Master of Science in Automation and Control Engineering
Master Thesis
A PASSIVITY-BASED APPROACH TO VOLTAGE
STABILIZATION IN DC MICROGRIDS
Candidate: Andrea Martinelli ID. 852539
Supervisor:
Prof. Riccardo Scattolini
Co-supervisor:
Prof. Giancarlo Ferrari-Trecate
Milan, 21/12/2017
This thesis was developed at ´Ecole Polytechnique F´ed´erale de Lausanne, where I was a guest of the Automatic Control Laboratory. First of all, I wish to express my gratitude to my supervisor, Prof. Scattolini, for the great opportunity he gave me. What made my experience so satisfactory, was surely the dedication of Prof. Ferrari-Trecate, who kindly hosted me at the lab and set everything up for a successful work, and the patience of Pulkit, who carefully guided me through the every-day jungle of maths. During my stay in Lausanne, I had the opportunity to meet remarkable people. In particular, I would like to thank my friends Michele and Martin for all the support and the carefree moments.
When I look back to my academic adventure, I am well aware that a large part of the credit is deserved by everyone that helped and sustained me. I feel very proud to say that I can share joys and sorrows with amazing friends. I wish to express my thanks to, more than anyone, my school mates Matteo P., Marcello, Danny, Matteo L., Luca, Jacopo and Riccardo, my automation buddies William, Andrea and Giuseppe, and my lifelong friends Matteo R. and Marco.
To my family I dedicate the culmination of this journey. To my parents and my sister, for their love and sacrifices, to my grandparents, for their wisdom, to my uncles and cousins, for inspiring me to do my best.
The last words are devoted to the person who prevented me to give up. You’ve al-ways been by my side and, through your love, you gave me passion and willpower. Heartfelt thanks Marta.
Abstract ix
1 Introduction 1
1.1 Introduction . . . 1
1.2 Thesis Contribution and Organization . . . 2
1.3 Additional Information . . . 3
2 Background 5 2.1 Algebraic Graph Theory and Communication Topology . . . 5
2.2 Lyapunov Stability . . . 7
2.3 Passivity Theory . . . 9
2.3.1 Definition and physical interpretation . . . 10
2.3.2 Output synchronization on strongly connected graphs . . . . 12
2.3.3 Interconnection of multiple passive systems . . . 13
2.4 DC Microgrids . . . 14
2.5 State-Space Averaging Method for DC/DC Power Converters . . . . 14
3 Low-Voltage DC Microgrids Stabilization 17 3.1 Model of the Electrical Network . . . 17
3.2 Model of the Agents . . . 19
3.3 Passivity of Open-loop Agents . . . 22
3.4 Design of Local Regulators . . . 23
3.5 Network Stabilization with Optimal Regulators . . . 24
3.5.1 Passivity of closed-loop agents via LMIs . . . 24
3.5.2 Network global asymptotic stability . . . 26
3.6 Network Stabilization with Explicit Regulators . . . 29
3.6.1 Passivity of closed-loop agents via explicit set analysis . . . . 30
3.6.2 Network global asymptotic stability . . . 33
4 Medium-Voltage DC Microgrids Stabilization 37 4.1 Model of the Electrical Network . . . 37
4.2 Model of the Agents . . . 39
4.2.1 MV-DGU agents . . . 39
4.2.3 Coupled agents . . . 41
4.3 Study of the Equilibria . . . 42
4.3.1 Equilibrium existence and uniqueness . . . 42
4.3.2 Equilibrium conditions . . . 46
4.4 Design of Non-linear Local Regulators . . . 48
4.5 Passivity of the Closed-loop Agents . . . 49
4.5.1 MV-DGU agents . . . 50
4.5.2 Line agents . . . 51
4.6 Network Global Asymptotic Stability . . . 51
5 Validation and Control 57 5.1 Averaged Models Validation . . . 57
5.1.1 LV agents validation . . . 57
5.1.2 MV agents validation . . . 59
5.2 Control Simulation . . . 60
6 Final Considerations and Conclusions 63 6.1 Energetic Interpretation of mG Stability . . . 63
6.2 Achievements and Future Developments . . . 64
2.1 RL circuit . . . 11
2.2 Buck converter . . . 14
3.1 LVDC Network Example . . . 18
3.2 i-th Buck converter connected to the mG network . . . 19
3.3 i-th DGU agent connected to the mG network . . . 21
3.4 i-th closed-loop DGU agent . . . 24
4.1 MVDC Network Example . . . 38
4.2 i-th Boost converter connected to the mG network . . . 39
4.3 Line agent connecting a couple of DGUs . . . 41
4.4 Graph G and the reverse graph Gr . . . 53
5.1 Simulink scheme for LV agents validation . . . 58
5.2 Validation of LV-DGU: step response comparison . . . 58
5.3 Simulink scheme for MV agents validation . . . 59
5.4 Validation of MV-DGU: step response comparison . . . 59
5.5 Sequence of plug-in operations (T1 = 1s, T2 = 1.5s) . . . 61
5.6 Control simulation of an islanded LVDC mG . . . 61
5.7 Control simulation of an islanded MVDC mG . . . 61
We consider the application of passivity theory to the problem of voltage stabiliza-tion in low and medium-voltage DC microgrids. Our objective is to derive control criteria to guarantee voltage reference tracking in DC microgrids, that we model as a set of dynamical agents able to exchange local information over a network. For the low-voltage scenario we propose, similarly to [20], a decentralized control architec-ture where the primary controller of each agent can be designed in a Plug-and-Play (PnP) fashion: regulators’ design relies just on local parameters and, independently from network topology, the addition and removal of nodes do not affect global stabil-ity. Differently from approach in [20] we provide, thanks to the passivity framework, explicit inequalities on control gains to design stabilizing local regulators. In this way we guarantee regulators to be always feasible, and we prevent the necessity of solving optimization problems. As concerns the medium-voltage case, we firstly provide a non-linear model for the network. Then we introduce a decentralized con-trol architecture, based on non-linear dynamic regulators, that is able to stabilize the global dynamics in a PnP fashion. Also in this case, regulators’ design does not require optimization tools. Each controller can be explicitly synthesized relying on single agent’s parameters, on the voltage reference value of its neighbours and on the resistance of power lines. Theoretical results are backed up by simulations in Simulink environment.
Key words: Passivity-based control, Decentralized control, DC microgrids, Voltage stability, Nonlinear stability analysis.
In questa tesi consideriamo l’applicazione della teoria della passivit´a al problema di stabilizzazione della tensione in microgrid DC a bassa e media tensione. Il no-stro obbiettivo ´e ricavare dei criteri di controllo che garantiscano l’inseguimento dei riferimenti di tensione nelle microgrid DC, modellizzate come un insieme di agen-ti dinamici che si scambiano informazioni locali attraverso una rete. Per il caso a bassa tensione proponiamo, in modo simile a [20], un’architettura di controllo de-centralizzata dove il controllore primario di ciascun agente viene progettato in modo cosiddetto Plug-and-Play (PnP): un singolo regolatore si basa solamente su para-metri locali, e l’inserimento e rimozione dei nodi di rete non ne influenza la stabilit´a globale, indipendentemente dalla topologia. Differentemente dall’approccio in [20] forniamo, grazie al framework della passivit´a, delle diseguaglianze esplicite per la scelta dei guadagni di controllo. In questo modo si garantisce sempre l’esistenza dei regolatori, e si previene la necessit´a di risolvere problemi di ottimizzazione. Per quanto riguarda il caso a media tensione, sviluppiamo innanzitutto un modello di rete non lineare. Quindi introduciamo un’architettura di controllo decentralizzata, basata su regolatori dinamici e non lineari, che garantisce la stabilit´a globale della rete in modo PnP. Anche in questo caso, il progetto dei controllori non richiede gli strumenti dell’ottimizzazione. Ciascun regolatore pu´o essere sintetizzato basandosi sui parametri del corrispettivo agente, oltre che sul valore di riferimento di tensione dei suoi vicini e sulla resistenza delle linee di connessione. I risultati teorici sono testati in ambiente Simulink.
Parole chiave: Controllo passivity-based, Controllo decentralizzato, Microgrids DC, Stabilit´a della tensione, Stabilit´a di sistemi non lineari.
