University of Groningen Distributed Control, Optimization, Coordination of Smart Microgrids Silani, Amirreza

Distributed Control, Optimization, Coordination of Smart Microgrids

Silani, Amirreza

DOI:

10.33612/diss.156215621

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Silani, A. (2021). Distributed Control, Optimization, Coordination of Smart Microgrids: Passivity, Output Regulation, Time-Varying and Stochastic Loads. University of Groningen.

https://doi.org/10.33612/diss.156215621

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3

Passivity Properties for Regulation of DC

Networks with Stochastic Load Demand

This chapter is based on our work presented in [125]. In this chapter, we present new (stochastic) passivity properties for Direct Current (DC) power networks, where the unknown and unpredictable load demand is modeled by a stochastic process. More precisely, the considered power network consists of distributed generation units supplying ZIP loads, i.e., nonlinear loads comprised of impedance (Z), current (I) and power (P) components. Differently from the majority of the results in the literature, where each of these components is assumed to be constant, we consider time-varying loads whose dynamics are described by a class of stochastic differential equations. Finally, we prove that an existing distributed control scheme achieving current sharing and (average) voltage regulation ensures the asymptotic stochastic stability of the controlled network.

3.1

Introduction

The recent wide spread of renewable energy sources, electronic appliances and batter-ies (including for instance EVs) motivates the design and operation of DC networks, which are generally more efficient and reliable than AC networks, attracting growing research interest [3] (see Subsection 1.2.1 for more details about the literature review of DC networks).

In [1, 5, 7–10, 31, 32], the load components are assumed to be constant. However, it is well known that electric loads are in practice time-varying and, due to the random and unpredictable diversity of usage patterns, it is more realistic to consider unknown time-varying loads described for instance by stochastic processes (see for instance [33–35]). Indeed, it is vital to model the loads by stochastic processes since the stochastic behavior of the loads may lead to fatal stability disruptions in power networks controlled by control schemes which do not consider stochastic load models [126]. In [42], a control scheme is proposed to regulate the renewable energy sources and energy storage devices of a DC network with stochastic Z loads. The control strategy requires the design of a filter for estimating the load resistance. A cascade control system for the energy management of DC microgrids with I loads is presented in [36], where the proposed control scheme includes an adaptive estimation of the quasi-stochastic load current profiles. In [37], a droop control scheme is designed for DC microgrids with stochastic Z loads. Moreover, in some papers, the Stochastic Differential Equations (SDEs) have been used for modeling the loads and other uncertainties in power system networks (see for instance [38–40]). In [41], the random load characteristic is considered to develop a stochastic model for voltage stability analysis. A stochastic power system model based on stochastic differential equations is presented in [39] to consider the uncertain factors such as load levels and system faults. In [40], a systematic and general approach to model power systems as continuous stochastic differential-algebraic equations is proposed and it justifies the need for stochastic models in power system analysis. In [127], the electric water heating loads are modeled by Markov processes and the transient and steady-state behavior of these loads are analyzed and predicted. A mean field game theoretic based control scheme is proposed in [128] as a load control mechanism, where electric water heating loads are modeled by a Markovian jump-driven model. In [129], a quantilized mean field game approach for energy pricing in a power network with stochastic load demand is presented. Differently, in this chapter the components of ZIP loads in a DC network are modeled as the sum of unknown constants and the solution to SDEs, then using the stochastic passivity of the network the asymptotic stochastic stability of the power network controlled by the distributed control scheme proposed by [14] is proved.

The main contributions of this chapter are as follows: (i) each component of the ZIP load is modeled as the sum of an unknown constant and the solution to an SDE describing the load dynamics; (ii) sufficient conditions for the stochastic passivity of the open-loop system are presented, facilitating the interconnection with passive control systems; (iii) the asymptotic stochastic stability of the power network controlled by the distributed control scheme proposed by [14] is proved.

3.2. Problem Formulation 33

is formulated. In Section 3.3, the stochastic dynamics of the ZIP loads are introduced and the stochastic passivity properties of the considered DC network are shown, then the stochastic stability of the closed-loop system with an existing controller is proved. The simulation results are presented and discussed in Section 3.4, while some concluding remarks are gathered in Section 3.5.

