Robust Hybrid Control for Demand Side Management in Islanded Microgrids

IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 6, NOVEMBER 2021 4865 Robust Hybrid Control for Demand Side Management in Islanded Microgrids Carolina Albea , Carlos Bordons , Senior Member, IEEE,andMiguelA.Ridao ,Member, IEEE Abstract—This paper focuses on designing a robust control law to manage the demand response of islanded microgrids composed of shifting and adjusting loads. On one side, Hybrid Dynamical System theory is suited here, because the microgrid model is composed of continuous-time dynamics (the energy stored in the battery and the adjustable loads), and discrete-time dynamics (the shifting loads). On the other side, Multi Agent System theory is used to control the adjusting loads in order to guarantee a consensus between them. Hence, non-uniform convergence of the State of Charge of the battery to a given reference is ensured. Robustness with respect to plug and play of any load and parameter variations is also ensured. Experimental results from a laboratory-scale microgrid validate the approach. Index Terms—Islanded microgrid, demanded side management, hybrid dynamical systems, multi-agent systems. I. INTRODUCTION AMICROGRID is a system that integrates loads, generators and storage units actuating as a single controllable entity. Microgrids can operate in islanded mode, connected to the distribution network or connected to other microgrids. Microgrids facilitate the penetration of renewable energy sources in the electrical system, reducing energy losses and leading to a new electrical power systems paradigm [1], [2]. Demand Side Management (DSM) is a significant capacity of electrical networks that allows customers to make decisions regarding their energy consumption, and helping operators to reduce the peak load demand and to flatten the load profile. The use of DSM provides profit in many areas, such as: Manuscript received October 19, 2020; revised January 19, 2021 and June 10, 2021; accepted July 28, 2021. Date of publication August 2, 2021; date of current version October 21, 2021. This work was supported in part by the Agence Nationale de la Recherche (ANR)–France under Grant ANR-18-CE40-0022-01 (Project HISPALIS); in part by the Agencia Estatal de Investigación (AEI)–Spain under Grant PID2019-105890RJ- I00/AEI/10.13039/501100011033 (Project HYCAM); and in part by the Local Ministry of Economy and Knowledge of Andalusia under Grant US-1265917. Paper no. TSG-01491-2020. (Corresponding author: Carolina Albea.) Carolina Albea was with CNRS, LAAS, Université de Toulouse, 31400 Toulouse, France. She is now with the Department of System Engineering and Automatic Control, Universidad de Sevilla, 41092 Seville, Spain (e-mail: [email protected]). Carlos Bordons is with the Department of System Engineering and Automatic Control and the Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, 41092 Seville, Spain (e-mail: [email protected]). Miguel A. Ridao is with the Department of System Engineering and Automatic Control, Universidad de Sevilla, 41092 Seville, Spain (e-mail: [email protected]). Color versions of one or more figures in this article are available at https://doi.org/10.1109/TSG.2021.3101875. Digital Object Identifier 10.1109/TSG.2021.3101875 improvement in the efficiency of the system, security of supply and reduction in overall operational costs and environmental impact. On the other hand, the microgrid control problem becomes more complex since new degrees of freedom appear. While DSM was utility-driven in the past, it is moving towards a customer-driven activity in the near future [3] and it can be used in microgrids which are not connected to the distribution network. In DSM framework, Demand Response (DR) refers to the actions taken by customers that use information (mainly prices) to adjust their loads. The concept of DR encompasses the discretionary changes by consumers of their usual consumption patterns in response to external conditions and has attracted the attention of many researchers [4]. This kind of schemes can be