1
Introduction
1.1
Introduction
The main target of this thesis is the application of passivity-based tools to direct-current microgrids (DC mGs), both in low-voltage (LV) and medium-voltage (MV) configuration. An mG is a group of spatially-distributed systems composed, for example, by loads and distributed generation units (DGUs), interconnected to each other though an electrical network [16]. MGs can connect and disconnect from the main grid to enable it to operate in both grid-connected or, as we focus in this thesis, in islanded-mode. The advantages of a distributed energy infrastructure with respect to the classical centralized one are numerous. They range from the capability of electrifying remote areas, islands, or large buildings, to the ability of improving resilience to faults and power quality in power networks [21]. Since 19th Century, the invention of transformers and poly-phase AC machines initiated the worldwide establishment of a complete AC generation, transmission and distribution grid. For this reasons, so far research mainly focused on AC mGs [9, 18]. Motivated by advances in power electronics, batteries and renewable DC energy sources, DC mGs find nowadays applications in various field such as high-efficiency households, electric vehicles, hybrid energy storage systems, data centres, avionics and marine systems [4]. The combination of DC distribution together with the mG concept becomes attractive, since [12]:
1. Renewable energy sources, electric vehicles and energy storage systems are naturally in DC, therefore efficiency is enhanced because of less number of power conversion stages;
2. The control and management of a DC system is much simpler than in AC, which makes DC mGs practically more feasible. For instance, control of react-ive power or unbalanced electric signals are not an issue. On the other hand, protection of DC systems is still a challenging problem;
3. Most consumer electronic appliances are in DC, such as computers, microwave-ovens, modern lighting systems, and so on.
A key challenge in DC mGs is to ensure voltage stability, that is achieved through decentralized control architectures at the primary level of each DGU [3]. Popu-lar solutions are based on droop controllers, but their stabilization properties has been shown either for specific mG topologies, or for general topologies, but relying on networked secondary regulators [5, 22]. An alternative class of controllers for LV mGs, called Plug-and-Play (PnP), has been proposed, amongst other, in [21] and [20]. The main feature of PnP approach is that, independently from mG size and topology, the addition and removal of DGUs do not require to retune all other local regulators. In particular, in [20], a control architecture based on PnP line-independent regulators has been developed. Local regulators are synthesized, once and for all, by solving an optimization problem based on Linear Matrix Inequal-ities (LMIs), whose parameters depend only on the specific DGU. Before plug -in and -out operations, a feasibility test is performed, based on the solution of the LMIs.
Passivity theory is the mathematical tool we exploit to study voltage stability in LV and MV DCmGs. It is a powerful framework to analyse complex systems, both with linear and non-linear dynamics [7]. It permits to design control actions based on sys-tem’s energetic considerations, and it has strong relationship with Lyapunov stability [8]. Moreover, when dealing with large-scale systems, passivity theory provides a compositional framework. In fact, the interconnection of passive dynamical systems present useful stability properties, depending on the nature of the communication channels and on the topology of the network [1, 16].
1.2
Thesis Contribution and Organization
In the thesis we recall a recently-developed passivity-based mathematical frame-work [16], through which we analyse the stability properties of islanded DC mGs. In particular, for the LV case, we rely on the mG model used in [20], which is based on linear DGUs. We propose, similarly to [20], a decentralized control architecture where the primary controller of each agent can be designed in a PnP fashion: regulat-ors’ design relies just on single agent’s parameters and, independently from network topology, the addition and removal of nodes do not affect global stability. The main improvement with respect to the approach in [20] is that we provide, thanks to the passivity framework, explicit inequalities on control gains to design stabilizing local regulators. In this way we prevent the necessity of solving optimization problems in the design phase. Moreover, within our architecture, plug -in and -out operations are always allowed, without retuning any other regulator or performing feasibility
tests.
As concern the MV environment, we firstly develop a novel non-linear model of the electrical network. Then we present a decentralized control architecture, based on non-linear dynamic regulators, that is able to stabilize the global dynamics in a PnP fashion. Also in this case, regulators’ design does not require optimization tools. Each controller can be explicitly synthesized relying on single agent’s para-meters, on the voltage reference value of its neighbours and on the resistance of power lines.
The thesis is organized as follows. In the first chapter, we provide a basic background on some mathematical tools that are extensively used in the rest of the work. The core topics are divided into two parts: in Chapter 2, the voltage stabilization prob-lem is addressed in the LV mGs framework, while Chapter 3 is devoted to study the MV mGs. Then, we validate our dynamical models and the developed control architectures in Chapter 4. Finally, conclusions are drawn in Chapter 5.
1.3
Additional Information
The thesis project has been entirely carried out at the Automatic Control Laboratory of ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), under the monitoring of Prof. Giancarlo Ferrari-Trecate. Prof. Riccardo Scattolini has been the supervisor of the project at Politecnico di Milano. Some of the thesis’ results have contributed to the writing of the following scientific papers that are, at the moment, under review:
• P. Nahata, R. Soloperto, M. Tucci, A. Martinelli, and G. Ferrari-Trecate. “A Passivity-Based Approach to Voltage Stabilization in DC Microgrids with ZIP Loads”. Automatica, submitted.
• R. Han, M. Tucci, A. Martinelli, J. M. Guerrero, and G. Ferrari-Trecate. “Plug-and-Play Voltage/Current Stabilization DC Microgrid Clusters with Grid Forming/Feeding Converters”. IEEE Transactions on Energy Conver-sion, submitted.
• R. Han, M. Tucci, A. Martinelli, J. M. Guerrero, and G. Ferrari-Trecate. “Global Stability Analysis of Primary Plug-and-Play and Secondary Leader-based Controllers for DC Microgrid Clusters”. American Control Conference, submitted.
Moreover, a technical report is already available online:
• R. Han, M. Tucci, A. Martinelli, J. M. Guerrero, and G. Ferrari-Trecate. “Hier-archical Plug-and-Play Voltage/Current Controller of DC Microgrid Clusters
with Grid-Forming/Feeding Converters: Line-independent Primary Stabiliza-tion and Leader-based Distributed Secondary RegulaStabiliza-tion”. Tech. Rep. [On-line], Available: arXiv:1707.07259, 2017.
2
Background
This chapter is devoted to the necessary background material for the rest of the thesis. In Section 2.1 we present a basic framework related to graph theory and communication topology. A short introduction to Lyapunov stability, invariant sets and LaSalle’s invariance principle is provided in Section 2.2. Then we introduce, in Section 2.3, the main concepts related to passivity theory and properties of the interconnection of passive dynamical systems. Moreover, we recall in Section 2.5 the details of the so called state-space averaging method, applied to a generic DC/DC power converter.
2.1
Algebraic Graph Theory and Communication
To-pology
In this first section we provide a basic introduction to graph theory and related notation [17]. A directed graph is a pair G = (V, E ), where the elements of the set V = {1, ..., N } are the nodes (or vertices), while the elements of E ⊂ V × V are the edges. An edge is therefore an ordered pair of distinct vertices. If, for all (i, j) ∈ E , one has (j, i) ∈ E , then the graph is said to be undirected. An edge (i, j) is said to be incoming with respect to j and outgoing with respect to i, and can be represented as an arrow with vertex i as its tail and vertex j as its head. A path of length r in a graph is a sequence (0, ..., r) of r + 1 distinct vertices such that for every i ∈ {0, ..., r − 1}, the edge (i, i + 1) belongs to E . A directed graph is strongly connected if any two distinct vertices can be joined by a path. If we ignore the directed nature of the edges, a graph is weakly connected when any two vertices are connected by a path, where paths are allowed to go either way along any edge. An unweighed graph is described by its N × N adjacency matrix
A = [aij], aij =
(
1 if (i, j) ∈ E
Notice that A is symmetrical if the network is undirected. If the graph is weighted (i.e. each edge has an associated weight), it can be described by a N × N weight matrix
W = [wij], wij =
(
wij if (i, j) ∈ E
0 otherwise. (2.2) A graph is called bipartite if the set of its vertices can be partitioned into two classes S1 (p vertices) and S2 (q vertices), such that edges can only connect vertices
of different classes. We describe a bipartite network with a rectangular p × q matrix called incidence matrix
B = [bij], i ∈ S1, j ∈ S2, bij =
(
1 if (i, j) ∈ E
0 otherwise. (2.3) In an undirected network, the set of neighbours Ni ⊂ V of a vertex i is the set of
vertices that share an edge with i. In the directed case, we can distinguish between the set of out-neighbors Ni+ = {j ∈ V : (i, j) ∈ E } and the set of in-neighbours Ni− = {j ∈ V : (j, i) ∈ E }, such that Ni = Ni+∪ N
−
i . We call k-length neighbours
of agent i those nodes that can be reached from i with a path of exactly length k. Notice that the neighbours correspond to the 1-length neighbours. The degree ki of
a vertex i is the number of edges connected to it (i.e. the number of its neighbours). For an undirected graph of N vertices, the degree can be written in terms of the adjacency matrix as ki = N X j=1 aij. (2.4)
We then define the degree matrix as D = diag(k1, ..., kN). In a directed network each
vertex has two degrees. The in-degree is the number of ingoing edges connected to a vertex and the out-degree is the number of outgoing edges. In the case of unweighed graphs they can be defined as
kinj = N X i=1 aij kiout= N X j=1 aij. (2.5)
The N × N Laplacian matrix is an alternative representation of the network, and it is defined as the difference between the degree matrix and the adjacency matrix
L = D − A. (2.6)
It can be shown that, if the graph is undirected and connected, the following prop-erties hold: L is symmetric, zero-row-sum, irreducible and positive semidefinite. Consider now N dynamical systems, called agents, whose internal dynamics is
de-scribed by
˙
xi = fi(xi), i = 1, ..., N, (2.7)
with f : Rn→ Rn. Hence, the global behaviour of the system may be described in
compact form by
˙
x = f (x), (2.8)
where x = [ x1 . . . xN ]T and f (x) = [ f1(x1) . . . fN(xN) ]T. Suppose now
that the agents can exchange information over a network. The communication network is represented by the graph G = (V, E ), where the node i ∈ V hosts the dynamical system i, and the communications happens through the edges. We assume that the global behaviour of the network is described by
˙
x = f (x) + ξ. (2.9)
The term ξ accounts for the coupling among the agents, and it takes the form
ξ = Ag(x), (2.10)
where A is the adjacency matrix associated to the graph G, and g(x) is a N ×N vector function that depends on the input-output characteristics of the agents, defined as
g(x) = 0 g12(x1, x2) · · · g1N(x1, xN) g21(x2, x1) 0 .. . . .. gN 1(xN, x1) 0 . (2.11)
We want to notice that each agent can communicate only with the subset Ni of its
neighbours, and therefore the exchange of information is a local process. In fact, the coupling term for the agent i is
ξi =
X
j∈Ni
aijgij(xi, xj). (2.12)
It is evident that the global behaviour of the N agents depends on the topology of the communication network, represented by the adjacency matrix.