3.2

Problem Formulation

In this chapter, we consider the DC network model introduced in Section 2.5 (see Table 2.1 for the description of the symbols and parameters) with a general nonlinear and stochastic load model including the parallel combination of the following unknown load components:

1. impedance component (Z): G∗ li+ ˆGli,

2. current component (I): Ili∗+ ˆIli,

3. power component (P): P∗ li+ ˆPli,

where G∗li, Ili∗, Pli∗∈ R>0are unknown constants (deterministic variables) and ˆGli, ˆIli,

ˆ

Pli: R≥0→ R are stochastic variables, the dynamics of which will be introduced in

Section 3.3. We use Z∗_{, I}∗_{and P}∗_{to denote the unknown constant components of the}

load1_{. Therefore, in presence of ZIP loads, I}

li(Vi)in (2.21) is given by

Ili(Vi) = (G∗li+ ˆGli)Vi+ Ili∗+ ˆIli+ Vi−1(P ∗

li+ ˆPli). (3.1)

Now, the dynamics of the overall network (2.24) can be rewritten as LgI˙g= −V + u

CgV = I˙ g+ AI − [Gl]V − Il− [V ]−1Pl

L ˙I = −A>V − RI,

(3.2)

where Gl= G∗l + ˆGl, Il= Il∗+ ˆIland Pl= Pl∗+ ˆPlwith G∗l, Il∗, Pl∗∈ R n

>0represent

the constant components of the load and ˆGl, ˆIl, ˆPl: R≥0 → Rnthe stochastic parts.

These dynamics are discussed in Section 3.3, where we assume that their steady-state solutions are equal to zero.

Now, we introduce the main control objectives of this chapter, i.e., current sharing and average voltage regulation. Assuming that the stochastic components of the

1_{For instance, in presence of a Z}∗

IP∗load, we have Ili(Vi) = G∗liVi+ Ili∗+ ˆIli+ Vi−1P

∗ li.

ZIP loads are zero at the steady-state, we notice that for a constant input u = u∗_{, the}

steady-state solution ( ¯Ig, ¯V , ¯I)to (3.2) satisfies

V = u∗ (3.3a) I_l∗+ [G∗_l] ¯V + [ ¯V ]−1P_l∗− Ig= AI (3.3b)

I = −R−1A>V . (3.3c) Note that in order to use Lyapunov stochastic stability analysis for the DC network affected by stochastic loads, the stochastic and deterministic terms of the SDEs de-scribing the stochastic load components at the equilibrium point should be identical to zero (see Definition 2.3). Otherwise, the stochastic stability of the overall system cannot be guaranteed. However, since the load components are the sum of stochastic variables and unknown constants, we do not lose the generality by assuming that the stochastic components of the loads are zero at the steady-state.

Because of different generation capacities, it is reasonable to require that the total load demand of the microgrid is fairly shared among all the different generation units as qiI¯gi = qjI¯gj, ∀i, j ∈ V, where qihas inverse relationship with the capacity

of DGU i. Then, we define the first objective concerning the steady-state value of the generated current as follows:

Objective 3.1. (Current sharing).

lim

t→∞Ig(t) = Ig= Q −1₁

ni∗g, (3.4)

where Q = diag(q1, . . . , qn), qi∈ R>0, ∀i ∈ Vand i∗gany constant value satisfying at the

steady-state i∗ g=

1_n>([G∗_l] ¯V +I_l∗+[ ¯V ]−1P_l∗) 1>

nQ−11n . Indeed, the latter implies that at the steady-state

the total generated current is equal to the total load demand, i.e., 1>nI¯g= 1>nIl( ¯V ).

