AbstractâA distributed multi-agent supervision strategy (S-. CG) based on Command Governor ideas is proposed to solve constrained voltage regulation ...
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
A Distributed Command Governor Approach for Voltage Regulation in Medium Voltage Power Grids with Distributed Generation. Francesco Tedesco and Alessandro Casavola controllable generators and Static Var Compensators (SVC) in order to maintain the voltage at each connection node within prescribed bounds. In this respect, the aim of this work is to propose a customization of the distributed Sequential Command Governor (S-CG) strategy recently presented in [3] to the problem at hand. Specifically, such a sequential strategy allows a single agent at a time to manipulate its own reference signal. See [3] for full details and proofs of its stability, feasibility and viability (liveness) properties. See also [4], [5], [6] for other relevant works on the topic.
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I. I NTRODUCTION
A. Casavola and F. Tedesco are with the Dipartimento di Ingegneria Elettronica, Informatica e Sistemistica (DIMES), Universit`a degli Studi della Calabria, Via Pietro Bucci, Cubo 42-c, Rende (CS), 87036, Italy.
978-1-4799-0178-4/$31.00 ©2013 AACC
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The problem of controlling voltages at the access nodes of Distributed Generators (DG) in standard Medium/Low Voltage (MV/LV) power grids has gained interest in recent years for its relevance in accommodating higher levels of DG penetration in actual power grids. In fact, the technological changes and restructuring processes necessary to implement novel smart-grid concepts shall involve a transition from highly passive transmission grids towards a more active management of them, facilitating bidirectional flows of energy and the introduction of small-scale DG units, even located at the end-user sites. Power grids are generally designed to operate radially without any generation on the distribution lines or at the customer sites. The introduction of DG units can significantly affect the power flows and the voltages, and deteriorate the effectiveness of standard voltage regulation systems in charge of handling load variations and adverse events. This work, by exploiting ideas presented in [1], extends the centralized solution proposed in [2] for the on-load tap changer (OLTC) set-point reconfiguration problem to the more general case where several OLTC devices and other smart devices (shunt devices, controllable renewable generators) act on the grid and need to be coordinated in order to achieve predetermined objectives. In particular, here the attention is focused on the development of a distributed constrained supervision strategy for the coordination of an arbitrary number of OLTC devices,
{casavola, ftedesco}@dimes.unical.it
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Abstract— A distributed multi-agent supervision strategy (SCG) based on Command Governor ideas is proposed to solve constrained voltage regulation problems in electrical Medium Voltage (MV) power grids in the presence of Distributed Generation (DG). The idea is that an active management of the set-points of some controllable variables of the grid, e.g. the power either provided by the distributed generators or consumed by the loads, can be an effective tool not only for maintaining relevant system variables within prescribed operative constraints in response to unexpected adverse conditions, but also for implementing some smart-grid functionalities. The supervision problem is formulated as a distributed constrained optimization problem by imposing that the voltages at certain nodes have, compatibly with all prescribed constraints and changed conditions, minimal deviations from their nominal values. Simulation results show that the proposed approach ensures, under certain conditions, feasible evolutions to the overall power grid under critical events.
Agent 4
Fig. 1. MV distribution grid. The colored regions represent parts of the grid handled by distinct agents which are assumed to be aware of the state of that part of the grid only (more details in Section II.b).