2.2
Lyapunov Stability
We want to recall the basics of Lyapunov theory and LaSalle’s invariance principle, since these concepts are extensively used in the thesis. With a reference to [7],
consider the following autonomous dynamical system
˙
x = f (x), (2.13)
where f : D → Rn is a locally Lipschitz map from a domain D ⊂ Rn into Rn. Suppose ¯x ∈ D is an equilibrium point of (2.13), that is f (¯x) = 0. Without loss of generality, we state all definitions and theorems for the case when the equilibrium point is the origin of Rn.
Definition 1 (Stability of an equilibrium). The equilibrium point x = 0 of (2.13) is
• stable if, for each > 0, there is δ = δ() > 0 such that
kx(0)k < δ ⇒ kx(t)k < , ∀t ≥ 0;
• unstable if it is not stable;
• asymptotically stable if it is stable and δ can be chosen such that
kx(0)k < δ ⇒ lim
t→∞x(t) = 0.
Then we are able to present the following result, that we state without proof
Theorem 1 (Lyapunov’s theorem). Let x = 0 be an equilibrium point for (2.13) and D ⊂ Rn be a domain containing x = 0. Let V : D → R be a continuously differentiable function, called Lyapunov function, such that
V (0) = 0 and V (x) > 0 in D − {0}, (2.14) ˙
V (x) ≤ 0 in D. (2.15) Then, x = 0 is stable. Moreover, if
˙
V (x) < 0 in D − {0}, (2.16)
then x = 0 is asymptotically stable.
From a practical point of view, when trying to prove the asymptotic stability of the origin, it may happen to use candidate Lyapunov functions whose derivative is only negative semidefinite. Since Lyapunov’s theorem provides only a sufficient condition for asymptotic stability, one can check the asymptotic stability of the origin even if
˙
Definition 2 (Positively invariant set). A set M is said to be a positively invariant set for (2.13) if
x(0) ∈ M ⇒ x(t) ∈ M, ∀ t ≥ 0. (2.17)
That is, if a solution belongs to M at some time instant, then it belongs to M for all future time.
Theorem 2 (LaSalle’s invariance principle). Let Ω ⊂ D be a set that is positively invariant with respect to (2.13). Let V : D → R be a continuously differentiable function such that ˙V (x) ≤ 0 in Ω. Let E = {x ∈ D : ˙V (x) = 0}. Let M be the largest invariant set in E. Then, every solution starting in Ω approaches M as t → ∞.
When our interest is in showing that x(t) → 0 as t → ∞, we need to establish that the largest invariant set M ⊂ E is the origin. This is done by showing that no solution to (2.13) can stay identically in E, other than the trivial solution x(t) = 0. A particular case hold when the Lyapunov stability theory is applied to a linear dynamical system of the form
˙
x(t) = Ax(t) + Bu(t). (2.18)
The following result can be stated [11]:
Theorem 3 (Lyapunov’s theorem for linear systems). A necessary and sufficient condition for the asymptotic stability of system (2.18) is that, given any matrix Q = Q0 > 0, there exists a matrix P = P0 > 0 verifying the following Lyapunov equation
A0P + P A = −Q. (2.19)
2.3
Passivity Theory
We first give a formal definition of the passivity property of a dynamical system, and recall how this property is linked with Lyapunov stability. Then we provide a physical interpretation of passivity and, finally, we highlight some properties of interconnections of passive systems.
2.3.1 Definition and physical interpretation
With reference to [7], let us introduce a dynamical system represented by the state model
˙
x = f (x, u) y = h(x, u),
(2.20)
where f : Rn× Rp → Rn is locally Lipschitz, h : Rn× Rp → Rp is continuous,
f (0, 0) = 0, and h(0, 0) = 0. The system is square, i.e. it has the same number of inputs and outputs.
Definition 3 (Passivity). The system (2.20) is said to be passive if there exists a continuously differentiable positive semidefinite function V(x), called the storage function, such that
uTy ≥ ˙V = ∂V
∂xf (x, u), ∀(x, u) ∈ R
nx Rp. (2.21)
Moreover, it is said to be strictly passive if
uTy ≥ ˙V + ψ(x), ∀(x, u) ∈ Rnx Rp, (2.22)
for some positive definite function ψ.
Notice that the passivity property depends on the specific choice of the input and output. We want now to describe the relationship that exists between strict passivity and Lyapunov stability [7].
Lemma 1. If the system (2.20) is strictly passive with a positive definite storage function V(x), then the origin of ˙x = f (x, 0) is asymptotically stable.
Proof. Take V (x) as a candidate Lyapunov function for ˙x = f (x, 0). Then the following inequality holds
0 ≥ ˙V + ψ(x). (2.23) Since ψ is positive definite, ˙V is a decreasing function in a neighbourhood of the origin, except in the origin itself where it is zero. Hence,
˙
V < 0. (2.24)
Notice that if the function ψ is only positive semidefinite, and therefore the system is passive, but not strictly passive, the asymptotic stability of the origin must be
u(t)
R x(t)
L
Figure 2.1: RL circuit
investigated using the LaSalle’s invariance principle, as we discussed in Section 2.2.
After giving a mathematical definition of passivity of dynamical systems, we aim to provide a physical interpretation with the following example.
Example 1. Consider the RL circuit in figure 2.1. Since the inductance is the only dynamic component, the behaviour of the circuit can be described by the first-order differential equation
L ˙x = −Rx + u, (2.25) where u is the input voltage and x the line current, and we omitted the dependence of variables on time. Let us now consider a positive definite storage function for the circuit, which represents the energy stored in the inductance
V (x) = 1 2Lx
2. (2.26)
The variation of stored energy over time can be computed by differentiating expres-sion (2.26) with respect to trajectories of (2.25):
˙
V (x) = Lx ˙x = ux − Rx2. (2.27)
If we consider the line current as output of the system (y = x), we may rewrite equation (2.27) as
uy = ˙V (x) + Rx2. (2.28) By making reference to Definition 3, we can state that dynamical system (2.25) is strictly passive with the storage function V (x) and the output transformation y = x. Beyond the mathematical description of the system, we observe that expres-sion (2.28) is actually the energy balance for the RL circuit. The term uy is the power generated by the voltage input, ˙V (x) is the variation of internal energy of the inductance, and Rx2 is the energy dissipated in the resistance due to Joule heating. Expression (2.28) suggests us that the RL circuit, due to Joule losses, behaves like
a dissipative component: the accumulated energy is always less or equal to the ex-ternal energy supplied by the input voltage. Asymptotic stability of (2.25) can be intuitively understood from the energetic behaviour. In fact, not only the circuit is not able to generate internal energy by itself, but also it dissipates the energy supplied.
