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

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below.

Document Version

Publisher's PDF, also known as Version of record

Publication date: 2021

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):

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

Copyright

Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policy

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum.

C

H

A

P

T

E

R

4

Output Regulation for Voltage Control in

DC Networks with Time-Varying Loads

This chapter is based on our work presented in [131]. In this chapter, we propose a novel control scheme for regulating the voltage in Direct Current (DC) power networks. More precisely, the proposed control scheme is based on the output regulation methodology and, differently from the results in the literature, where the loads are assumed to be constant, we consider time-varying loads whose dynamics are described by a class of nonlinear differential equations. We prove that the proposed control scheme achieves voltage regulation ensuring the stability of the overall network.

4.1

Introduction

Nowadays, the ever increasing electrification of transportation and buildings may indeed increase the demand fluctuations, putting a strain on the system stability [33, 132] (see Subsection 1.2.1 for more details about the literature review of DC networks). For this reason, the resilience and reliability of the power grid may benefit from the design and analysis of control strategies that theoretically guarantee the system stability in presence of time-varying loads [42,43]. Indeed, [33] and [132] show that loads can be described for instance by stochastic processes (e.g. Ito calculus) or dynamical models, respectively, attracting much research attention (see for instance

[41] for AC networks and [42, 43] for DC networks). However, [42, 43] do not provide any stability or convergence guarantee. In Chapter 3, the load components in DC networks are described by a class of stochastic differential equations. Differently, in this chapter we adopt a load model similar to but more general than the one in [41], and propose a control scheme based on the output regulation methodology [77], guaranteeing voltage regulation in presence of time-varying loads.

The main contributions of this chapter are as follows: (i) the voltage control problem in DC networks including time-varying loads is formulated as a standard output regulation problem; (ii) we consider time-varying impedance and current load components; (iii) we describe each load component as the output of a large class of nonlinear dynamical exosystem, as it is customary in output regulation theory [77]; (iv) the proposed control scheme achieves voltage regulation ensuring the stability of the overall network.

The rest of this chapter is organized as follows. The control problem is formulated in Section 4.2. In Section 4.3, the output regulation methodology is recalled and the control scheme designed, guaranteeing voltage regulation and stability of the overall network. The simulation results are presented and discussed in Section 4.4, while some concluding remarks are gathered in Section 4.5.

4.2

Problem Formulation

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

1. impedance component (Z): Gli,

2. current component (I): Ili,

where Gli, Iliare time-varying signals. More precisely, we assume that Gli, Iliare the

outputs of nonlinear dynamical exosystems, whose dynamics are introduced in the next subsection. Therefore, in presence of ZI loads, Ili(Vi)in (2.21) is given by

Ili(Vi) = GliVi+ Ili. (4.1)

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

CgV = I˙ g+ AI − [Gl]V − Il

L ˙I = −A>V − RI,

(4.2)

where Gland Ilare the output vectors of the exosystems that we introduce in the

4.2. Problem Formulation 49

4.2.1

Exosystems model

The electric loads in DC networks are in practice time-varying [132]. In this chapter, we consider the dynamics of the components of the ZI load, i.e., Gl, Il, as the outputs

of (known) nonlinear dynamical exosystems, as it is customary in output regulation theory [77, 78]. Let y denote G or I in case of Z or I loads, respectively. Then, the exosystem dynamics can be expressed as follows:

˙ dayi= 0 ˙ db_yi= syi(dbyi) yli= Γyicol dayi, dbyi
, (4.3) where da yi : R≥0 → R, d b yi : R≥0 → R

nd_{are the states of the exosystem describing}

the constant and time-varying components of yli, respectively, syi: Rnd → Rnd, and

Γyi∈ R1×(nd+1). Then, (4.3) can be written compactly as

˙

dy = Sy(dy)

yl= Γydy,

(4.4) where dy : R≥0→ Rn(nd+1)is defined as dy:= col(day1, dby1, . . . , dayn, dbyn), yl: R≥0→

Rn, Sy : Rn(nd+1) → Rn(nd+1)is defined as Sy := col(0, sy1, . . . , 0, syn), and Γy :=

blockdiag(Γy1, . . . , Γyn) ∈ Rn×n(nd+1).