II. D ISTRIBUTION S YSTEM M ODELING Consider in Fig. 1 a radial power grid obtained by suitably simplifying a typical portion of the MV Italian distribution network. Two HV/MV transformers equipped with on-load tap changers (OLTC) supply the feeders to which small size generators and/or loads are connected. Both loads and generators, the latter supposedly consisting of small-size renewable energy generators (wind, sun, hydro, etc.), are characterized by a high degree of uncertainty because they follow the customer behaviour and the primary source availability. For exemplification purposes, it is assumed that one distributed generator (only one for simplicity) is endowed with ”smart” functionalities. Specifically, it is possible to specify the desired amount of generated power, and in particular the provided current, by setting a suitable set-point. The voltages
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at the secondary sides of the HV/MV transformers can be i modelled as ideal voltage sources (EM V , i = 1, 2), whose amplitude can be controlled by the OLTC tap positions. Furthermore, the grid is provided with a shunt SVC device, modelled as a current source injecting active/reactive current according to an external supervision logic. By assuming the balanced loads, symmetrical generation and a linear network behavior, one can derive the singlephase equivalent circuit depicted in Fig. 2. The grid under consideration consists of: • Two 132 kV HV networks with a short-circuit power Asc of 6000 M V A; • Two HV/MV S1 and S2 stations, each one connected to a 132 kV HV busbar through a 132/20 kV, 40 M V A, transformer, equipped with On Load Tap Changer (OLTC), at the primary side and to 20 kV MV busbars, B1 and respectively B2 , at their secondary sides; • The feeder F1 connected to B1 , is split into three 3 Km long line sections, namely (L11 , L12 L13 ), over which the loads C1 , C2 and C3 are connected. Notice in particular that a SVC device acts on C1 ; • A 30 Km long transmission line connecting busbars B1 to B3 • The feeder F2 connected to B2 , composed by a single 3 Km long line section L21 supplying 20 kV MV to busbar B3 . • The feeder F3 connected to B2 , where the generation unit DG1 is connected; • The sub-feeder F4 connected to B3 , split into two 3 Km long line sections (L41 , L42 ) over which only loads are presents; • The sub-feeder F5 connected to B3 , split into two 3 Km long line sections (L51 , L52 ) over which the generation units DG2 and DG3 are connected. Notice that DG3 is the smart DG unit. Twelve bus voltages result which have to be maintained within prescribed constraints. The following notation ZLi := RLi + jXLi , ZCi = RCi + jXCi , ¯i (t) := ED (t) + jEI (t), I¯i (t) := ID (t) + jII (t), ∀i ∈ [1, 12], E i i i i i i ¯M E V (t) = EM V D (t), i = 1, 2 (1)
will be adopted where the electrical parameters: ZLi , RLi and XLi denote respectively the line impedances, resistances and reactances, ZCi , RCi and XCi the load impedances, ¯i (t) and I¯i (t) complex numbers resistance and reactances, E representing respectively the voltages on and currents abi sorbed by the loads Ci and EM V (t) the HV/MV transformer secondary voltages. In what follows, it will be assumed for simplicity that the distribution grid evolves along linear regimes. Such a choice is usually considered in the literature for local studies around a feasible equilibrium point of the distribution system. Moreover, it is worth pointing out that non-linear behaviours of loads may be modelled by adding additional time-varying disturbances affecting the nominal load currents (7). Following the same reasoning, also the DG units will be treated as linear components and represented as current
EMV2 (t)
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Equivalent ciruit
generators, namely J¯i (t) = JDi (t) + jJIi (t), i = 1, 2, 3. (2) that correspond respectively to DG1 , DG2 and DG3 . Also in this case the generated currents are represented in complex form. In particular, notice that while J¯1 (t) and J¯2 (t) represent non-manipulable disturbances for the power grid, the generator J¯3 (t) is assumed to be a smart device, that is, its generated power (current) is manipulable. Finally, the SVC device on bus 1 is considered as an additional manipulable control input, representable as a current source δ¯SV C (t) = δ¯SV C (t) + jδ SV C (t) (3) 1