2.3.2 Output synchronization on strongly connected graphs
At this point, we want to extend the stability analysis to a network of agents whose dynamics is passive. As presented in [1], consider N agents whose dynamics can be written, for i = 1, ..., N , as
˙
xi = fi(xi) + gi(xi)ui
yi = hi(xi),
(2.29)
with fi : Rn → Rn, gi : Rp → Rn, hi : Rn → Rp, fi(0) = 0 and hi(0) = 0. Notice
that each agent has p inputs and p outputs. Moreover, consider that the agents are coupled together using the control
ui=
X
j∈Ni
Kji(yj− yi), (2.30)
where Kji is a positive constant, and Ni is the set of neighbours agents of i. We
assume that if i ∈ Nj, then also j ∈ Ni and Kij = Kji. Note that sets Ni define an
undirected graph G = (V, E ), where nodes in V = {1, ..., N } represent the agents and edges E represent communication channels between agents. The channels enable the computation of ui as in (2.30), thanks to the transmission of variables yj.
Definition 4 (Output synchronization). Consider to have a network of N agents. In the absence of communication delays, the agents are said to output synchronize if
lim
t→∞kyi(t) − yj(t)k ∀i, j = 1, ..., N. (2.31)
Theorem 4 [1]. Consider the dynamical system described by (2.29) with the control (2.30). If the agents dynamics are passive with radially unbounded positive definite storage functions Vi(xi), and the communication graph G is strongly connected, then
the coupled non-linear system (2.29),(2.30) is globally stable and the agents output synchronize.
Let’s define with x = [ x1 · · · xN ]T the global state of the network. It is proved
in [1] that, if conditions of Theorem 4 are satisfied, it is always possible to find a global Lyapunov function V (x), which is a linear combination of the single agent’s
storage functions, such that ˙V (x) ≤ 0. As a consequence, by virtue of the LaSalle’s invariance principle (see Theorem 2), the global dynamics converges to the largest invariant set M contained in
E = n xi ∈ Rn×1, i = 1, ..., N : ˙V (x) = 0 o =nxi ∈ Rn×1, i = 1, ..., N : ψi(xi) = 0, (yi− yj)T(yi− yj) = 0 ∀j ∈ Ni, ∀i = 1, ..., N o . (2.32)
2.3.3 Interconnection of multiple passive systems
The interconnection model introduced in the previous section is useful to analyse stability for certain classes of mGs, such as those operating in LV (see Chapter 3). The problem is that, as we will discuss in Chapter 4, the MV mG model we consider does not fit with the coupling structure (2.30). For this reason we refer to [16], where a different interconnection paradigm is examined. Thereafter we summarize the main results in [16]. Consider again a set of control-affine dynamical systems (2.29). The interconnection among agents is now described through the weighed directed graph G = (V, E ), with weight matrix W . An alternative representation is its reverse graph Gr = (V, Er), with weight matrix Wr, such that (i, j) ∈ Er if and only if (j, i) ∈ E . Notice that, in general, W 6= Wr. Let the agents be coupled
together by the control variables
ui = X j∈Ni+ wijyj+ X j∈Ni− wrijyj, i = 1, ..., N. (2.33)
We are now in the position to enunciate the following result.
Theorem 5 [16]. Consider a set of dynamical systems defined by (2.29), coupled with each other through input (2.33). If each system is passive with a radially un-bounded positive definite storage function Vi(xi), i.e.
uiyi= ˙Vi(xi) + ψi(xi), ψi(xi) ≥ 0, (2.34)
then, for all initial conditions xi(0), the origin of the interconnected system is simply
stable and the state x = [ x1 · · · xN ]T converges, as t → ∞, to the largest
invariant set contained in
E =nx = [ x1 · · · xN ]T : ψi(xi) = 0, ∀i ∈ V
o
. (2.35)
In fact, due to interconnection (2.33) and agents’ passivity, it is guaranteed the existence of a Lyapunov function V (x), which is a linear combination of the N
Vin
D R L
I
C V Load
Figure 2.2: Buck converter
storage functions, such that ˙V (x) ≤ 0. Notice that, if agents’ are strictly passive, then the origin x = 0 is a globally asymptotically stable equilibrium. On the other hand, if agents’ dynamics are passive, but not strictly passive, the stability of the origin must be investigated with the LaSalle’s invariance principle, by checking the largest invariant set contained in (2.35).
2.4
DC Microgrids
We model islanded DC mGs as a set of dynamical agents that exchanges informations through a network with arbitrary topology. The nodes represent the DGUs and the edges are the electric transmission lines. A DGU is modelled as a DC/DC power converter, where the DC voltage source (representing a generic renewable energy source) sustains a local DC load. The type of power conversion depends on the specific task that the mG has to accomplish. In fact, the local load may need to be operated with a voltage magnitude which is lower or higher than the renewable voltage source. In the first case, we include in the DGU a Buck converter [20], while in the second case a Boost converter. In Figure 2.2 is depicted the circuit model of a generic Buck converter. When we model DGU agents as Buck converters, we say that the mG is a low-voltage (LV) one, while with Boost converters the mG is a medium-voltage (MV) one. The principal difference in modelling LV and MV mGs, a part from the DC/DC converter, are the transmission lines. In the LV case we approximate the lines as purely resistive, while we cannot do the same with MV mGs [15]. Therefore, we consider RL lines in MV mGs modelling.
2.5
State-Space Averaging Method for DC/DC Power
Converters
In this section we refer to [13] to address the problem of obtaining a single state-space representation of DC/DC power converters. From Figure 2.2, we observe that the basic DC/DC power conversion is achieved by repetitive switching between two linear networks. On the assumption that the circuit operates in the so called
continuous conduction mode (CCM), in which the inductor current I does not fall to zero at any point in the cycle, there are only two different “states” of the circuit. Hence, converter’s dynamics can be described by means of a hybrid system
˙
x = Apx + bp, p = {on, of f }, (2.36)
where “on” and “off” represent the two discrete states in which the switch D is close and open, respectively. The state x is made by the electric dynamical quantities, which is our case are the inductor current I and the capacitance voltage V . Ap is
the dynamical matrix, and bp accounts for the exogenous terms, such as the voltage
input and the load. Let’s consider a period of time T , where in the first part the switch is closed, and in the second is opened. The switching happens at a certain time t0 ∈ [0, T ]. We introduce a new variable d, that we call duty cycle, which
represents the fraction of time where the switch is closed
d = t0
T. (2.37)
The duty cycle complement is therefore the fraction of time where the switch is opened
d∗= 1 − d = T − t0
T . (2.38)
Our objective now is to replace the state-space description of the two linear circuits, emanating from the two successive phases of the switching cycle , by a single state-space description, which represents approximately the behaviour of the circuit across the whole period T . We therefore perform the following operation
˙
x = (dAon+ d∗Aof f) x + dbon+ d∗bof f, (2.39)
that represents the averaging between the “on” and “off” dynamics, weighted over the fraction of time spent in each configuration (i.e. the duty cycle and its com-plement). We must be aware of the fact that the previous operation, in practice, neglects the higher frequency components of system’s dynamics. To be more precise, let the two linear systems be described by
˙
x(t) = Aonx(t), t ∈ [0, t0] (2.40a)
˙
x(t) = Aof fx(t), t ∈ [t0, T ], (2.40b)
where terms bon and bof f are neglected for simplicity. The exact solutions of these
state space equations are
x(t) = eAontx(0), t ∈ [0, t
x(t) = eAof f(t−t0)x(t
0), t ∈ [t0, T ]. (2.41b)
Since x(t) is continuous across the switching instant t0, the solution over the entire
period T is
x(T ) = ed∗Aof fTedAonTx(0). (2.42)
On the other hand, the exact solution over the period T of the system
˙
x(t) = (dAon+ d∗Aof f)x(t) (2.43)
is
x(T ) = e(dAon+d∗Aof f)Tx(0). (2.44)
From (2.42) and (2.44) it follows that, by performing the state space averaging operation, we introduce the following approximation
ed∗Aof fTedAonT ≈ e(dAon+d∗Aof f)T (2.45)
which, as shown in [13], is well satisfied if the following linear approximation holds
edAonT ≈ I + dA
onT
ed∗Aof fT ≈ I + d∗A
of fT
(2.46)
Finally, this latter linear approximation is equivalent to require the converter switch-ing frequency fs = 1/T to be significantly greater than the cutoff frequency of the
3
Low-Voltage DC Microgrids
Stabilization
In this chapter, we apply the passivity framework to an islanded low-voltage DC microgrid (LVDC mG), in order to derive stability criteria for the network voltages. We firstly provide a model for the communication graph (Section 3.1) and for the agents (Section 3.2). Notice that we are revising existing models [19, 20], adapting them to our environment. Then, in Section 3.4, we equip the agents with state-feedback PI regulators for primary voltage control. The last section is devoted to prove global asymptotic stability of the closed-loop networked dynamics.