Now, we observe that achieving objective 3.1, does not generally allow to per-form also voltage regulation. Indeed, the achievement of current sharing generally implies voltage differences between the nodes of the microgrid. As a consequence, a particular form of voltage regulation has been proposed in the literature, where the (weighted) average of the steady-state voltages is regulated towards the (weighted) average of the voltage references [13, 14]. Then, the second objective concerning the steady-state value of the voltages is defined as follows:

Objective 3.2. (Average voltage regulation).

lim

t→∞1 >

nQ−1V (t) = 1>nQ−1V = 1>nQ−1V∗, (3.5)

3.3. Stochastic Passivity of DC Networks 35

Before showing the stochastic properties of the controlled network, we assume that there exists a steady-state solution to (3.2):

Assumption 3.1. (Steady-state solution). There exists a constant input u∗and a

steady-state solution ( ¯Ig, ¯V , ¯I)to (3.2) satisfying (3.3) and achieving objectives 3.1 and 3.2.

3.3

Stochastic Passivity of DC Networks

In this section, we introduce the dynamics of the stochastic components of the load, i.e., ˆIl, ˆPland ˆGl, respectively, and verify the stochastic passivity of the open-loop

system. Finally, we prove that the control scheme proposed by [14] ensures the asymptotic stochastic stability of the controlled network.

We notice that the stochastic terms in (3.2) enter in a multiplicative manner. For this reason, an appropriate mathematical framework such as the Ito calculus frame-work introduced in Section 2.1, should be adopted to analyze such a model. In the following subsections we introduce the dynamics of the stochastic load components, which we model via an SDE in the Ito calculus framework. More precisely, we adopt the well-known Ornstein-Uhlenbeck process [130], which is indeed widely used for the description of physical phenomena (see for instance [33]). For the sake of exposition, we first consider Z∗IP∗loads, i.e., only the load current is described by a stochastic process. Later, we extend the results also to Z∗IP and ZIP loads, respectively.

3.3.1

Z

IP

loads

In this subsection, we consider Z∗IP∗loads type, i.e., equation (3.1) becomes Ili(Vi) =

G∗_liVi+ Ili∗+ ˆIli+ Vi−1Pli∗, where the SDE describing the dynamics of ˆIliis given by

d ˆIli= −µIliIˆlidt + σIliIˆlidwi, (3.6)

where µIli and σIli are positive constants. As mentioned in Section 3.2, it can be

noticed that the deterministic and stochastic parts of (3.6) at the equilibrium point are identical to zero. Now, in order to verify a passivity property of the considered network, we introduce an assumption on the parameters of the SDE (3.6) and restrict the trajectories to be inside a subset of the state-space.

Assumption 3.2. (Condition on the parameters of (3.6)). For all i ∈ V, the stochastic

parameters of (3.6) satisfy µIl> 1 2σ 2 Il− 1 2In, (3.7) where µIl = diag(µIl1, . . . , µIln)and σIl= diag(σIl1, . . . , σIln).

Let us define the set ΩZ∗_IP∗:= {(I_g, V, I, ˆI_l) : 1

2([G ∗

l]−Π−1)−[Pl∗][V ]−1[ ¯V ]−1> 0}.

Then, the stochastic passivity property of system (3.2) with Z∗IP∗loads is verified via the following lemma:

Lemma 3.1. (Stochastic passivity of(3.2), (3.6)). Let Assumption 3.2 hold and ΩZ∗_IP∗

be nonempty. System (3.2) with Z∗_IP∗_{loads and ˆI}

lgiven by (3.6) is stochastically (shifted)

passive with respect to the storage function SZ∗_IP∗(I_g, V, I, ˆI_l) = 1 2(Ig− ¯Ig) >_L g(Ig− ¯Ig) + 1 2(V − ¯V ) >_C g(V − ¯V ) +1 2(I − ¯I) >_{L(I − ¯I) +} 1 2 ˆ I_l>Π ˆIl, (3.8)

and supply rate (Ig− ¯Ig)>(u − u∗)for all the trajectories (Ig, V, I, ˆIl) ∈ ΩZ∗_IP∗, where

Πis a (suitable) positive definite diagonal constant matrix of appropriate dimensions and ( ¯Ig, ¯V , ¯I)satisfies (3.3).