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Hence, by taking into account the single-phase equivalent circuit of Fig. 2 and denoting by I¯Ti (t) the currents drawing into the buses at each i-th node, the following relations result I¯T1 (t) := I¯1 (t) − δ¯1SV C (t), I¯T4 (t) := I¯4 (t) − J¯1 (t) I¯T6 (t) := I¯6 (t) − J¯2 (t), I¯T9 (t) := I¯9 (t) − J¯3 (t) (4) I¯Ti (t) := I¯i (t), i = 2, 3, 5, 7, 8, 10, 11, 12 Then, by resorting to the Kirchhoff’s circuit laws, the following impedance matrix description can be achieved for the distribution network
¯M V (t) E ¯ E Z1,1 , ¯1 (t) E2 (t) .. = . .. Z13,1 , . ¯12 (t) E
..., .. . ...,
¯ I¯M V (t) Z1,13 IT1 (t) I¯T2 (t) .. . .. Z13,13 . I¯T12 (t)
(5)
where I¯M V (t) denotes the HV/MV transformer current and Zi,i = φ(ZLi ) the entries of the impedance matrix (see [7] for details). Finally, the load characteristic equations lead to ¯i (t) + ZC I¯i (t) = 0, i = 1, ..., N −E (6) i We shall now consider the presence of additional instrumental disturbances acting on the system for modelling load changes, injected active power and injected/absorbed reactive power. It is well-known that the injection of active/reactive power alters the loads behaviour in a nonlinear way. In order to take these aspects into account, we allow the load disturbances to vary to a large extent. Specifically, we assume that the module of the current absorbed by the loads may vary within the following range [Ii − δIi , Ii + δIi ] (7) where δIi = γIi accounts for both measurement errors over the voltage profile of all system buses and for load power
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variations. Allowable ranges corresponding to the choice γ ≈ 0.25 are typically considered in the literature, see [9]. Therefore, from eqs. (5)-(6) and definitions (1)-(4), the following discrete-time state space description can be derived through straightforward algebraic manipulations { xp (t + 1) = Gg(t) + GI δI (t) + GJ J(t) (8) y(t) = Hxp (t) where xp(t)=[ED1 (t)ID1 (t)...ED12 (t)ID12 (t)EI1 (t)II1 (t)...EI12 (t)II12 (t)]T SV C C g(t) = [EM V1 (t)EM V2 (t)JD3 (t)JI3 (t)δD (t)δISV (t)]T , 1 1 T δI (t) = [δID1 (t)δII1 (t)...δID12 (t)δII12 (t)] J(t) = [JD1 (t)JI1 (t)JD2 (t)JI2 (t)]T (9)
and
y(t) = [ED1 (t)...ED12 (t)EI1 (t)...EI12 (t)]T (10) The explicit structures of matrices H, GJ and GI are not included here for space reasons but details can be found in[8]. Remark - 1 It is worth remarking that we consider the DG presence as a power disturbance added on the bus. As a consequence, the only information we need to know for design purposes is its maximum allowable range and not its time profile. 2 A. Constraints and control requirements The system described by (8) is subject to the following set of constraints: ¯i (t), i = 1, 2, ..., N cannot be imposed • Load voltages E to equal the reference value En , because the voltage drops are different each others in the various distribution lines. Therefore, the following constraints must be fulfilled at each time instant: ¯i (t)|2 ≤ (1+α)En2 , i = 1, ..., 12, ∀t (11) (1−α)En2 ≤ |E ¯i (t)|, i = 1, 2, ..., N within ±4% (α ∈ Deviations of |E (0 0.04]) are reasonable in practice. Actually, such load voltage constraints can be (conservatively) replaced by constraints on the real parts EDi (linear constraints). The involved approximations are usually acceptable because in typical distribution networks the line reactances XLi and the line resistances RLi have negligible values with respect ¯i (t)|. In fact, because the to the load voltage modules |E ¯i (t) is almost zero [7], phase displacement of the phasor E ¯ the imaginary part EIi (t) of Ei (t) can be neglected and ¯i (t) ∼ one approximately has E = EDi (t), i = 1, ..., 12, ∀t. Then, constraint (11) can be replaced by ¯D (t) ≤ (1 + α)En , α ∈ (0, 0.04] (1 − α)En ≤ E i i = 1, ..., 12,∀t ∈ ZZ+ •
(12)
As it is well-known, OLTC alters the power transformer turns ratio in a number of predefined steps and the secondary side voltage is identically changed, each step usually representing a change of 0.25 − 1.25% in the LV side voltage. Because the switching step is small with respect to the load tolerance (see (11)), one can usually take into account a continuous approximation of the OLTC dynamics