3.1
Model of the Electrical Network
The LVDC mG is modelled as an undirected weighted graph (see Section 2.1): GLV =
(VLV, ELV). Each node of the set VLV = {1, ..., N } hosts a dynamical system that
we call Distributed Generation Unit (DGU), or simply agent. Any edge (i, j) ∈ ELV
has an associated weight R1
ij ∈ R
+, such that 1 Rij =
1
Rji. The numerical values of
the weights represent the inverse of the resistances of the power lines connecting the DGUs. The network is described by the N × N weight matrix
W = [wij], wij =
( 1
Rij if (i, j) ∈ E
LV
0 otherwise. (3.1)
Moreover, assuming that there are no isolated DGUs, and that any line connect exactly a couple of DGUs, we are allowed to state that the LV network is strongly connected. We suppose the dynamics of each agent is described by
˙
xi = fi(xi) + ξi, i ∈ VLV, (3.2)
with fi : Rn→ Rn. Precise models of functions fi will be given in the next section.
DGU1 DGU2 DGU3 DGU4 DGU5 1 R12 1 R23 1 R34 1 R45 1 R13 1 R24
Figure 3.1: LVDC Network Example
with its neighbours Ni, giving rise to the following coupling terms
ξi =
X
j∈Ni
wijgij(xi, xj). (3.3)
The term gij(xi, xj) depends on the input-output characteristics of the agents. The
whole mG behaviour can be represented as
˙
x = f (x) + ξ, (3.4)
where x = [ x1 . . . xN ]T, f (x) = [ f1(x1) . . . fN(xN) ]T, ξ = W g(x) and
g(x) is a N × N vector function that depends on agents’ input-output character-istics, whose general expression has been defined in (2.11). We remind that the global dynamics is influenced, through the weight matrix W , by the topology of the communication network GLV. We depict in Figure 3.1 an example of LVDC network.
The example network has 5 agents, and so it can represented by the symmetric 5 × 5 weight matrix W = 0 R1 12 1 R13 0 0 1 R12 0 1 R23 1 R24 0 1 R13 1 R23 0 1 R34 0 0 R1 24 1 R34 0 1 R45 0 0 0 R1 45 0 .
For instance, the coupling term for the DGU agent 1 is
ξ1 = X j∈N1 w1jg1j(x1, xj) = 1 R12 g(x1, x2) + 1 R13 g(x1, x3). (3.5)
Vin,i + − Di Ri I Li i Ci Vi Iload,i Inet,i mG Network
i-th Buck Converter
PCCi
Figure 3.2: i-th Buck converter connected to the mG network
3.2
Model of the Agents
In this section, we describe the dynamical equations of a LVDC agent. As discussed in Section 2.4, we model the i-th LV-DGU as a DC/DC power converter, which steps down the input voltage Vin,i to obtain an output voltage Vi such that
Vi < Vin,i.
A converter with this feature is the Buck converter. Figure 3.2 provides a circuital representation of the Buck converter. The output voltage Vi supply a local DC
load, modelled as an ideal current generator Iload,i. The term Inet,i represents the
sum of currents that agent i receives from the rest of the network. As concerns circuit nonidealities, we consider the internal resistance Ri, but neglect other minor
elements such as the switch resistance and the diode voltage drop. We have to point out that the following approximations are introduced:
1. The DC voltage source is constant, and represents a generic renewable source. We consider this approximation reasonable since renewable power fluctuations take place at a slow timescale, compared to the one we are inter-ested in for stability analysis. Moreover, renewables are usually equipped with storage units damping stochastic fluctuations;
2. Loads are connected locally at the Point of Common Coupling (PCC) of each DGU. It has been shown that general interconnections of loads and DGUs can always be mapped into this topology using a network reduction method known as Kron reduction [2];
3. Quasi-stationary-Line (QSL) approximation. We consider the lines con-necting any couple of DGUs as purely resistive;
4. We assume the Buck converter to work only in Continuous Conduction Mode (CCM), which means that the inductance current Ii does not fall to
zero in any point of the cycle. This assumption is supported by the fact that modern switches are based on MOSFET or IGBT transistors that permit very high-frequency switching behaviour.
The Buck converter is equipped with a high-frequency switch Di, that enables
trans-itions into two possible discrete states, that we call “ON” and “OFF”. Hence, the dynamic of a DGU is described by means of a hybrid system (see Section 2.5)
˙
xi = Apxi+ bp, p = {on, of f } (3.6)
where xi = [ Vi Ii ]T is the state, made by the output voltage and the inductance
current. We aim to characterise the behaviour of a DGU in each of the two states: • “ON” STATE, when the switch Di is closed. In this configuration the
in-ductance accumulates energy coming from the input voltage, and the value of the inductance current increases. The (continuous) behaviour of the converter during this phase is
(
CiV˙i = Ii− Iload,i+ Inet,i
LiI˙i = −Vi− RiIi+ Vin,i
. (3.7)
Hence, the matrices that describe the system are
Aon,i= 0 C1 i −1 Li − Ri Li , bon,i= − 1 Ci(Iload,i− Inet,i) Vin,i Li .
• “OFF” STATE, when the switch Di is open. The inductance releases energy towards the load, and inductance current decreases. Converter’s behaviour is described by
(
CiV˙i = Ii− Iload,i+ Inet,i
LiI˙i = −Vi− RiIi
. (3.8)
Hence, the matrices that describe the system are
Aof f,i= 0 C1 i −1 Li − Ri Li , bof f,i= −1 Ci(Iload,i− Inet,i) 0 .
If the switching frequency fs of the converter is sufficiently higher than the cutoff
frequency of the LC filter, then the discontinuous model can be approximated by a continuous averaged model, by means of the so called state space averaging method. The reader may refer to Section 2.5, where a description of the technique and its approximations is provided. We introduce a new continuous variable di ∈ (0, 1),
diVin,i Ri I Li i Ci Vi Iload,i Inet,i mG Network
i-th DGU agent
PCCi
Figure 3.3: i-th DGU agent connected to the mG network
which constitutes the converter’s duty cycle, and its complement d∗i = 1 − di. Then
we introduce the following model
˙
xi= (diAon,i+ d∗iAof fi) xi+ dibon,i+ d
∗
ibof f,i, (3.9)
that represents the averaging between the “ON” and “OFF” dynamics, weighted over the fraction of time spent in each configuration (i.e. the duty cycle). The resulting averaged dynamics is
(
CiV˙i = Ii− Iload,i+ Inet,i
LiI˙i = −Vi− RiIi+ diVin,i.
(3.10)
We depicted in Figure 3.3 the linear circuit that represents equations (3.10), con-nected to the LVmG network.
From a control perspective, we notice that the only manipulable input is the duty cycle di. Since the duty cycle multiplies the constant voltage source, we can consider
directly Vin,ias the control variable. So, from now on, we incorporate the duty cycle
into Vin,i, which becomes the control input. We are aware of the fact that the duty
cycle is saturated between 0 and 1, but in this first stage we do not take into account saturations. Moreover, by referring to the notation we introduced in Section 3.1, we can describe the coupling term as
Inet,i = X j∈Ni wijgij(xi, xj) = X j∈Ni 1 Rij (Vj− Vi). (3.11)
In fact, the current circulating in a transmission line is proportional to the difference of voltages at the PCCs, and the proportionality coefficient is the inverse of the cable resistance. Now we have a complete description of a single DGU agent dynamics
ΣDGUi : ( CiV˙i = Ii− Iload,i+ P j∈Ni 1 Rij(Vj− Vi) LiI˙i = −Vi− RiIi+ Vin,i. (3.12)
We may also represent system (3.12) in compact form
ΣDGUi : x˙i = Aiixi+ Biui+ Miri+ ξi, (3.13)
where xi= [ Vi Ii ]T is the state, ui= Vin,iis the control input, ri = [ Iload,i 0 ]T
is the exogenous term, ξi=Pj∈NiAij(xj−xi) is the coupling term, and the matrices
are Aii= 0 C1 i −1 Li − Ri Li , Bi= 0 1 Li , Mi = −1 Ci 0 0 0 , Aij = 1 RijCi 0 0 0 .