4.2.2

Control objective

In this section, we introduce the main control objective of this chapter, i.e., voltage regulation. First, we notice that for a constant input u, the steady-state solution (Ig, V , I, dI, dG)to (4.2) and (4.4) satisfies V = u (4.5a) ΓIdI+ [ΓGdG]V − Ig= AI (4.5b) I = −R−1A>V (4.5c) 0 = SI(dI) (4.5d) 0 = SG(dG). (4.5e)

The main control objective concerning the steady-state value of the voltage is defined as follows:

Objective 4.1. (Voltage regulation).

lim

t→∞V (t) = V

∗_{, (4.6)}

4.3

The Controller Design Based on Output

Regulation Problem

In this section, we formulate the voltage control problem as a standard output regulation problem [77] in order to design a control scheme achieving Objective 4.1. Let the network state x : R≥0 → Rm+2n and the exosystems state d : R≥0 →

R2n(nd+1) _{be defined as x := col(I}

g, V, I) and d := col(dI, dG), respectively, and

u : R≥0 → Rn be the control input. Then, we can rewrite (4.2) and (4.4) as the

following composite system: ˙

x = f (x, d) + g(x, d)u (4.7a) ˙

d = S(d) (4.7b)

h(x, d) = V − V∗, (4.7c) where h(x, d) is the output mapping, S(d) := col SI(d), SG(d)
, g(x, d) := col(L−1g ,

0n×n, 0m×n)and f (x, d) :=   −L−1 g V C_g−1 Ig+ AI − [ΓGdG]V − ΓIdI
L−1 − A>_{V − RI}
 .

Now, we compute the relative degree of system (4.7), which will be used in the following subsections for analyzing the zero dynamics of system (4.7). Let

fa(x, d) := col(f (x, d), S(d))

ga(x, d) := col(g(x, d), 02n(nd+1)×n),

(4.8) then, based on the definition of relative degree for Multi Input Multi Output (MIMO) systems [77, Definition 2.47], the relative degree of the system (4.7) is computed in the following lemma.

Lemma 4.1. (Relative degree of system(4.7)). For each i = 1, . . . , n, the i-th output hi

of system (4.7) has relative degree ri = 2for all the trajectories (x, d).

proof. System (4.7) satisfies

Lgah(x, d) = 0n×nIn0n×m0n×2n(nd+1)
ga(x, d) = 0n×n Lfah(x, d) = 0n×nIn0n×m0n×2n(nd+1)
fa(x, d) = C_g−1(Ig+ AI − [ΓGdG]V − ΓIdI) LgaLfah(x, d) = Cg−1 ∗ ∗ ∗
ga(x, d) = C_g−1L−1_g , (4.9)

4.3. The Controller Design Based on Output Regulation Problem 51

which concludes the proof.

Before introducing the output regulation methodology, we show in the following lemma that there exists a state-feedback controller that asymptotically stabilizes system (4.7a) with constant loads.

Lemma 4.2. (Stabilizability of system(4.7a) with constant loads). Consider system (4.7a) with d = d∗_{, d}∗ _{= col(d}∗

I, d∗G) ∈ R

2n(nd+1) _{being any constant vector, such that}

[Gl] = [ΓGd∗G]and Il= ΓId∗I. Let Kx:= −K 0n×n 0n×m
, where K ∈ Rn×nis a

positive definite diagonal matrix. Then, system (4.7a) in closed-loop with the state-feedback controller

u = Kxx (4.10)

asymptotically converges to the equilibrium point (Ig, V , I), satisfying (4.5a)–(4.5c).

proof. Consider the following Lyapunov function

S(x) = (Ig− Ig)>Lg(Ig− Ig) + (V − V )>Cg(V − V )

+ (I − I)>L(I − I). (4.11) Then, the derivative of the Lyapunov function (4.11) along the solutions to (4.7a) satisfies

˙

S(x) = − (Ig− Ig)>K(Ig− Ig) − (V − V )>[ΓGd∗G](V − V )

− (I − I)>_{R(I − I)}

≤ 0, (4.12)

where the inequality follows from K, R > 0 and [ΓGd∗G] ≥ 0. Then, as a preliminary

result we can conclude that the solutions to the closed-loop system (4.7a), (4.10) are bounded. Moreover, according to LaSalle’s invariance principle, these solutions converge to the largest invariant set contained in Ω := {Ig, I, V : Ig = Ig, I = I}.

Hence, the behavior of the closed-loop system (4.7a), (4.10) on the set Ω can be described by

0 = −V − KIg (4.13a)

C ˙V = Ig+ AI − [ΓGd∗G]V − Il (4.13b)

0 = −A>V − RI. (4.13c) Then, it follows from (4.13a) that V is also constant on the largest invariant set Ω, concluding the proof.