used to avoid undesirable peaks in the demand curve that take place in some time periods along the day, resulting in a more beneficial rearrangement [5]. The same concept can be used inside a microgrid, where the loads take the role played by customers in the electrical network. In this way, the different loads in a microgrid can be adjusted in order to try to adapt the demand to the available generation, allowing the operation of the microgrid even in low-generation situations. Although DSM is a general concept and DR is related to the control actions taken on the load side, in many occasions the terms are used indistinctly. In microgrids, a responsive and controllable demand behavior can compensate the volatility introduced by intermittent resources on the generation side [6]. In this way, some loads can be adjusted (both in amplitude or in connection times) to contribute to fulfil microgrid operation objectives. This can be achieved by the introduction of new manipulated variables (both continuous and binary) in the problem formulation, which makes the control problem more complex. In a microgrid, loads can be manipulated up to a certain point. There are some critical loads whose demand must always be met. These uncontrollable loads have to be operated at a certain power and a certain time period that cannot be deferred. But, on the other hand, there exist controllable loads whose total consumption or occupied time duration can be modified, such as Electric Vehicles (EV) or Heating, Ventilating and Air-Conditioning (HVAC). Then, some loads can be reduced, shed or postponed during lack of supply or emergency situations or just to optimize the operation of the microgrid. Controllable loads can be classified as: i) deferable/shiftable if their connection can be stopped, restarted or shifted to other time slots, ii) adjustable/curtaible/variable if their consumption can be adjusted to a lower level if 1949-3053 c 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. 4866 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 6, NOVEMBER 2021 necessary [7]. However reducing consumption or shifting the load to some other points in time can affect the normal operation of the load, so a certain penalty might be associated to load variation/shifting when implementing the DR procedure and, possibly, also a priority of disconnection of the different loads. Two main methods can be used for demand management in a microgrid: •Shifting: this strategy considers the shifting of certain amounts of energy demand from some time periods to others, with lower expected demand, either in response to price signals or to generation forecast. The Energy Management System (EMS) has flexibility to defer some energy packets, but the total amount of power required by the load must be fulfilled for a certain period. •Curtailment/Shedding/Variation: this strategy consists of adjusting the magnitude of the power that can be demanded by loads when necessary. Accordingly, considering that loads have a certain degree of manipulation, the demanded power can be reduced during certain times in order to improve the operation of the microgrid or during contingencies. A maximum allowable level of curtailment must be specified for each load. Both mechanisms can be used in grid-connected or in islanded mode in order to improve the economic benefit. However, in the case of islanded mode they can be crucial, since the grid is not available to supply the loads in case of generation deficit. In this case, the required amount of curtailed load must be chosen according to the forecasted deficit and the available storage. There are several research works that address DR in smart grids or microgrids. The paper [5] presents a strategy based on a load-shifting technique formulated as a minimization problem and solved with a heuristic-based evolutionary algorithm. Another load-shifting application is done in [8] using Genetic Algorithms, minimizing the combination of generation cost and the inconvenience caused to the customer, which is modelled as a polynomial function of the shifting