3.3
Passivity of Open-loop Agents
As a preliminary analysis, we show that a single DGU, when disconnected from the network, is a passive agent. Consider the model (3.12), neglecting the coupling term Inet,i and the load current Iload,i, and introducing an output transformation yi = Ii:
CiV˙i= Ii LiI˙i= −Vi− RiIi+ Vin,i yi = Ii. (3.14)
We define the storage function VA,i(Vi, Ii), that represents the real energy stored in
the electric circuit
VA,i(Vi, Ii) = 1 2CiV 2 i + 1 2LiI 2 i, (3.15)
and we compute its derivative along the system’s trajectories ˙ VA,i(Vi, Ii) = CiViV˙i+ LiIiI˙i = Vi(Ii) + Ii(−Vi− RiIi+ Vin,i) = uiyi− RiIi2 Therefore uiyi= ˙VA,i+ RiIi2 | {z } ψA,i(xi) , (3.16)
where ψA,i(xi) ≥ 0 ∀xi ∈ R2. As a consequence, according to Definition 3 in
the background material, dynamical system (3.14) is passive (even if not strictly) with respect to input Vin,i and output Ii. This result can be intuitively understood
2.3.1. In fact, expression (3.16) is the energy balance of the DGU electrical circuit. We remember that the control objective is to derive suitable criteria to guarantee stability of each DGU, in spite of coupling terms. Hence, firstly we need to add a control loop to each DGU to assure reference tracking. Moreover, as we will explain in Sections 3.5 and 3.6, in order to prove network stability, we have to make sure that each closed-loop agents is passive with respect to output Vi and input ξi.
3.4
Design of Local Regulators
Make reference to system (3.13), and let Vref,i be the constant desired reference
trajectory for the output voltage Vi. In order to asymptotically track Vref,iwhen ri
is constant, we proceed, as in [20], by augmenting the model with an integrator
˙vi = Vref,i− Vi. (3.17)
We obtain in this way the following augmented dynamics
˙ˆxi= ˆAiixˆi+ ˆBiui+ ˆMirˆi+ ˆξi, (3.18)
where ˆxi = [ Vi Ii vi ]T is the augmented state, ui = Vin,i is the input, ˆri =
[ Iload,i Vref,i ]T is the exogenous term, ˆξi = Pj∈NiAˆij(ˆxj − ˆxi) is the coupling
term, with matrices
ˆ Aii= 0 C1 i 0 −L1 i − Ri Li 0 −1 0 0 , Bˆi = 0 1 Lti 0 , ˆ Mi= −C1 i 0 0 0 0 1 , Aˆij = 1 RjiCi 0 0 0 0 0 0 0 0 .
Now we equip the system with the following state-feedback proportional controller
Ci : ui= Kixˆi =
h
k1,i k2,i k3,i
i ˆ
xi, (3.19)
that, together with the integral action (3.17), defines a multivariable PI regulator (see Figure 3.4). Therefore, the closed-loop augmented system is
ˆ
Vin,i Ri I Li i Ci Vi Iload,i Inet,i mG Network
i-th DGU agent
PCCi
Ci
Figure 3.4: i-th closed-loop DGU agent
where ˆ Fi = ( ˆAii+ ˆBiKi) = 0 C1 ti 0 k1,i−1 Lti k2,i−Rti Lti k3,i Lti −1 0 0 . (3.21)
We want to point out that the overall control architecture is decentralised, since the computation of each control law depends only on the state of the specific DGU agent. Furthermore notice that, if coupling between agents is neglected when designing local regulators, asymptotic stability of the whole network might not hold (see Appendix A in [20]).
3.5
Network Stabilization with Optimal Regulators
Here the aim is to describe how to design local gains Ki so as to guarantee local
passivity of each DGU. After that, we exploit results in Section 2.3.2 to guarantee stability of the whole mG. We point out that the same stability result is presented in [20]. In this section we provide a different argument, based on passivity of local agents and properties of interconnected passive systems.
3.5.1 Passivity of closed-loop agents via LMIs
We want to prove passivity of the closed-loop DGU agents (3.20). As discussed in Section 2.3.1, the passivity property for a dynamical system is defined for a specific input-output choice. We consider the coupling term as the input
ˆ
ui = ˆξi, (3.22)
which represents the sum of currents that agent exchanges with the network. Notice that (3.22) is a fictitious control variable, because we cannot directly modulate how
it forces system (3.20). As concern the output transformation, we set ˆ yi= h 1 0 0 i ˆ xi = Vi. (3.23)
Moreover, since system (3.20) is linear, we can neglect the exogenous term ˆdi. Hence,
the dynamical system for which we want to prove passivity is ˙ˆxi = ˆFixˆi+ ˆui ˆ yi= Vi. (3.24)
Now we introduce the following structured storage function
VB,i(ˆxi) = ˆxTi Pixˆi = ˆxTi 1 2 0 0 0 p22,i p23,i 0 p23,i p33,i xˆi, (3.25)
where p22,i, p23,i, p33,i are arbitrary entries, but such that Pi > 0. The derivative of
VB,i(ˆx) along (3.24) trajectories is
˙ VB,i(ˆx) = ( ˆFixˆi+ ˆui)TPixˆi+ ˆxTi Pi( ˆFixˆi+ ˆui) = ˆxTi( ˆFiTPi+ PiFˆi | {z } −Qi )ˆxi+ 2ˆuTi Pixˆi = ˆxTi(−Qi)ˆxi+ 2 h Inet,i 0 0 i 1 2 0 0 0 p22,i p23,i 0 p23,i p33,i Vi Ii vi = ˆxTi(−Qi)ˆxi+ Inet,iVi = ˆxTi(−Qi)ˆxi+ ˆuTi yˆi. Equivalently, ˆ uTi yˆi = ˙VB,i(ˆxi) + ˆxTi Qixˆi | {z } ψB,i(ˆxi) . (3.26)
In order to guarantee passivity for (3.24), we must ensure that ψB,i(ˆx) is positive
semidefinite. This latter condition requires matrix Qi to be positive semidefinite.
Matrix Qi is defined as
ˆ
FiTPi+ PiFˆi= −Qi. (3.27)
The problem is then to find a triplet of gains (k1,i, k2,i, k3,i), such that equation
(3.27) is verified with ˆFi defined in (3.21), Pi in (3.25) and Qi ≥ 0. The same
problem has been addressed in [20], where the gains Ki are obtained via numerical
Each local controller Ci is then designed such that the inequality
ˆ
FiTPi+ PiFˆi+ Γ−1i ≤ 0 (3.28)
is verified. The term Γ−1i is intended to embody a margin of robustness, and it is defined as Γi = diag(γ1i, γ2i, γ3i) > 0. The optimization problem in [20] is the
following Oi : min Yi, Gi, γ1i, γ2i, γ3i, βi, ζi α1iγ1i+ α2iγ2i+ α3iγ3i+ α4iβi+ α5iζi (3.29a) subject to Yi> 0 (3.29b) " YiAˆTii+ GTi BˆiT + ˆAiiYi+BiˆGi Yi Yi −Γi # ≤ 0 (3.29c) " −βiI GTi Gi −I # < 0 (3.29d) " Yi I I ζiI # > 0 (3.29e) γ1i, γ2i, γ3i≥ 0, βi > 0, ζi > 0, (3.29f)
where αji, j = 1, ..., 5 are positive weights, Pi = Yi−1, Ki = GiYi−1and, by
construc-tion, kKik2 <
√
βiζi. Constraint (3.29b) guarantees matrix Pi to have the positive
definite structured form (3.25), while constraint (3.29c) guarantees Qito be positive
semidefinite. As concern the last two constraints (3.29d) and (3.29e), they allow to achieve a trade-off between the magnitude of Γi and the aggressiveness of the
control action, governed by the magnitude of βi and ζi. Since all the constraints in
(3.29) are Linear Matrix Inequalities (LMIs), the optimization problem Oi is convex
and can be solved in polynomial time. It is shown in [20] that problem (3.29) has a solution for a wide range of electrical coefficients Ri, Li and Ci.
Finally, we can conclude that, if gains Ki are computed via optimization problem
(3.29), then system (3.24) is passive with storage function (3.25).