[7] for control synthesis purposes. Because standard tap changers approximately offer a change of 10% of the rated ¯M V (t) can vary in percentage with voltage, the value of E respect to the nominal voltage En , as follows ¯M V (t)| ≤ (1 + β)En , ∀t, β ∈ (0 0.1] (13) E n ≤ |E • The module of the current absorbed/injected by the SVC device can be at most 25% of the nominal current absorbed by bus 1 [9], i.e., 0 ≤ δISV C ≤ 0.25I1 (14) Hence, the coordination problem to solve consists of determining the voltage set-points EM Vi and current set points J¯3 (t), δ¯1SV C such that: ¯i (t), i = 1, ..., 12 satisfy the con• the load voltages E straints (12); ¯M V (t) satisfies inequality (13); • E SV C ¯ • δI (t) satisfies inequality (14); 1 2 ¯SV C are the best approximations of • EM V , EM V , J¯3 (t), δ 1 their related nominal references. III. C ENTRALIZED AND D ISTRIBUTED C OMMAND G OVERNOR (CG) P ROBLEM F ORMULATION Let us consider a set of N subsystems A = {1, . . . , N }, each one being a LTI closed-loop dynamical system regulated by a local controller which ensures stability and good closedloop performance in linear regimes when the constraints are not active (small-signal regimes when the coordination is effective). Let the i-th closed-loop subsystem be described by the following discrete-time model ∑ Φij xj (t) xi (t+1)=Φii xi (t)+ Gi gi (t) + Gdi di (t)+ j∈A−{i}
y (t) =Hiy xi (t) i ci (t) =Hic x(t) + Li g(t) + Ldi d(t)
(15) where: t ∈ ZZ+ , xi ∈ IRni is the state vector (which includes the controller states under dynamic regulation), gi (t) ∈ IRmi the CG action, which, if no constraints were present, would essentially coincide with the reference ri (t) ∈ IRmi . The [ ]T vector di (t) = δIi (t)T , Ji (t)T ∈ IRndi is an exogenous bounded disturbance satisfying di (t) ∈ Di , ∀t ∈ ZZ+ with Di a specified convex and compact set such that 0ndi ∈ Di . Moreover, Gdi = [GIi GJi ] is used in (15); yi (t) ∈ IRm thec output, viz. a performance related signal. Finally, ci ∈ IRni represents the local constrained vector which has to fulfill the set-membership constraint ci (t) ∈ C i , ∀t ∈ ZZ+ , (16) C being a convex and compact polytopic set. It is worth pointing out that, in order to possibly characterize global (coupling) constraints amongst states of different subsystems, the vector ci in (15) is allowed to depend on the aggregate state and manipulable reference ∑N vectors x = [xT1 , . . . , xTN ]T ∈ IRn , with n = n , and ∑N i=1 i T T g = [g1T , . . . , gN ] ∈ IRm , with m = m i, d = i=1 ∑N [dT1 , . . . , dTN ]T ∈ IRnd , with nd = i=1 ndi . Moreover, we T T T T denote by r = [r1T , . . . , rN ] ∈ IRm , y = [y1T , . . . , yN ] ∈ ∑N c m nc T T T c IR and c = [c1 , . . . , cN ] ∈ IR , with n = i=1 ni ,
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i
the other relevant aggregate vectors. The overall system arising by the composition of the above N subsystems can be described as x(t + 1) = Φx(t) + Gg(t) + Gd d(t) y(t) = H y x(t) (17) c(t) = H c x(t) + Lg(t) + Ld d(t) where Φ = [Φij ]i,j=1,...,N G = diag(G1 , ..., GN ), Gd = y ), diag(Gd1 , ..., GdN ) = [GI GJ ], H y = diag(H1y , ..., HN c c T c T T T T T H = [(H1 ) , ..., (HN ) ] , L = [(L1 ) , ..., (LN ) ] , and Ld = [(Ld1 )T , ..., (LdN )T ]T . It is further assumed that: (A1) the overall system (17) is asymptotically stable. Roughly speaking, the CG design problem is that of locally determining, at each time instant t and for each master agent i ∈ A associated to each subsystem, a suitable reference signal gi (t) which is the best feasible approximation of ri (t) and such that its application never produces constraint violations, i.e., ci (t) ∈ C i , ∀t ∈ ZZ+ , i ∈ A. A centralized solution Before introducing the distributed CG approach, let us briefly recall how the centralized CG works. A centralized solution of the above stated CG design problem have been proposed in [11]. The solution is based on the following arguments: by linearity, one is allowed to separate the effects of the initial conditions and inputs from those of disturbances, e.g. x(t) = x(t)+˜ x(t), where x(t) is the disturbance-free component and x ˜(t) depends only on the disturbances. Then, in the sequel we adopt the following notation xg := (In − Φ)−1 Gg, y g := Hy xg , cg := Hc xg +Lg (18) for the disturbance-free equilibrium solutions of (17) to a constant command g(t) ≡ g, with g ∈ IRm . Consider next the following set recursion C0:=C ∼ Ld D, Ck:=Ck−1 ∼ Hc Φ
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∞ ∩
Ck
(19)
k=0
where, for given sets A, E ⊂ Rn , A ∼ E ⊂ A is a proper restriction of A denoted in the literature as the P-difference between sets: A ∼ E := {x ∈ IRn : x + e ∈ A, ∀e ∈ E} (20) In [10] it has been proved that the sets Ck are nonconservative restrictions of C such that c(t) ∈ C∞ , ∀t ∈ ZZ+ , implies that c(t) ∈ C, ∀t ∈ ZZ+ . Thus, one is allowed to consider the disturbance-free system evolutions only and adopt a “worst case” approach. For reasons which have been clarified in [11], it is convenient to introduce the following sets for a given δ > 0 { } C δ := C∞ ∼ Bδ , Wδ := g ∈ IR2m : cg ∈ C δ (21) where Bδ is a ball of radius δ centered at the origin and Wδ represents the set of all commands whose corresponding steady-state solutions cg satisfy the constraints with margin δ > 0. Then, if we consider the following family of constant virtual command sequences g(·) = {g(k) ≡ g ∈ Wδ , ∀k ∈ ZZ+ } (22) the disturbance-free state x ¯(t) and c-vector evolutions emanating from x ¯(t) under a whatever constant command g are given by