time. A scenario-based stochastic optimization approach is developed in [9] for real-time price-based DR management of residential appliances, which can be embedded into smart meters, considering time-varying electricity price uncertainties. A multiobjective optimization method and a Linear Matrix Inequality (LMI) approach are used in a case study of three microgrids connected to the grid in [7]. The minimization of the operation cost and the inconvenience caused due to shifting or curtailment of loads is a multiobjective optimization problem that may include binary variables associated to the connection/disconnection of loads. Thus, a mixed integer optimization problem must be solved. This can be done in an open-loop fashion or using feedback as done in Model Predictive Control (MPC) [10]. A DR method based on MPC is proposed in [11] considering constraints on the peak shift effect, and an event-triggered mechanism using aggregates. A strategy based on MPC to manage the energy resources of a residential microgrid using load curtailment, that promotes short term reduction of electricity demand in pre-defined hours is presented in [12]. In this work, a DR is proposed here for active power balancing islanded microgrids. The DR is dealt with both shifting and adjusting loads, that can model any demand, by using the Hybrid Dynamical System (HDS) theory given in [13]. On one hand, a control loop manages the shifting loads, through an optimization problem, which looks for leading the State of Charge (SOC) of the battery to a neighborhood of a given reference. On the other hand, another control loop drives the adjustable loads, by using Multi-Agent System (MAS) theory [14], achieving the convergence of these adjustable loads to any leader-follower consensus [15] with the aim at reducing the mentioned neighborhood. This approach guarantees a maximum of supplied loads such that the demand follows the generation. Some properties of the system are revealed, as the characteristic of a non-uniform convergence, and even, the scalability and robustness with respect to any parameter variation (e.g., any adjustable load, which is required to be constant) or a plug-and-play event. MAS and HDS theory have already been employed to control microgrids. Indeed, a hybrid cascade control algorithm was designed in [16]. Besides this, in [17], a multi agent algorithm based on a hybrid system was considered. A control law focussed on HDS and MAS theory, ensuring stability and robustness is proposed in [18]. These mentioned control works do not deal with a DR. Likewise, [19]–[21] provide a DR in microgrids by applying multi agent system theory. However, these references only consider shifting loads or adjusting loads, without dealing with both at the same time. Moreover, in [22], a DR algorithm based on hybrid multi-agent theory for microgrids was presented, but without stability and robustness guaranties. Different to many previous works, a DR mechanism that considers both kind of loads (shiftable and adjustable), which models any demand, is proposed here by using HDS and MAS theory. In particular, [5], [8], [9] deal with DR just for shiftable or adjustable loads. References [10]–[12] do not model the demand, as we do here with a discrete-time and continuoustime dynamics (shiftable and adjustable loads, respectively). Moreover, in [5], [7]–[12] is not given stability properties as the non-uniform convergence property given here. Moreover, focusing on the applied theory, HDS and MAS, we mention the works [16]–[18] that do not consider a DR as control problem and [19]–[22] that do not provide stability and robustness guaranties. Then, we highlight that HDS and MAS theory allow to model discrete-time and continuous-time dynamics and ensure stability and robustness of a DR problem. The rest of the paper is organized as follows: Section II presents the considered islanded microgrid and states the problem. Section III provides some preliminaries definitions. The proposed control is given in Section IV, which is validated through experiment results Section V. A conclusion section closes the paper. Notation: Throughout the paper Ndenotes the set of natural numbers and Rthe set of real numbers, Rnthe n-dimensional Euclidean space and Rn×mthe set of all real n×mmatrices. The set of non-negative real numbers is denoted by R≥0. 