3.5.2 Network global asymptotic stability
Our aim now is to use the results presented in Section 2.3.2, concerning the output synchronization on strongly connected graphs, to characterize the stability properties of the entire mG. To be more specific, we want to check if the global state ˆx = [ ˆx1 . . . xˆN ]T converges to ˆx = 0 as t → ∞. In order to apply Theorem 4, the
1. Each agent’s dynamics can be written as
˙
xi= fi(xi) + gi(xi)ui
yi = hi(xi).
(3.30)
Recalling that a single closed-loop DGU is described by equations (3.24), it is sufficient to impose fi(xi) = ˆFixˆi, gi(xi) = h 1 Ci 0 0 iT , and hi(xi) = h 1 0 0 i
to obtain the model (3.30).
2. Agents are coupled together using the control
ui =
X
j∈Ni
Kji(yj − yi).
In our case, the coupling term has the form
ui= X j∈Ni 1 Rji (Vj − Vi).
Therefore, we can impose
Kji=
1 Rji
. (3.31)
3. Agent’s dynamics are passive with a radially unbounded positive definite stor-age function.
The main result of the previous section is that, if feedback gains Ki are
com-puted via optimization problems (3.29), agents (3.24) are passive with radially unbounded positive definite storage functions (3.25). In fact, we can write
ˆ
uTi yˆi = ˙VB,i(ˆxi) + ψB,i(ˆxi)
| {z }
≥0
.
4. The communication graph is strongly connected
In Section 3.1, we modelled the LV mG as a weighed, undirected, strongly connected graph.
As a consequence of Theorem 4, it is guaranteed the existence of a global Lyapunov function V (ˆx) = N X i=1 ρiψB,i(ˆxi), ρi∈ R ∀i ∈ VLV, (3.32)
which is a linear combination of the single storage functions (3.25), such that ˙V (ˆx) ≤ 0. Therefore, by virtue of the LaSalle’s invariance principle (see Section 2.2), the solution of all dynamical systems (3.24) converge to the largest invariant set M contained in
E =nxˆi∈ R3, i ∈ VLV : ψB,i(ˆxi) = 0, (yi− yj)T(yi− yj) = 0 ∀j ∈ Ni, ∀i ∈ VLV
o .
In order to prove the asymptotic stability of the origin, we must show that the largest invariant set M is exactly ˆx = 0. We firstly need to characterize the set E. Recalling that ψB,i(ˆxi) = ˆxTi Qixˆi, we know that set E is composed by those vectors
that nullify the quadratic form of matrix Qi. It is shown in [20] that those vectors
take the form
ˆ xi = αi βi δiβi , with αi, βi∈ R and δi = − k2,i− Ri k3,i . (3.33)
Moreover, due to the output synchronization property (see Definition 4)
(yi− yj)T(yi− yj) = 0,
we are allowed to conclude that
αi= ¯α ∀i ∈ VLV. (3.34)
Then the set E has the following structure
E = ˆ xi ∈ R3 : ˆxi = h ¯ α βi δiβi iT , ∀i ∈ VLV . (3.35)
Now we use the definition of invariant set (see Section 2.2), which in our case takes the expression
ˆ
xi(0) ∈ E ⇒ ˆxi(t) ∈ E ∀t ≥ 0, (3.36)
to determine the set M . Suppose to solve state-space equations (3.24), starting from a point ˆxi(0) = [ ¯α βi δiβi ]T which is contained in the set E. We have
˙ˆxi(0) = ˆFixˆi(0) + ˆui(0) = 0 C1 i 0 k1,i−1 Li k2,i−Ri Li k3,i Li −1 0 0 ¯ α βi δiβi + X j∈Ni ˆ Aij(ˆxj(0) − ˆxi(0)) | {z } =0
= βi Ci k1,i−1 Li α¯ − ¯α .
In order to keep the trajectory inside the set E for any future time, we require
˙ˆxi(0) = βi Ci k1,i−1 Li α¯ − ¯α = ¯ α βi δiβi , (3.37)
which is a set of linear equations in ¯α and βi. We distinguish two cases:
1. ¯α = 0; therefore, the unique solution is ¯α = βi= 0 ∀i ∈ VLV;
2. ¯α 6= 0; it follows that βi= Ciα¯ βi= k1,i −1 Li α¯ βi= −δ1iα¯ =⇒ Ci = k1,i− 1 Li = −1 δi .
The condition k1,i−1
Lti = −
1
δi, as shown in [20], cannot be verified if the gains are
computed via optimization problems (3.29). Therefore, the unique feasible solution to the linear systems (3.37) is ¯α = βi = 0 ∀i ∈ VLV. This result allows us to conclude
that the largest invariant set M ⊂ E is the origin ˆx = 0, and thus the global dynamic is asymptotically stable. Moreover, since the global Lyapunov function (3.32) is radially unbounded, the origin ˆx = 0 is globally asymptotically stable.
3.6
Network Stabilization with Explicit Regulators
In the previous section, we showed that dynamical systems (3.18), equipped with regulators (3.19) with gains computed via optimization problems (3.29), produce a globally asymptotically stable networked dynamics. We were able to prove network stability from the passivity property of the single DGU agents. The proof of local passivity is based on the fact that local regulators are computed via LMIs. In this section, we present an alternative way to show passivity for the agents. In particular, we eliminate the necessity to solve optimization problems in the synthesis of the controllers. In fact, we directly compute an explicit set of the regulator gains that guarantee passivity for agents (3.24). After that, we show that also in this case the global asymptotic stability of the network is verified.
3.6.1 Passivity of closed-loop agents via explicit set analysis
In this section, we show that a single closed-loop agent (3.24) can be made pass-ive through a proper choice of the regulator’s gains. We report the state-space form of agents (3.24), where the input is the usual coupling term ui = Inet,i =
P
j∈Ni
1
RijCi(Vj− Vi), and we neglect the index i for simplicity of notation
ˆ ΣDGU : C ˙V = I + u L ˙I = (k1− 1)V + (k2− R)I + k3v ˙v = −V y = V. (3.38)
We define the following parametrized storage function
VC(ˆx) = ˆxTP ˆx = ˆxT 1 2C 0 0 0 L(k2−R) h Lk3 h 0 Lk3 h (k1−1)k3 h ˆ xi, (3.39)
where h = Lk3− (k1− 1)(k2 − R). The reader may refer to Appendix A, where
he/she will find the procedure we developed to obtain the form of storage function (3.39). Now we are able to state the following:
Theorem 6 (Passivity of a MV-DGU). Consider a storage function of the form (3.39). Then, if a triplet of gains (k1, k2, k3) is selected belonging to the set
Z = k1 < 1, k2 < R, 0 < k3 < L1(k1− 1)(k2− R) , (3.40)
the dynamical system (3.38) is passive.
Proof. We compute the derivative of VC(ˆx) along trajectories of (3.38):
˙ VC(ˆx) = ˙ˆxTP ˆx + ˆxTP ˙ˆx = 2 C 2V 1 CI + 1 Cu + L(k2− R) h I + Lk3 h v k1− 1 L V + + k2− R L I + k3 Lv k3I + (k1− 1)k3 L v (−V ) = uV + 2 h (k2− R) 2I2+ k2 3v2+ 2(k2− R)k3Iv = uy + 2 h (k2− R)I + k3v 2 .
Equivalently, uy = ˙VC(ˆx) + 2 (k2− R)I + k3v 2 −h | {z } ψC(ˆx) . (3.41)
At this point, in order to guarantee passivity (see Definition 3), we must ensure that the parametrized storage function (3.39) is positive definite, and that the function ψC(ˆx) is positive semidefinite, i.e.
VC(ˆx) > 0, (3.42a)
ψC(ˆx) ≥ 0. (3.42b)
The condition (3.42a) is equivalent to require matrix P to be positive definite. Since P is a real symmetric matrix, we can invoke Sylvester’s criterion [6], which states that a real symmetric matrix is positive definite if and only if all of its leading principal minors are positive. In our case, this requires that
• det 1
2C > 0, which is always satisfied;
• det 1 2C 0 0 L(k2−R) h = 1 2CL(k2− R)
h > 0, which is satisfied by k2 and h belonging to the set
Z1 = n {k2− R > 0} ∩ {h > 0}o∪n{k2− R < 0} ∩ {h < 0}o; • det 1 2C 0 0 0 L(k2−R) h Lk3 h 0 Lk3 h (k1−1)k3 h = 1 2C(L(k2− R)(k1− 1)k3− L 2k2 3) h2 = = − 1 2CLk3(Rk1+ k2− k1k2+ Lk3− R) h2 = − 1 2CLk3 h > 0, which is satisfied by k3 and h belonging to the set
Z2 = n {k3 > 0} ∩ {h < 0} o ∪n{k3 < 0} ∩ {h > 0} o .