x ¯(k, x(t), g) :=
k
Φ x(t) +
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Φk−i−1 Gg
(23)
c¯(k, x(t), g) := Hc x ¯(k, x(t), g) + Lg where c¯(k, x(t), g) has to be understood as the disturbancefree virtual evolution at the virtual time instant k (opposite to the real time t) of the constrained vector c, from the initial condition x(t) (applied at virtual time zero) under the constant command g. Finally, for any k¯ ≥ 0 we define the convex and closed set of admissible virtual sequences ¯ = Wδ ∩ G(x, k) ¯ V(x, k) (24) where { } ¯ = g ∈ IRm : c¯(k, x, g) ∈ C ¯ , ∀k ∈ ZZ+ (25) G(x, k) k+k Therefore, provided that V(¯ x(t), 0) is nonempty, the centralized CG action can be chosen as the solution of the following constrained optimization problem gˆ(t)=arg min ∥g − r(t)∥2Ψg (26) g∈V(x(t),0)
where Ψg = ΨTg > 0m and ∥v∥Ψ := v T Ψv. Such a solution represents the best feasible approximation of r(t) which, if constantly applied from t onwards never produces constraints violation. A. Distributed Sequential CG (S-CG) Here we introduce a distributed CG scheme based on the above centralized CG approach. We assume the agents in Fig. 1 are connected via a communication network. Such a network is modeled by a communication graph: an undirected graph G = (A, B), where A denotes the set of the N subsystems and B ⊂ A × A the set of edges representing the communication links among agents. We will assume that each agent acts as a gateway in redistributing data among the other, no directly-connected, agents. Then, at each time instant t, each i-th agent is aware of the following vectors: T ξi (t − 1)=[g1T (t − di,1 ), . . . , giT (t − 1), . . . , gN (t − di,N )]T T T T ϑi (t − 1)=[x1 (t − di,1 ), . . . , xi (t), . . . , xN (t − di,N )]T
As a consequence, the most recent common information regarding the measurement of the overall state available to each agent is x(t − dmax,i ) where dmax,i = maxj∈A di,j . Next, let assume, without loss of generality, that the sequence H = {1, 2, ..., N − 1, N } is a Hamiltonian cycle defined on G. The idea behind the approach is that only one agent per decision time is allowed to manipulate its local command signal gi (t) while all others are instructed to keep applying their old commands. After a new CG computation, the agent in charge transmits its local command and state to the next updating agent that is necessarily a neighbor. Such a polling policy implies that, eventually after a preliminary initialization cycle, at each time instant t, the following information are available to the generic i-th ”agent in charge” • the history of the aggregate vectors applied in the last N steps g(t − N + j), j = {1, . . . , N − 1} • the measurement of the state at time t − dmax,i , i.e., x(t − dmax,i ) By exploiting this information, it is possible to compute the estimation x ˆ(t) of the current free-disturbance state at time t by means of the following recursions
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Voltage on bus 3
In the distributed case we assume that four agents supervise the system by acting on specific components of g(t) and are provided with the communication network depicted in Figure 3. Specifically: • Agent 1 acts on EM V1 and directly receives state measurements about feeders F1 F4 and F5 . • Agent 2 acts on EM V2 and directly receives state measurements about the feeders F2 and F3 . • Agent 3 acts on JD1 (t) and JI1 (t) and directly receives state measurements about the feeders F4 and F5 . SV C C • Agent 4 acts on δD (t) and δISV (t) and directly re1 1 ceives state measurements about the feeders F1 . AGENT 1
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IV. S IMULATIONS In this section, both the centralized (CG) and distributed (S-CG) formulations related to the above voltage regulation problem will be applied. In the centralized case a single CG device is superimposed to the system. At each time t, it gets the state xp (t) of the overall system and computes the reference g(t) on the basis of the desired reference r(t) collecting the nominal set-points for the OLTC devices, the nominal current set-point for DG3 and the injected/absorbed current set-point for SVC, i.e., SV C,n SV C,n T n r(t) = [En , En , JD , JIn3 , δD , δI1 ] (29) 3 1
Voltage on bus 4
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where Ψi > 0 are weighting matrices, t mod N is the remainder of the integer division t/N . 2 More theoretical details about the S-CG can be found in [3].