0n×mand 1n×mare the zero and ones matrices, respectively, of n×mdimension. Inis the identity matrix of dimension n. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. ALBEA et al.: ROBUST HYBRID CONTROL FOR DEMAND SIDE MANAGEMENT IN ISLANDED MICROGRIDS 4867 diag{a1,a2,...,an}is a diagonal matrix whose elements are a1,a2,...,an. rank(M)denotes the rank of the matrix M.x means the transpose vector of xand x+represents the value of xafter an instantaneous change. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION An islanded microgrid is a system composed of sources, loads and batteries interconnected by a bus line. Microgrids can have several renewable sources as well as different ways of energy storage. The methodology presented in this paper can be applied to any microgrid but, for the sake of simplicity, a microgrid with photovoltaic generation and hydrid storage by means of electricity (batteries and ultracapacitor) and hydrogen (electrolyzer, fuel cell and hydrogen tanks) will be considered. The different powers flows are: Pbat Battery power. PdDemanded power. Pgen Netted generated power. Pcl,nPower of the n-constant power load. Pvl,iPower of the i-variable power load. This microgrid presents a set of Constant Power Loads (CPLs) and Variable Power Loads (VPLs), that means, loads whose power is fixed or can change, respectively. Notice that Pgen represents the netted generated power including some power losses, which cannot be neglected. Moreover, the microgrid power balance equation can be written as: Pd(σ, t)=αPvl +γσPcl(t) Pbat(σ, t)=Pd(σ, t)−Pgen(t) where Pvl(t)=[Pvl,1(t), Pvl,2(t),...,Pvl,i(t),...,Pvl,I(t)]∈ RI ≥0is a vector composed of the powers of the IVPLs, α∈ {0,1}1×Iis an exogenous vector that gathers whether the i- VPL is connected (α(i)=1)or disconnected (α(i)=0), Pcl =[Pcl,1,Pcl,2,...,Pcl,n,...,Pcl,N]∈RN ≥0is a vector composed of the power of the NCPLs and γσ∈{0,1}1×Nis an input vector with σ∈N:={1,2,...,2N}, which means that the load n-CPL is connected (γσ(n)=1)or disconnected (γσ(n)=0). By some abuse of notation in the different powers, the dynamic in the SOC, which provides the level of energy in the storage unit, is given by ˙x=−η(Pbat)Pbat(σ, t) Cmax (1) η(Pbat)=1/η, if Pbat ≥0 η, if Pbat <0(2) where for readability, one adopts the notation x=SOC. Likewise, the maximum capacity of the battery in energy units is Cmax and the efficient coefficient of storage is η, which takes two different values, if the battery is discharging (1/η) or charging (η). The battery and the i-VPLs are designed such that SOCm≤x≤SOCM,(3) being SOCmand SOCMthe lower and upper bounds of the SOC, respectively, and Pm,i≤Pvl,i≤PM,i,(4) being Pm,iand PM,ithe lower and upper bounds respectively, for each i-VPL. This saturated model of loads can represent systems as EV charge or heating systems without violating temperature constraints, for instance. These conditions, (3) and (4), mean that the variables, xand Pvl,iare saturated by an upper and lower bounds. Hence, we define xs:=sat(x,SOCm,SOCM),(5) Pvl,s:=satPvl,1,Pm,1,PM,1,satPvl,2,Pm,2,PM,2,..., satPvl,I,Pm,I,PM,I,(6) where sat(s,sm,sM)=max{min{s,sM},sm}leads to saturate the variable ssuch that sflows in [sm,sM]. In the sequel, we will adopt, with abuse of notation, xand Pvl, instead of, xs and Pvl,s. Likewise, Pvl :=Pns vl +Ps vl,(7) with Ps vl :=Ps vl,1,Ps vl,2,...,Ps vl,i(8) Pns vl :=Pvl −Ps vl,(9) being Ps vl,i∈{0,Pm,i,PM,i}, i.e., each component of Ps vl is either the upper and lower bound of Pvl,iif the i-VPL is saturated, or 0 if the i-VPL is non-saturated. Besides, Pns vl is a vector composed of either the Pvl,iflowing in [Pm,i,PM,i], if the i-VPL is non-saturated, or 0 if the i-VPL is saturated. Moreover, the following conditions define a scenario characterized by a generated power that can supply some loads without getting empty or full the battery. Definition 1 (Balanced-SOC Mode): A microgrid defined by Eq. (1) and constrained by 11,NPcl + I  i=1 PM,i≥Pgen (10) 01,NPcl + I  i=1 Pm,i≤Pgen.