The condition (3.42b) is verified when h ≤ 0, which is represented by the set
Z3 = {h ≤ 0}.
to the set Z, that is the intersection of the previous sets
Z = Z1∩ Z2∩ Z3. (3.43)
Now we aim to find an explicit expression for Z. We proceed by simplifying the set as follows Z =n{k2− R > 0} ∩ {h > 0}o∪n{k2− R < 0} ∩ {h < 0}o | {z } Z1 ∩ ∩n{k3> 0} ∩ {h < 0} o ∪n{k3< 0} ∩ {h > 0} o | {z } Z2 ∩nh ≤ 0 o | {z } Z3 =n{k2− R < 0} ∩ {h < 0} o ∩n{k3 > 0} ∩ {h < 0} o ∩nh ≤ 0o = {k2− R < 0} ∩ {k3> 0} ∩ {h < 0}.
So we find a first explicit condition, which is
k2− R < 0, (I)
and a second
k3 > 0. (II)
Additionally, we have that h < 0, and hence
h = Lk3− (k1− 1)(k2− R) < 0,
which implies
k3 <
1
L(k1− 1)(k2− R). (III) It has to be noticed that condition (III) is compatible with (I) and (II) if and only if
k1− 1 < 0. (IV)
By merging conditions (I),(II),(III) and (IV), we find an explicit expression of the set Z = k1 < 1, k2 < R, 0 < k3 < L1(k1− 1)(k2− R) ,
which concludes the proof of the theorem.
In conclusion we can state that, in order to make system (3.38) passive, it is sufficient to select a triplet of gains belonging to set Z. The only parameters we need to
synthesise the controller are the inductance L and the inductor resistance R. In this way, controllers can be designed a-priori, with no need to solve any optimization problem. As a last comment, we want to point out that the capacitance C does not affect the passivity property of DGU agents.
3.6.2 Network global asymptotic stability
Now, as in Section 3.5.2, we want to exploit Theorem 4 to prove global asymptotic stability of the whole mG. We already know that the theorem is applicable, because both the network and the agents are the same of Section 3.5.2. So we omit to show that conditions 1-4 hold also in this case. As a consequence of Theorem 4, we know that it exists a positive definite radially unbounded global Lyapunov function of the form (3.32), such that ˙V (ˆx) ≤ 0. Then, the overall dynamics ˆx converges to the largest invariant set M contained in
E =nxˆi ∈ R3, i ∈ VLV : ψC,i(ˆxi) = 0, (yi− yj)T(yi− yj) = 0 ∀j ∈ Ni, ∀i ∈ VLV
o .
Differently from Section 3.5.2, the passivity condition is here expressed by
uiyi = ˙VC,i(ˆxi) + 2 (k2,i− Ri)Ii+ k3,ivi 2 −h | {z } ψC,i(ˆxi) , (3.44)
where (k1,i, k2,i, k3,i) ∈ Zi. The function ψC,i(ˆxi) is nullified by state vectors with
Ii = −k2,ik3,i−Rivi and any Vi, that is by vectors of the form
ˆ xi = h αi −k2,ik3,i−Riγi γi iT .
Moreover, due to the output synchronization property (yi− yj)T(yi − yj) = 0, we
know that
αi= ¯α ∀i ∈ Vlv. (3.45)
Hence, set E has the following structure:
E = ˆ xi ∈ R3 : ˆxi = h αi −k2,ik3,i−Riγi γi iT , ∀i ∈ VLV . (3.46)
Now we use again the definition of invariant set (see Section 2.2) to determine the set M . Suppose to solve state-space equations (3.24) starting from a point ˆ
xi(0) =
h
αi −k2,ik3,i−Riγi γi
iT
˙ˆxi(0) = ˆFixˆi(0) + ˆui(0) = 0 C1 i 0 k1,i−1 Li k2,i−Ri Li k3,i Li −1 0 0 ¯ α − k3,i k2,i−Riγi γi + X j∈Ni ˆ Aij(ˆxj(0) − ˆxi(0)) | {z } =0 = − k3,i Ci(k2,i−Ri)γi k1,i−1 Li α¯ − ¯α . (3.47)
Then we impose trajectories to remain confined in set E, obtaining the following set of linear systems in ¯α and γi
− k3,i Ci(k2,i−Ri)γi k1,i−1 Li α¯ − ¯α = ¯ α − k3,i k2,i−Riγi γi . (3.48)
We analyse two cases:
1. γi = 0; then the system reduces to
h 0 k1,i−1 Li α¯ − ¯α iT =h α¯ 0 0 iT , (3.49)
which has ¯α = 0 as unique solution;
2. γi 6= 0; so it must hold, for example, the equivalence
− k3,i Ci(k2,i− Ri) γi= ¯α. This implies − k3,i Ci(k2,i− Ri) γi = −γi =⇒ k3,i = Ci(k2,i− Ri). (3.50)
Notice that curve (3.50) doesn’t belong to set Zi. In fact, (k2− Rt) is always
strictly negative in Zi and k3 strictly positive.
Since gains are needed to belong to Zi to guarantee passivity, we conclude that the
unique solution for systems (3.48) is ¯α = γi = 0 ∀i ∈ VLV. This result allows us
to state that the largest invariant set M ⊂ E is the origin ˆx = 0, and therefore the networked dynamics ˆx(t) is asymptotically stable. Furthermore, due to radially
unboundedness of Lyapunov function (3.32), the origin is a globally asymptotically stable equilibrium.
4
Medium-Voltage DC Microgrids
Stabilization
This chapter is devoted to the application of passivity theory to MVDC mGs. We firstly provide a model for the communication network and the non-linear agents in Section 4.1 and 4.2. Since we deal with non-linear systems, we study the property of mG equilibria in Section 4.3. After that, we introduce in Section 4.4 the PnP control architecture based on non-linear local regulators. Section 4.5 is devoted to show the passivity property for closed-loop dynamical agents, while in Section 4.6 the global asymptotic stability of the entire mG is proven.
4.1
Model of the Electrical Network
The model of the MV mG is different from the one we provided for the LV case. In fact, the LV network is represented by an undirected weighted graph (see Section 2.4), where each node hosts a DGU dynamical agent. In the MV case, we do not approximate the transmission lines as purely resistive (see Section 4.2.2), but we do consider the presence of inductors. As a consequence, we have to include in the modelling phase also the lines dynamics. We model the MV mG as a directed, unweighed, bipartite graph GM V = (VM V, EM V). The set of nodes VM V = {1, ..., N }
is partitioned into two subsets S1 (q nodes) and S2 (p nodes), such that any edge
(i, j) ∈ EM V connects only nodes of different subsets. The nodes of subset S 1 host
DGU dynamical agents, while nodes in subset S2host Line dynamical agents. Hence,
we can describe the communication network with the following p×q incidence matrix
B = [bij], i ∈ S1, j ∈ S2, bij = 1 if (i, j) ∈ EM V −1 if (j, i) ∈ EM V 0 otherwise. (4.1)
Notice that any Line agent has one edge that enters and one edge that exits the node. This topological condition is due to the fact that in a real mG the transmission lines
DGU1
DGU3
DGU2
DGU4
Line1 Line4 Line3
Line5
Line2
Figure 4.1: MVDC Network Example
allow, in a defined time instant, the current to flow between a couple of DGUs in a unique direction. We want also to stress out that, due to the edges directionality, the strong connectivity property does not necessarily hold for the MV network. In fact, in the example network of Figure 4.1, there not exist, for instance, any path from the DGU4 to any other node. If we, instead, ignore the directed nature of the
edges, it is always possible to find a path between any couple of nodes. In fact, any DGU is connected to at least another DGU, and any line connects exactly a couple of DGUs. Therefore, we conclude that the MV network is weakly connected. The example network can be described by the following 4 × 5 incidence matrix
B = 1 0 0 0 −1 0 0 −1 −1 1 −1 1 0 1 0 0 −1 1 0 0 ,
where the 4 rows represent the DGU agents, and the 5 columns the Line agents. We observe that, since any line is positioned between two DGUs, matrix B is zero-column-sum. In fact, any column has only one positive and one negative unitary element. Moreover, we can describe the coupling terms in the same way we did in Section 3.1. For DGU agents we have
ξi=
X
j∈Ni
bijgij(xi, xj), (4.2)
and for line agents
ξj =
X
k∈Nj
bkjgjk(xk, xj), (4.3)