proof. The Ito derivative of the storage function (3.8) satisfies

LSZ∗_IP∗ =(I_g− ¯I_g)>(−V + u + ¯V − ¯V ) + (V − ¯V )>(I_g+ AI − I_l∗− ˆI_l− G∗_lV − [V ]−1Pl∗+ ¯Ig− ¯Ig) + (I − ¯I)>(−A>V − RI + R ¯I − R ¯I) − ˆI_l>µIlΠ ˆIl+ 1 2 ˆ I_l>σ>σΠ ˆIl =(Ig− ¯Ig)>(u − u∗) − (I − ¯I)>R(I − ¯I) − ˆIl>(µIl− 1 2σ 2 Il− 1 2In)Π ˆIl −1 2 Π −1 2(V − ¯V ) + Π 1 2Iˆ_l
> Π− 1 2(V − ¯V ) + Π 1 2Iˆ_l
− (V − ¯V )> 1 2([G ∗ l] − Π −1_{) − [P}∗ l][V ] −1_{[ ¯V ]}−1
_{(V − ¯V ),} (3.9) along the solutions to (3.2), (3.6). Then, we can conclude that LSZ∗_IP∗(x) ≤ (u −

u∗)>(Ig− ¯Ig)for all the trajectories (Ig, V, I, ˆIl) ∈ ΩZ∗_IP∗. 

Remark 3.1. (Z∗_{I load). We observe that in presence of only Z}∗_{I loads, the result provided}

in Lemma 1 can be strengthened. Indeed, the absence of P loads implies that the system (3.2), (3.6) is stochastically passive for any sufficiently large Π.

3.3.2

Z

IP loads

In this subsection, we consider Z∗_{IP loads type, i.e., equation (3.1) becomes I} li(Vi) =

G∗_liVi+ Ili∗+ ˆIli+ Vi−1(P ∗

li+ ˆPli), where the SDE describing the dynamics of ˆPliis

given by

3.3. Stochastic Passivity of DC Networks 37

where µPli and σPli are positive constants. Also in this case, we notice that the

equilibrium points of the deterministic and stochastic parts of (3.10) are zero. Now, in order to verify a passivity property of the considered network, we introduce an assumption on the parameters of the SDE (3.10) and restrict the trajectories to be inside a subset of the state-space.

Assumption 3.3. (Condition on the parameters of (3.10)). For all i ∈ V, the stochastic

parameters of (3.10) satisfy

µPl> σ

2

Pl, (3.11)

where µPl= diag(µPl1, . . . , µPln)and σPl= diag(σPl1, . . . , σPln).

Let us define the set ΩZ∗_IP:= {(I_g, V, I, ˆI_l, ˆP_l) : 1

2([G ∗

l] − Π−1) − [Pl∗][V ]−1[ ¯V ]−1− 1

2ΣµPl[V ]

−2_{> 0}. Then, the stochastic passivity property of system (3.2) with Z}∗_IP

loads is verified via the following lemma:

Lemma 3.2. (Stochastic passivity of (3.2), (3.6), (3.10)). Let Assumptions 3.2, 3.3 hold

and ΩZ∗_IPbe nonempty. System (3.2) with Z∗IP loads and ˆI_l, ˆP_lgiven by (3.6), (3.10) is

stochastically (shifted) passive with respect to the storage function SZ∗_IP(I_g, V, I, ˆI_l, ˆP_l) = S_Z∗_IP∗(I_g, V, I, ˆI_l) +

1 2

ˆ

P_l>Σ ˆPl, (3.12)

and supply rate (Ig− ¯Ig)>(u − u∗)for all the trajectories (Ig, V, I, ˆIl, ˆPl) ∈ ΩZ∗_IP, where

Σis a (suitable) positive-definite diagonal constant matrix of appropriate dimensions. proof. The Ito derivative of the storage function (3.12) satisfies