(11) describes a functioning mode where the generated power can supply the demanded one, such that ˙xcan go to zero. This work is oriented to provide control laws to an islanded microgrid working in both: a functioning mode that satisfies conditions (10) and (11), and a microgrid that does not fulfil these conditions. In the sequel, one can consider Pns vl,i(t)as the variable to be controlled such that all Pns vl,i(t)converge to a consensus from a leader-follower consensus algorithm, through a distributed control. That means there is, at least, a VPL, with the role of leader, which provides the reference for the others Pns vl,i(t)as followers, leading the variable xto any objective. Moreover, this consensus algorithm provides robustness with respect to any parameter variation or plug-and-play mechanism given by α, that is, robustness with respect to a failure or connection/disconnection of any adjustable load, VPL. Moreover, xwill be also controlled by the variable input σ, which decides the connected CPLs. These N-CPLs will have a priority level Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. 4868 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 6, NOVEMBER 2021 to be connected to the microgrid, given by the following weight vector: β=[β1,β 2,...,β N].In particular, the controlled objective is that xconverges to a neighborhood of xr,k, which is a sampled reference that follows the variation of Pgen,kthat is, it seems reasonable that the higher solar generation, the bigger SOC reference and reciprocally. Indeed, the reference is selected as follows χPgen,k=Pgen,k−Pes gen,av 2Pes gen,amp Kxr+SOCM+SOCm 2 xr,k=satχPgen,k,SOCm,SOCM,(12) where Pgen,k=Pgen(tk)is an exogenous signal sampled with a sample interval Ts=tk+1−tk, with k∈N.Pes gen,av and Pes gen,amp define the average and amplitude of the estimated generated power, respectively. The SOC response will depend on this suited estimation, but is not the scope of this work. Note that the reference is composed of a bias given by SOCM+SOCm 2in order to have enough amplitude for the SOC response and an offset defined by Pgen,k−Pes gen,av 2Pes gen,amp Kxrin order to make better use of the microgrid. Finally, Kxr∈(0,1] is a constant parameter that decides the offset, i.e., how the variation of Pgen,kwill be taken into account by the reference. Note that if Kxris close to 0, the offset is 0 and xr,kis the bias SOCM+SOCm 2. Likewise, if Kxris close to 1 the offset is different to 0, i.e., when Pgen,k is close to its maximum (minimum) value, xr,kwill be near SOCM(SOCm) and when Pgen,kis approximately Pes gen,av,xr,k will be close to SOCM+SOCm 2. Likewise, it is necessary to avoid a Zeno behavior, guaranteeing a minimum dwell time, T,for the connected CPLs. After this system description, the problem can be stated. Problem 1: The objective is to ensure a DSM of Eq. (1) for abalanced SOC, through a hybrid formulation of the system with two control levels. 1) A closed loop managing the connected loads ensuring that the solutions to Eq. (1) converge to the set A:=(x,Pvl,σ,τ):x−xr,k<ε,, where xr,kis a sampled reference and τis a variable that represents a timer. This reference is computed from the microgrid capacities and the measured Pgen,k. Moreover, εis a parameter that describes the error given by the CPLs. 2) Another closed loop that establishes a consensus of all Pns vl,i, under a leader-follower criteria, such that αPvl + γσPcl(t)converges to Pgen,k. This fact implies that x converges to xr,k, ensuring robustness of the distributed system with respect to any plug-and-play mechanism or parameter variation. Note that the problem deals with a shared-load management, providing a scalable and robust controlled system. The first problem objective is to control the CPLs such that xconverges to a neighborhood of xr,k,|x−xr,k|<ε,and then, the second problem objective looks for controlling the connected and nonsaturated VPLs