In the following, we briefly recall for the readers’ convenience some concepts of the output regulation methodology. Then, we propose a control scheme for the problem of voltage regulation in DC networks including time-varying loads.

4.3.1

Output regulation methodology

Following [77, Section 3.2] and Section 2.2, we define the nonlinear output regulation problem for system (4.7) as follows:

Problem 4.1. (Nonlinear output regulation). Let the initial condition x(0), d(0)
of system (4.7) be sufficiently close to the equilibrium point x, d
satisfying (4.5). Then, design a static state feedback controller

u(t) = k(x(t), d(t)), (4.14) such that the closed-loop system (4.7), (4.14) has the following two properties:

Property 4.1. The trajectories col x(t), d(t)
of the closed-loop system exist and are bounded for all t ≥ 0,

Property 4.2. The trajectories col x(t), d(t)
of the closed-loop system satisfy

limt→∞h(x, d) = 0n, achieving Objective 4.11.

Now, in analogy with [77, Assumption 3.10_{], we introduce the following}

assump-tion.

Assumption 4.1. (Stability of exosystem). The equilibrium d of the exosystem(4.7b) is Lyapunov stable and there exists an open neighborhood D of d = d in which every point is Poisson stable [77, Remark 3.2].

Then, according to [77, Theorem 3.8], the solvability of Problem 4.1 is established in the following theorem.

Theorem 4.1. (Solvability and regulator equation). Let Assumption 4.1 hold.

Prob-lem 4.1 is solvable if and only if there exist smooth functions x(d) and u(d) defined for d ∈ D such that

∂x(d)

∂d S(d) = f (x(d), d) + g(x(d), d)u(d) (4.15a) 0n= h(x(d), d). (4.15b)

proof. See [77, Theorem 3.8].

1_{Note that Property 4.2 implies x = col(Ig, V}∗

4.3. The Controller Design Based on Output Regulation Problem 53

4.3.2

Controller design for the power network

In this subsection, a novel control scheme is designed for solving Problem 4.1 and, consequently, achieving Objective 4.1 in presence of time-varying loads. More pre-cisely, we first analyze the zero dynamics of system (4.7) in order to make the regula-tor equation (4.15) simpler. Then, inspired by the output regulation theory [77], we present the proposed control scheme.

Let x(d) in (4.15) be partitioned as x(d) := col(xa_{(d), x}b_{(d)), with x}a_{(d) :=}

col Ig(d), V (d)
and xb(d) := I(d). Then, consider the following PDE:

∂xb(d)

∂d S(d) = % x

b_{(d), d
, (4.16)}

where

% xb(d), d
= −L−1_(A>_V∗_{+ RI(d)). (4.17)}

Moreover, let Ig(d), V (d) be given by

V (d) = V∗

Ig(d) = −AI(d) + [ΓGdG]V∗+ ΓIdI.

(4.18)

Recalling that for each i = 1, . . . , n, the i-th output hi of system (4.7) has relative

degree equal to 2 (see Lemma 4.1), equation (4.18) follows from considering the output and its first-time derivative being identically zero. In the following theorem, we propose a controller solving Problem 4.1.

Theorem 4.2. (Controller design). Let Assumption 4.1 hold. Consider system(4.7) in closed-loop with u = u∗_e x(d), d
+ Kx x − x(d)
, (4.19) where u∗_e x(d), d
= V∗+ LgAL−1A>V∗+ LgAL−1RI(d) + LgΓISI(dI) + Lg[V∗]ΓGSG(dG), (4.20)

and I(d) is the solution to (4.16). Then, the trajectories of the closed-loop system (4.7), (4.19) starting sufficiently close to (Ig, V∗, I, dI, dG)are bounded and converge to the set where

the voltage is equal to the corresponding desired reference value V∗_{, achieving Objective 4.1.}

proof. In analogy with [77, Theorem 3.26], we first compute the following matrix: He(x, d) =  h(x, d) Lfah(x, d)  . (4.21)

Then we notice that the solution to He(x, d) = 02nfor system (4.7) can be expressed

as follows:

V = V∗

Ig= −AI + [ΓGdG]V∗+ ΓIdI.