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1.2 ELSE 1.2.1 APPLY gi (t) = gi (t − 1) 1.3 TRANSMIT ALL THE INFORMATION TO THE NEIGHBORS
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1.1.3 APPLY gi (t) AND T (t−1)]T 1.1.4 UPDATE g(t) = [g1T (t−1), ..., giT (t), ..., gN
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gi (t) = arg min(wi ) ∥ wi − ri (t) ∥2Ψi SUBJECT TO : T [g1T (t−1),...,wiT,...,gN (t−1)]T∈ V(ˆ x(t), dmax,i)
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Fig. 4. Voltage on loads C3 , C4 C6 C7 : (black dashed) without CG, (blue) CG, (green dash-dotted) S-CG
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Sequential-CG Algorithm (S-CG) - i-th Agent AT EACH TIME t 1.1 IF(t mod N ) == i 1.1.1 RECEIVE g(t − 1) AND x(t − 1) FROM THE PREVIOUS AGENT IN THE CYCLE H 1.1.2 SOLVE
have been considered for the network model (8). Moreover, for the sake of simplicity, all variables will be considered in pu units. The command g(t) is computed at each sampling step t, the latter having a period of 15 min. The following power profiles have been considered for the non-manipulable DG generators: • DG1 injects active power: PDG1 = 0.1; • DG2 injects active power: PDG2 = 0.15 and absorbs reactive power: QDG2 = 0.0726 at a constant power factor cos ϕDG = 0.9. The disturbance δI (t) acting on the loads Ci , i = 1, ..., 12 has been generated as a uniformly distributed sequence of random values satisfying the bounds defined in (7). The objective of these simulations is to show that the proposed distributed strategy is effective in maintaining the prescribed coordination constraints on relevant variables of the grid (see Figure 1) despite any allowable disturbance sequence d(t) possibly occurring. For space reasons, only few constrained variables are analyzed hereafter.
E MV [p.u.]
x ˆ(t − dmax,i ) = x(t − dmax,i ) (27) x ˆ(k + 1) = Φˆ x(k) + Gg(k), k = t − dmax,i , ..., t Then, by setting the parameter k¯ in (24) equal to the timedelay dmax,i , we can formulate the following distributed SCG algorithm:
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AGENT 3
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Fig. 3.
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The following data RCi = 14.4, XCi = 7,
RLi = 0.0678, XLi = 0.1152,
i = 1, ..., 12 i = 1, ..., 12.
Fig. 5.
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OLTC voltages: (blue) CG, (green dash-dotted) S-CG
Generator commands
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−0.06
Fig. 6. Injected/absorbed current on generator SVC device (left) and Injected/absorbed current on generator DG3 (right): (blue) CG, (green dashdotted).