LSZ∗_IP=(I_g− ¯I_g)>(u − u∗) − (I − ¯I)>R(I − ¯I) − ˆI_l>(µ_I l− 1 2σ > IlσIl− 1 2In)Π ˆIl −1 2(Π −1 2(V − ¯V ) + Π 1 2Iˆ_l)>(Π− 1 2(V − ¯V ) + Π 1 2Iˆ_l) −1 2(V − ¯V ) >_(G l− Π−1)(V − ¯V ) − ˆPl>ΣµPlPˆl +1 2 ˆ P_l>σ>_PlσPlΣ ˆPl+ (V − ¯V ) >_{(−[V ]( ˆP} l+ Pl∗) + [ ¯V ] −1_P∗ l) =(Ig− ¯Ig)>(u − u∗) − (I − ¯I)>R(I − ¯I) − ˆIl>(µIl− 1 2σ 2 Il− 1 2In)Π ˆIl −1 2 Π −1 2(V − ¯V ) + Π 1 2Iˆ_l
> Π− 1 2(V − ¯V ) + Π 1 2Iˆ_l
−1 2 ˆ P_l>(µPl− σ 2 Pl)Σ ˆPl− 1 2  Σ12µ 1 2 Pl ˆ Pl− Σ− 1 2µ− 1 2 Pl (−1 > n + ¯V >_{[V ]}−1₎>>  Σ12_µ 1 2 Pl ˆ Pl− Σ− 1 2_µ− 1 2 Pl (−1 > n + ¯V >_{[V ]}−1₎>_{− (V − ¯V )}>1 2([G ∗ l] − Π −1₎ − [P_l∗][V ]−1[ ¯V ]−1−1 2ΣµPl[V ] −2_{(V − ¯V ),} (3.13)

along the solutions to (3.2), (3.6), (3.10). Then, we can conclude that LSZ∗_IP(x) ≤

(u − u∗)>(Ig− ¯Ig)for all trajectories (Ig, V, I, ˆIl, ˆPl) ∈ ΩZ∗_IP. 

3.3.3

ZIP loads

In this subsection, we consider the full ZIP loads, where the SDE describing the dynamics of ˆGliis given by

d ˆGli= −µGliGˆlidt + σGliGˆlidwi, (3.14)

where µGliand σGliare positive constants. Also in this case, the deterministic and

stochastic parts of (3.14) at the equilibrium point are identical to zero. Note that the SDEs (3.6), (3.10), (3.14) satisfy the conditions of Theorem 2.1, i.e., Lipschitz and linear growth conditions. Therefore, there exist unique solutions to the SDEs (3.6), (3.10), (3.14). Now, in order to verify a passivity property of the considered network, we introduce an assumption on the parameters of the SDE (3.14) and restrict the trajectories to be inside a subset of the state-space.

Assumption 3.4. (Condition on the parameters of (3.14)). For all i ∈ V, the stochastic

parameters of (3.14) satisfy

µGl> σ

2

Gl, (3.15)

where µGl = diag(µGl1, . . . , µGln)and σGl= diag(σGl1, . . . , σGln).

Note that the Assumptions 3.2–3.4 are required to show the stochastic passivity of (3.2), (3.6), (3.10), (3.14). Indeed, the SDEs (3.6), (3.10), (3.14) can be stochastically unstable without considering any assumptions on their parameters. Thus, it is hard to show the stochastic passivity of (3.2), (3.6), (3.10), (3.14) and design a control scheme guaranteeing the stochastic stability of the network with stochastically unstable loads. Also, it completely makes sense in practical applications to consider that the loads’s behavior is stochastically stable.

Let us define the set ΩZIP:= {(Ig, V, I, ˆIl, ˆPl, ˆGl) : ₂1([G∗l]−Π−1)−[Pl∗][V ]−1[ ¯V ]−1− 1

2ΣµPl[V ]

−2_{− [V − ¯V ]}2_Λµ

Gl > 0}. Then, the stochastic passivity property of

sys-tem (3.2) with ZIP loads is verified via the following lemma.

Lemma 3.3. (Stochastic passivity of (3.2), (3.6), (3.10), (3.14)). Let Assumptions 3.2–

3.4 hold and ΩZIPbe nonempty. System (3.2), with ˆIl, ˆPl, ˆGlgiven by (3.6), (3.10), (3.14) is

stochastically (shifted) passive with respect to the storage function SZIP(Ig, V, I, ˆIl, ˆPl, ˆGl) = SZ∗_IP(I_g, V, I, ˆI_l, ˆP_l) +

1 2 ˆ

G>_l Λ ˆGl, (3.16)

and supply rate (Ig− ¯Ig)>(u − u∗)for all the trajectories (Ig, V, I, ˆIl, ˆPl, ˆGl) ∈ ΩZIP, where