(4.22)

Therefore, there exist the partition xa_{:= col(I}

g, V ), xb := Iand sufficiently smooth

function δ(xb_{, d) := col(−AI + [Γ}

GdG]V∗+ ΓIdI, V∗)such that He(x, d)|xa=δ(xb,d)=

02n. Recalling that for each i = 1, . . . , n, the i-th output hiof system (4.7) has relative

degree equal to 2 (see Lemma 4.1), we compute the equivalent control input ue(x, d)

by posing the second-time derivate of the output mapping (4.7c) equal to zero, i.e., L2_f ah(x, d) + LgaLfah(x, d)ue(x, d) = 0n, (4.23) that is, ue(x, d) = V + LgCg−1[ΓGdG] Ig+ AI − [ΓGdG]V − ΓIdI
+ LgAL−1 A>V + RI
+ LgΓISI(dI) + Lg[V ]ΓGSG(dG). (4.24) Now, let u∗e(x, d) := ue(x, d)|xa=δ(xb,d). By replacing V and Ig in (4.24) with the

right-hand side of (4.22), we obtain

u∗e(x, d) = V∗+ LgAL−1A>V∗+ LgAL−1RI + LgΓISI(dI)

+ Lg[V∗]ΓGSG(dG).

(4.25) According to Lemma 4.1, the zero dynamics of (4.7) can be expressed as

L ˙I = − A>V∗− RI ˙

d = S(d). (4.26)

Now, we have %(xb_{, d) = −A}>_V∗_{− RI, with x}b_{= I. Then, we replace x}b_{with the}

solution xb_{(d)to (4.16); therefore, %(x}b_{(d), d)can be given by (4.17). Moreover, by}

observing that the matrix ∂%(x_∂xbb,d)|(x,d)=(x,d)= −Ris negative definite, then from [77]

we know that the solution to (4.15) exists and can be given by

x(d) =   −AI(d) + [ΓGdG]V∗+ ΓIdI V∗ I(d)   u(d) = u∗_e x(d), d
, (4.27)

where I(d) is the solution to (4.16) and u∗e x(d), d
is given by (4.20). Consequently,

4.4. Simulation Results 55 V1 V4 V2 V3 L12 R12 L14 R14 R34 L34 R23 L23 DGU 1 z }| { Ig1− Il1 Ig2− Il2 Ig4− Il4 Ig3− Il3

Figure 4.1: Scheme of the considered network with 4 power converters [13, 14].

imply that the trajectories of the closed-loop system (4.7), (4.19) starting sufficiently close to (Ig, V∗, I, dI, dG)are bounded and converge to the set where the voltage is

equal to V∗, achieving Objective 4.1.

Note that Assumption 4.1 is considered in Theorems 4.1 and 4.2 since [77, rem 2.28] is exploited in the proof of [77, Theorems 3.8 and 3.26]. Indeed, [77, Theo-rem 2.28] shows the boundedness of the difference between the system trajectories x(t)and the solution x(d) to the regulator equation (4.15) even if there exists no deterministic equilibrium point in Assumption 4.1. Thus, the results in Theorems 4.1 and 4.2 can be adopted even if there exists no deterministic equilibrium point in Assumption 4.1.

Remark 4.1. (Controller properties). Note that the structure of the control scheme

we propose in this chapter is more complex than other control schemes proposed in the literature [1, 5, 7–10, 31, 32]. More precisely, the proposed control scheme is distributed and requires some information about the network parameters. However, this higher complexity is associated with the more challenging control objective we achieve. Indeed, differently from [1, 5, 7–10, 31, 32], the proposed control scheme achieves voltage regulation in DC networks including time-varying rather than constant loads. Moreover, we notice that I(d) can be approximated via the approximation methods proposed for instance in [77, Chapter 4], [133, 134].

4.4

Simulation Results

In this section, the performance of the proposed method is assessed in simulation. We consider a DC network composed of 4 nodes as illustrated in Fig. 4.1, whose electric parameters are equal to those reported in [13, Tables II, III] and are identical or very

similar to those used in [10, 14, 31, 32, 135–137] for simulations and in [9, 138, 139] for experimental validation in DC networks test facilities. In the following, we consider two different scenarios. In the first one we assume that there is no mismatch between the actual load profile and the one generated by the exosystem. Then, in the second scenario we assume that there is a mismatch between the actual load profile and the one generated by the corresponding exosystem, showing that the controlled system is input-to-state stable (ISS) with respect to such a mismatch and the voltages are kept very close to the desired references.