Figure 4 (black dashed lines) shows the grid evolutions under the effect of the above mentioned exogenous signals without the action of any centralized or distributed CG supervising scheme. In this case, the OLTC device has a constant voltage reference value, no SVC unit is present and the DG3 generator is not furnished by any smart functionality. It is worth noticing that the constraints on loads C4 and C6 are fulfilled because of the presence of DG generators, while on feeder F1 and sub-feeder F4 , where DG is absent, the corresponding voltage constraints are violated. Figure 4 (blue and green dash-dotted lines) and Figure 6 depict the system evolutions under the same scenario when all smart devices are enabled and either the CG or the S-CG logics operate with δ = 10−6 and r(t) = [1, 1, 0.05, 0, 0, 0]. As a result, coordination always arises during the simulation, this being certified by the fact that all relevant variables remain always constrained inside their bounds. In fact, for both the CG and S-CG cases, the OLTC devices are able to keep the LV voltages within the prescribed ranges, having been assisted in this task by the actions of both the SVC and DG3 units. In particular, SVC injects a certain level of active/reactive current while DG3 introduces active power according to the load demands and voltage level on bus-bar B2 . In this respect, in Figs. 5,6 it is possible to note how the voltage level EM V2 is related to the current injected by DG3 : let focus on the green dash-dotted lines that represent the S-CG action. At the beginning, the EM V2 value in Fig. 5 is kept just under 1.02p.u. and the DG3 active current JD3 is 0.06p.u. in Fig. 6. After 5 hours, the EM V2 value is increased up to 1.03p.u.. In these conditions DG3 can decrease its injected active current value. After 15 hours EM V2 is gradually decreased down to 1p.u.. This action would produce lower voltages on sub-feeders F3 and F4 if they were not compensated. However, such a situation is avoided by a progressive increasing of the active current injected by DG3 . Finally, as far as the computational burdens are con-
cerned, the on-line CPU-times pertaining to the centralized and distributed CG schemes are 0.55 and 0.08 seconds per steps respectively. Note that the CPU-times related to the distributed algorithm corresponds to the computational time of a single agent. It is worth pointing out that the distributed S-CG scheme features a CPU-time reduction of approximatively one order of magnitude with respect to the centralized CG one. V. C ONCLUSIONS This paper has presented a study concerning voltage regulation problems in MV distribution grids in the presence of high penetration of DG. A multi-master/multi-slave distributed Command Governor (CG) approach has been used to design a supervisor scheme which was shown to be effective in providing a secondary voltage supervision layer for an effective and safe integration of green power generators into the distribution grid in spite of the variability and intermittent behavior of the renewable sources. It has been shown that an active orchestration of the set-points of available control devices and/or the modulation of the power amounts either provided by the distributed generators or consumed by the loads is an effective tool, under certain conditions, for maintaining the system operability in spite of unpredictable adverse conditions. R EFERENCES [1] F. Bignucolo, R. Caldon, and V. Prandoni, “Radial MV networks voltage regulation with distribution management system coordinated controller”, Electric Power Systems Research 78:(4), April 2008. [2] A. Casavola, G. Franz`e, D. Menniti and N. Sorrentino, “Voltage regulation in distribution networks in the presence of distributed generation: A voltage set-point reconfiguration approach”. Electric Power Systems Research 81: (1), January 2011. [3] F. Tedesco, A. Casavola, E. Garone, “Distributed Command Governor Strategies for Constrained Coordination of Multi-Agent Networked Systems” American Control Conference 2012, Montreal (CA), 2012, http://tedescof.wordpress.com/publication/. [4] Franz`e G., Tedesco F. , “Constrained Load/Frequency Control Problems in Networked Multi-Area Power Systems”. Journal of the Franklin Institute, 2011, Vol. 348, pp. 832-852. [5] A. Casavola , E. Garone , F. Tedesco , “Distributed Coordination-byConstraint Strategies for Multi-agent Networked Systems”, IEEE CDC 2011, Orlando, USA, Dec. 2011. [6] A. Casavola, E. Garone , F. Tedesco , “A Liveliness Analysis of a Distributed Constrained Coordination Strategy for Multi-Agent Linear Systems”, IEEE CDC 2011, Orlando, USA, Dec. 2011. [7] M. Iliˇc, J. Zabourszky, Dynamics and Control of Large Electric Power Systems, Wiley Interscience, 1997. [8] Casavola A. , Franz`e G. , Carelli N., “ Voltage regulation in networked electrical power systems for distributed generation: a constrained supervisory approach”. Proceedings of ”NOLCOS 2007”, Pretoria, South Africa, 2007. [9] L.V. Barboza, G.P. Dimuro, R.H.S. Reiser, “Towards interval analysis of the load uncertainty in power electric systems”, 8th International Conference on Probabilistic Methods Applied to Power Systems, Ames, Iowa, 2004, pp. 538-544. [10] E.G. Gilbert, I. Kolmanovsky and K. Tin Tan. “Discrete-time Reference Governors and the Nonlinear Control of Systems with State and Control Constraints”. Int. Journ. on Robust and Nonlinear Control, 5, pp. 487-504, 1995. [11] A. Bemporad, A. Casavola and E. Mosca “Nonlinear control of constrained linear systems via predictive reference management”. IEEE Transaction on Automatic Control,42(3), 340-349, 1997.
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