3.3. Stochastic Passivity of DC Networks 39

proof. The Ito derivative of the storage function (3.16) satisfies LSZIP=(Ig− ¯Ig)>(u − u∗) − (I − ¯I)>R(I − ¯I) − ˆIl>(µIl−

1 2σ 2 Il− 1 2In)Π ˆIl −1 2(Π −1 2(V − ¯V ) + Π 1 2Iˆ_l)>(Π− 1 2(V − ¯V ) + Π 1 2Iˆ_l) −1 2 ˆ P_l>(µPl− σ 2 Pl)Σ ˆPl −1 2  Σ12µ 1 2 Pl ˆ Pl− Σ− 1 2µ− 1 2 Pl (−1 > n + ¯V>[V ]−1)> > Σ12µ 1 2 Pl ˆ Pl− Σ− 1 2µ− 1 2 Pl (−1 > n + ¯V>[V ]−1)>−1 2 ˆ G>_l (µGl− σ 2 Gl)Λ ˆGl− 1 2  Λ12µ 1 2 Gl ˆ Gl+ Λ −1 2 µ −1 2 Gl(V − ¯V ) (V − ¯V )>1n > Λ12µ 1 2 Gl ˆ Gl+ Λ −1 2 µ −1 2 Gl(V − ¯V )(V − ¯V ) >₁ n  − (V − ¯V )> 1 2([G ∗ l] − Π −1_{) − [P}∗ l][V ] −1_{[ ¯V ]}−1₋1 2ΣµPl[V ] −2_{− [V − ¯V ]}2_Λµ Gl  (V − ¯V ), (3.17) along the solutions to (3.2), (3.6), (3.10), (3.14). Then, we can conclude that LSZ∗_IP∗(x)

≤ (u − ¯u)>(Ig− ¯Ig)for all trajectories (Ig, V, I, ˆIl, ˆPl, ˆGl) ∈ ΩZIP. 

Remark 3.2. (Sufficient conditions) Note that the matrices Π, Σ, Λ are (suitable)

positive-definite diagonal constant matrices corresponding to the stochastic states in the Lyapunov functions (3.8), (3.12), (3.16), respectively. More precisely, since the SDEs (3.6), (3.10), (3.14) are added to the overall system, the terms 12Iˆ

>

l Π ˆIl,1₂Pˆl>Σ ˆPl,1₂Gˆ>l Λ ˆGlare considered

in the Lyapunov functions (3.8), (3.12), (3.16), respectively, in order to deal with the stochas-ticity. Then, we observe that for large Π and small Σ, Λ, the sufficient conditions for ΩZIPto

be nonempty are similar to the (well-known) sufficient conditions provided in the literature for DC networks with (non-stochastic) Z∗I∗P∗loads, i.e., high voltage and large values of the load conductance [6, 9, 10, 15].

3.3.4

Closed-loop analysis

In this subsection, we consider the distributed controllers proposed by [14] and show that the closed-loop system is asymptotically stochastically stable, achieving at the steady state objectives 3.1 and 3.2.

Now, for the sake of completeness, we report below the controller proposed by [14] for i ∈ V, i.e., τξiξ˙i= − X j∈Ncom i ρij(qiIgi− qjIgj) τηiη˙i= − ηi+ Igi (3.18) ui= − Ki(Igi− ηi) + qi X j∈Ncom i ρij(ξi− ξj) + Vi∗.

where τξi, τηi, Ki ∈ R>0 are design parameters and Nicomis the set of DGUs

com-municating with DGU i via a communication network (possibly different from the electric network), where ρij∈ R>0are the edge weights. The distributed controller

(3.18) can be written compactly for all i ∈ V as

τξξ = − L˙ comQIg (3.19a)

τηη = − η + I˙ g (3.19b)

u = − K(Ig− η) + QLcomξ + V∗, (3.19c)

where τξ, K, τηare positive definite diagonal matrices of appropriate dimensions and

Lcom_{is the weighted Laplacian matrix associated with the communication network.}

We notice that the port-Hamiltonian framework can be utilized to generalize the proposed approach in this chapter for more general power network models [123, 124]. Indeed, the the controller design and stability analysis of the closed-loop system can be accomplished via shifted (nonlinear) Hamiltonians (Bregman functions) for more general power network models. However, the port-Hamiltonian based control design and stability analysis are left for future research.

