy Society General Meeting, 1–5. IEEE. Abur, A. and Exposito, A.G. (2004). Power system state estimation: theory and implementation. CRC press. Allemong, J.J.; Radu, L.S.A. (1982). A fast and reliable state estimation algorithm for aep’s new control center. IEEE Transactions on Power Apparatus and Systems, PAS-101. doi:10.1109/tpas.1982.317159. Avila-Becerril, S. and Espinosa-P´erez (2016). A hamiltonian approach for stabilization of microgrids including power converters dynamic. Congreso Latinoamericano de Control Autom´ atico CLCA, 2016, 246–251. Avila-Becerril, S., Espinosa-P´erez, G., and Fernandez, P. (2016). Dynamic characterization of typical electrical circuits via structural properties. Mathematical Problems in Engineering, 2016. Barklund, E., Pogaku, N., Prodanovi´c, M., HernandezAramburo, C., and Green, T.C. (2008). Energy management in autonomous microgrid using stabilityconstrained droop control of inverters. Transactions on Power Electronics, IEEE, 23(5), 2346–2352. Bollob´as, B. (1998). Modern graph theory, volume 184. Springer Science & Business Media. Crow, M. (2009). Computational methods for electric power systems. Fang, X., Misra, S., Xue, G., and Yang, D. (2012). Smart grid—the new and improved power grid: A survey. Communications Surveys & Tutorials, IEEE, 14(4), 944–980. G´omez-Exp´osito, A., Conejo, A.J., and Ca˜ nizares, C. (2016). Electric energy systems: analysis and operation. CRC Press. Kundur, P., Balu, N., and Lauby, M. (1994). Power system stability and control, volume 7. McGraw-hill New York. Noriega-Pineda, D., Espinosa-P´erez, G., Cardenas, V., and Alvarez-Ram´ırez, J. (2010). Passivity-based control of multilevel cascade inverters: High-performance with reduced switching frequency. International Journal of Robust and Nonlinear Control, 20(9), 961–974. Pedrasa, M.A. and Spooner, T. (2006). A survey of techniques used to control microgrid generation and storage during island operation. In Proceedings of the 2006 Australasian Universities Power Engineering Conference (AUPEC’06), 1–6. Pogaku, N., Prodanovi´c, M., and Green, T.C. (2007). Modeling, analysis and testing of autonomous operation of an inverter-based microgrid. Power Electronics, IEEE Transactions on, 22(2), 613–625. Schiffer, J., Ortega, R., Astolfi, A., Raisch, J., and Sezi, T. (2014). Conditions for stability of droop-controlled inverter-based microgrids. Automatica, 50(10), 2457– 2469. Simpson-Porco, J.W., D¨ orfler, F., and Bullo, F. (2013). Synchronization and power sharing for droop-controlled inverters in islanded microgrids. Automatica, 49(9), 2603–2611.
allows, in addition to the stabilization of the microgrid, a dispatch of power flows. The controller is evaluated numerically for load disturbances. Also, a state estimator that allows inclusion of an error in the measurement, has been incorporated, making the scenario more realistic. 6852