to guarantee the convergence of xto xr,k. The robustness guarantees mentioned in item 2), take into account any Pvl,ithat for any reason needs to be constant. III. PRELIMINARIES Highlighting that the microgrid is composed of i∈IVPLs, the idea is to control locally every one of these i-subsystems, in such a way that the system enjoys of suited convergence and robustness properties. The next definition about the Laplacian matrix is introduced to define a compact hybrid system that gathers all dynamics. Definition 2: The Laplacian matrix represents the undirected graph G(L,E), being L={1,2, .., I}the set of nodes and E⊆L×Lthe set of interconnections between agents. The neighbors of agent iare denoted by Ni(α) ={h∈ L:(i,h)αiαh∈E}. The graph Laplacian of the network is given by L(α) =[lih(α)] with lih(α) :−αiαh,if i∈Ni |Ni(α)|,if i=h,(13) where, |Ni(α)|denotes the number of neighbors of node i. Note that the Laplacian matrix L(α) is a symmetric positive semi-definite matrix [23]. Moreover some definitions about HDS theory given in [13] are reviewed here. Definition 3 [13]: The dynamical behavior described by differential equation (˙x=f(x)) or inclusion (˙x∈f(x))is referred as flow. Definition 4 [13]: The dynamical behavior described by difference equation (x+=g(x)) or inclusion (x+∈g(x))is referred as jump. Definition 5 [13]: A hybrid system H=(C,f,D,g)is defined as follows: H:˙x∈f,x∈C x+∈g,x∈D, where fand grepresent the flow and jump maps, respectively, and Cand Drepresent the flow and jump sets, respectively. In other words, the state xcan change according the differential inclusion ˙x∈fand difference inclusion, ˙x∈g, while in the flow set, Cor jump set, D, respectively. Note that an equation is a particular case of an inclusion. Definition 6 [13, Definition 2.3]: A subset dom ζ⊂ R≥0×Nis a compact hybrid time domain if dom ζ:=J−1 k=0([tk,tk+1,k])for some sequence of times 0 ≤t0≤ t1≤t2≤···≤tJ. Definition 7 [13]: Hybrid arcs φ(t,k)are solutions to a hybrid system (C,f,D,g), i.e., for each jump k∈N, the function t→ φ(t,k)is locally absolutely continuous on the interval Ik=t:(t,k)∈dom(ζ), being dom(ζ) the hybrid time domain defined in [13, Definition 2.3]. IV. HYBRID CONTROL Let us adopt here the paradigm given in [13], since the system is composed of continuous-time dynamics, (xand Pvl,i for all iloads) and discrete-time dynamics, (α). The first problem is focussed on proposing a solution for item 1 of Problem 1. To this end, one can define the time of the battery to achieve xr,k: T(x,σ):=x−xr,k η(Pbat)Pbat(σ,t) Cmax .(14) Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. ALBEA et al.: ROBUST HYBRID CONTROL FOR DEMAND SIDE MANAGEMENT IN ISLANDED MICROGRIDS 4869 Note, this definition means that T(x,σ) is the elapsed time when xreaches xr,kif the battery is getting empty as well as getting full. However Pbat(σ, t)=0 means that T=∞. Indeed, in this situation xcan flows forever, without reaching any bound of the battery. Now, one can define the cost function which is desirable to minimize: J(x,σ):=K T(x,σ) −βγσ,(15) where K>0 is a weight constant. It is worth noting that this cost function penalizes both a fast battery time of xto achieve, xr,k, and a higher number of disconnected CPLs. The last goal is related with the load priorities, given by the weight vector, β, which minimizes the penalties induced by switching off any CPL. The objective about minimizing the convergence time of xto a neighborhood of xr,kmay seem contradictory, because it is usual to minimize system responses. However, in a microgrid, it is not desirable that the battery gets neither empty, nor full very fast. In particular, if one considers a high xand a low Pgen,k, which implies a low xr,k, it is not desirable that xconverges to the low level xr,kvery fast, but that the battery gets empty slowly, maximizing its response time. The cost function J(x,σ) is updated in each jump, then we can manage the jumps establishing that no jump occurs whereas J(x,σ+)>J(x,σ) or a minimum dwell time, T, between two consecutive switching of σ, has not elapsed yet. This condition will be implemented in the flow and jump sets, below. Likewise, the control law will be encoded by a condition that improves the issues that induce the system to jump. Then, let us select the index jas follows: j∗:=argmin jK T(x,j)−βγj.