In the following theorem, we show that the closed-loop system is asymptotically stochastically stable and the average voltage regulation and current sharing objectives are attained.

Theorem 3.1. (Closed-loop analysis). Let Assumptions 3.1–3.4 hold and ΩZIPbe

non-empty. System (3.2) with ˆIl, ˆPl, ˆGlgiven by (3.6), (3.10), (3.14) and controlled by (3.19) is

asymptotically stochastically stable and achieves objectives 3.1 and 3.2 for all the trajectories (Ig, V, I, ˆIl, ˆPl, ˆGl) ∈ ΩZIP.

proof. Let x := (Ig, V, I, ˆIl, ˆPl, ˆGl, ξ, η)and consider the following storage function

S(x) = SZIP(Ig, V, I, ˆIl, ˆPl, ˆGl) + 1 2(ξ − ¯ξ) >_τ ξ(ξ − ¯ξ) + 1 2(η − ¯η) >_τ η(η − ¯η), (3.20)

3.3. Stochastic Passivity of DC Networks 41

The Ito derivative of the storage function (3.20) satisfies LS =(Ig− η)>K(Ig− η) − (I − ¯I)>R(I − ¯I) − ˆIl>(µIl−

1 2σ 2 Il− 1 2In)Π ˆIl −1 2(Π −1 2(V − ¯V ) + Π 1 2Iˆ_l)>(Π− 1 2(V − ¯V ) + Π 1 2Iˆ_l) −1 2 ˆ P_l>(µPl− σ 2 Pl)Σ ˆPl −1 2  Σ12µ 1 2 Pl ˆ Pl− Σ− 1 2µ− 1 2 Pl (−1 > n + ¯V >_{[V ]}−1₎>>_Σ1 2µ 1 2 Pl ˆ Pl− Σ− 1 2µ− 1 2 Pl (−1 > n + ¯V>[V ]−1)>−1 2 ˆ G>_l (µGl− σ 2 Gl)Λ ˆGl− 1 2  Λ12µ 1 2 Gl ˆ Gl+ Λ− 1 2µ− 1 2 Gl (V − ¯V ) (V − ¯V )>1n > Λ12µ 1 2 Gl ˆ Gl+ Λ− 1 2µ− 1 2 Gl (V − ¯V )(V − ¯V ) >₁ n  − (V − ¯V )> 1 2([G ∗ l] − Π−1) − [Pl∗][V ]−1[ ¯V ]−1− 1 2ΣµPl[V ] −2_{− [V − ¯V ]}2_Λµ Gl  (V − ¯V ), (3.21) along the solutions to the closed-loop system (3.2), (3.6), (3.10), (3.14), (3.19). Then, it follows that LS ≤ 0, for all the trajectories (Ig, V, I, ˆIl, ˆPl, ˆGl) ∈ ΩZIP. Then, as a

preliminary result we can conclude that the solutions to the closed-loop system (3.2), (3.6), (3.10), (3.14), (3.19) are bounded. Moreover, according to LaSalle’s invariance principle, these solutions converge to the largest invariant set contained in Ψ := {Ig, I, V, ˆIl, ˆPl, ˆGl, ξ, η : Ig= η, I = ¯I, V = ¯V , ˆIl= ˆPl= ˆGl=0}. Hence, the behavior

of the closed-loop system (3.2), (3.6), (3.10), (3.14), (3.19) on the set Ψ can be described by LgI˙g= − ¯V + QLcomξ + V∗ (3.22a) 0 = Ig+ AI − Il∗− [G ∗ l] ¯V − [ ¯V ] −1_P∗ l (3.22b) 0 = − A>V − RI¯ (3.22c) d ˆIl=0 (3.22d) d ˆPl=0 (3.22e) d ˆGl=0 (3.22f) τξξ = − L˙ comQIg (3.22g) τηη = 0.˙ (3.22h)