Scenario 1.Let the system initially be at the steady-state with Il(0) = col 30, 15, 30,

26
A and Gl(0) = col 0.07, 0.05, 0.06, 0.08
Ω−1. Then, consider at the time instant

t = 1s the following load variations: variations ∆Il = 1.43 sin(0.08t − 0.12) +

0.45 sin(1.37t − 3.5) + 1 A for Nodes 1, 2 and 3, ∆Il = 12.41 sin(0.477t − 1.1) +

11.98 sin(0.495t + 1.97) + 0.5A for Node 4, and ∆Gl= 0.005∆IlΩ−1, i.e, the

exosys-tem (4.3) can be expressed as

˙ dayi= 0 ˙ dbyi=      0 −ωα yi 0 0 ωα yi 0 0 0 0 0 0 −ω_yiβ 0 0 ωβ_yi 0      dbyi yli= Γyicol dayi, d b yi
, (4.28) where da yi : R≥0 → R, d b yi : R≥0 → R

4_{are the states of the exosystem, ω}α yi, ω

β yiare

equal to 0.08 and 1.37 rad/s for Nodes 1, 2 and 3, and 0.477 and 0.495 rad/s for Node 1, respectively. Moreover, the elements of the matrix Γyican be obtained by

the amplitude and phase of the sinusoidal terms in ∆Iland ∆Gl, where y denotes G

or I in case of impedance or current loads, respectively. Also, consider the unknown variations ∆Il= col(10, 7, −10, 5)A and ∆Gl= 0.01 col(−1, 1, −1, 1) Ω−1at the time

instant t = 5 s. We can observe in Figure 4.3 that the voltages at each node converge to their corresponding desired reference values, achieving voltage regulation (see Objective 4.1) and the system is robust with respect to unknown constant load variations. Thus, the robustness against uncertain constant loads is guaranteed. Also, we can notice from Figure 4.4 that the generated current at each node is stable.

4.4. Simulation Results 57 0 2 4 6 8 10 time (s) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Load (A)

Real data of Node 1 Exosystem of Node 1

(a) Higher approximation

0 2 4 6 8 10 time (s) 0.5 1 1.5 2 2.5

Load (A)

Real data of Node 1 Real data of Node 2 Real data of Node 3 Real data of Node 4 Exosystem of Nodes 1,2,3 Exosystem of Node 4

(b) Lower approximation

Figure 4.2: Comparison between the load profile from the database [140] and the load profile produced by the considered exosystems. In (a) we show that by increasing the state dimension, our exosystem can approximate very well real data. In (b) we keep on purpose the exosystems’ dimension lower in order to emulate possible uncertainties in practice between the day-ahead load forecast and the actual profile.

Figure 4.3: Scenario 1: time evolution of the voltages at each node together with the corresponding desired values (dashed lines).

Scenario 2. At the time instant t = 1 s the loads vary according to the real data in [140] while the controller uses the information of the exosystems, which differ from the real data (see Figure 4.2 (b)). We can observe from Figure 4.5 that the voltage at each node is kept very close to the corresponding voltage reference, showing that the controlled system is ISS with respect to the possible mismatch between the actual load profile and the one generated by the exosystem, achieving in practice voltage regulation (Objective 4.1). Also, we can note from Figure 4.6 that the generated current at each node is stable. Moreover, we would like to notice that if we approximate the real load profile with an exosystem whose state dimension is higher (see Figure 4.2), voltage regulation is achieved with an higher accuracy.

Moreover, we compare our controller with the one proposed in [135], which is designed to deal with constant loads only. We can clearly observe from Figure 4.7 that the controller in [135] is not capable to achieve voltage regulation.

4.4. Simulation Results 59

Figure 4.4: Scenario 1: time evolution of the currents generated at each node.

Figure 4.5: Scenario 2: time evolution of the voltages at each node together with the corresponding desired values (dashed lines).

Figure 4.6: Scenario 2: time evolution of the currents generated at each node.

Figure 4.7: Scenario 2 (Controller [135]): time evolution of the voltages at each node together with the corresponding desired values (dashed lines).

4.5. Concluding Remarks 61

4.5

Concluding Remarks

In this chapter, we have considered time-varying dynamics for the load components of a DC power network. Then, we have proposed a control scheme based on the output regulation methodology to achieve voltage regulation and guarantee the stability of the overall network. In the next chapter, we use robust output regulation theory to tackle the problem of voltage regulation in DC networks comprising also uncertain constant power loads.