It follows from (3.22h) that η is constant on the largest invariant set. Then, Ig

is also constant. Since Igis constant, it follows from (3.22g) that LcomQIgis also

constant, implying that ξ would increase unbounded if Lcom_QI

g6= 0, contradicting

the preliminary result on the boundedness of the solutions. Then, Lcom_{Q ¯I} gmust

necessarily be equal to zero, implying that the objective of current sharing is achieved (see objective 3.1). Since Igis constant, we can pre-multiply (3.22a) by 1>nQ−1and

Node 1

Node 2

Node 4

Node 3

Figure 3.1: Topology of a DC microgrid. The arrows denote the positive direction of Ig; the dashed lines represent the communication links.

obtain 1>nQ−1V = 1¯ >nQ−1V∗, i.e., average voltage regulation (see objective 3.2).



Note that, in analogy with Theorem 1 and by virtue of Lemmas 1 and 2, similar results can be proved for the cases of Z∗_IP∗_{and Z}∗_{IP loads, respectively.}

3.4

Simulation Results

In this section, the performance of the proposed method is evaluated via a case study example. We consider the closed-loop system (3.2), (3.6), (3.10), (3.14), (3.19) describ-ing a DC network composed of 4 nodes (i.e., DGUs) depicted in Figure 3.1. The parameters of the DC network are taken from [13]. The parameters of the controllers are chosen as Ki = 0.4, τηi= 0.005, τξi= 1, i = 1, ..., 4. The initial values of Pl∗are

(25, 10, 25, 20)W. Their variations at the time instant t = 1s are ∆P∗

l = (8, 5, −8, 3)W,

while the initial values of G∗

l, and Il∗are (0.07, 0.045, 0.06, 0.08) Ω−1and (8, 4, 5, 12) A,

respectively. The stochastic parameters related to the load current, power and con-ductance are selected as µIli = 2.5, σIli = 1, µPli = 2, σPli = 0.7, µGli = 1.3, σGli =

0.2, i = 1, ..., 4, respectively. Figure 3.2 illustrates the currents generated by each DGU. We can see that the generated currents converge to the values that ensure fair current sharing (dashed lines). Moreover, Figure 3.3 shows the voltage at the PCC of each node. It can be seen that the weighted average voltage (dashed line) converges

3.4. Simulation Results 43

Figure 3.2: Time evolution of the currents generated at each node.

to the weighted average of the desired voltages, i.e., 380 V. The current flows through the transmission lines are represented in Figure 3.4.

To investigate the size in practice of the set ΩZIP, we consider L(V, Pl∗) := 1 2([G ∗ l]− Π−1) − [P_l∗][V ]−1[ ¯V ]−1− 1 2ΣµPl[V ] −2_{− [V − ¯V ]}2_Λµ Gl, with Π = 10 3 In, Σ = Λ =

10−7_In, [G∗l] = 0.045In. Figure 3.5 illustrates Lii(Vi, Pli∗)for different values of Vi

and P∗

li, where i = 2. It can be seen that Lii(Vi, Pli∗) > 0for a wide range of Viand

P_li∗, i.e., Vi ∈ [60, 800] V, and Pli∗ ∈ [5, 200] W. Therefore, the stability of the system

is guaranteed for a wide range of operating conditions. Moreover, we notice that by increasing the value of Πiior decreasing the values of Σiiand Λii, the range of

Viand Pli∗can be enlarged. Furthermore, we notice that these constant matrices are

parameters of the Lyapunov function and do not affect the parameters of the used controller.

Figure 3.3: Time evolution of the voltages at each node together with the correspond-ing average desired value (dashed lines).

3.5. Concluding Remarks 45

Figure 3.5: The simulation of Lii(Vi, Pli∗)for different Viand Pli∗.

3.5

Concluding Remarks

In this chapter, we have considered stochastic dynamics for the impedance (Z), current (I) and power (P) components of ZIP loads in DC power network. Then, we have verified the stochastic passivity of the considered system. In order to achieve average voltage regulation and current sharing, we have used an existing distributed control scheme, proving the asymptotic stochastic stability of overall system. In the next chapter, we model the components of loads as exosystems and use output regulation methodology to tackle the problem of voltage regulation in DC networks.