(16) This argmin function looks for changing the functioning mode of each CPL: connected, disconnected; minimizing the xslope in order to obtain a smooth SOC evolution to reduce the battery degradation as well as maximizing the number of connected loads. Note that these objectives are opposite. Nevertheless, the first one is weighted by K. Indeed, as Kis larger, it will be more important to reach a reduced xslope in order to induce that xconverges slower to A. Then, it is possible to define the hybrid model scheme according to Definition 5, H: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎣ ˙x ˙ Pvl ˙σ ˙τ ⎤ ⎥ ⎥ ⎦ =f(ξ), ξ ∈C ⎡ ⎢ ⎢ ⎣ x+ P+ vl σ+ τ+ ⎤ ⎥ ⎥ ⎦ ∈g(ξ), ξ ∈D, (17) where ξ=[xP vl στ]∈Hsuch that H:=[SOCm,SOCM]× [Pm,1,PM,1]×[Pm,2,PM,2]×···×[Pm,I,PM,I]×N×{0,2T}. Remember that σand Pvl are the control inputs. The maps fand gare f(ξ) :=⎡ ⎢ ⎢ ⎣ −η(Pbat)αPvl+γσPcl−Pgen,k Cmax −KcL(α)Pvl −KcLPns vl −P∗ k1I×1 0N,1 rτ T ⎤ ⎥ ⎥ ⎦ , g(ξ) :=⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x Pvl argmin jK T(x,j)−βγj 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ,(18) x,Pvl and Pns vl follow the definitions given in (5)–(9). r(τ T)=min(1,2−τ T)induces a minimum dwell time, defined by the positive parameter, T. In this way, a Zeno behavior is prevented. Kc=diag{Kc,i,Kc,2,...,Kc,I}, where Kc,i>0 defines the convergence speed of each Pvl,i,to a consensus objective between the VPLs. Likewise, L:= diag{1, 2,..., I}, where i∈{0,1}, which means that if i=1, the i-VPL is a leader and if i=0, the i-VPL is a follower. Moreover, P∗ k(Ps vl,σ,Pgen,k)=Pgen,k−γσPcl−αPs vl αsα>0, with αs:=α1swPvl,1,Pm,1,PM,1,α 2swPvl,2,Pm,2,PM,2,..., αlswPvl,I,Pm,I,PM,I, denoting sw(s,sm,sM)=max{(s−sM)(s−sm), 0}, which switches the variable sfrom 0 to 1, when s/∈[sm,sM], and reciprocally. Note that this is a binary vector variable that detects if Pvl,iis saturated. In this way, the leader Pvl,iwill enforce that αPvl +γσPcl − Pgen,kgoes to 0. Note that the leader takes the information about the reference such that it is going to fulfil two objectives: ensure that the leader’s VPL converges to the reference and reach a consensus with its neighbors. Referring to leader(s) selection, on one hand the user has to select the VPL(s) with maximum power and reliability, ensuring that it will be the last one to be saturated and it will not be disconnected. Indeed, if the leader is saturated, the other nodes, that has not saturated yet, will converge to this value, blocking them to achieve a larger power. On the other hand, each potential leader can present different transient times. However, this transient time is tuned by the parameters Kc,i.AsKc,iis larger, the transient time will be shorter independently of the selected leader. Assumption 1: The microgrid has been designed such that rank(L)≤1, i.e., at least, there is one leader, that ensures the convergence of αPvl +γσPcl to Pgen,k. Moreover, the leaders can not be disconnected and will be the last ones to be saturated. Then, the so-called flow and jump sets are selected, C:=ξ:Jx,σ+≥J(x,σ) ∪{ξ:τ∈[0,T]}(19) D:=ξ:Jx,σ+≤J(x,σ)∩{ξ:τ∈[T,2T]}.(20) This regularization is motivated by the fact that jumps cannot happen when the solution cannot improve the cost function Jafter the jump, and the solution does not flow, at least, T ordinary time after the last jump. A solution to the item 2 of Problem 1 is included in the hybrid scheme (17)–(20). Note that the VPL dynamics are Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. 4870 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 6, NOVEMBER 2021 Fig. 1. Block diagram of the control mechanism, where fσ= argmin j(K T(x,j)−βγj)and fP=−KcL(α)Pvl −KcL(αPvl +γσPcl −Pgen,k). defined in the map f(ξ), by using the Laplacian matrix L(α), which takes into account the fact that a particular VPL is connected or disconnected. Moreover, matrix Lgathers the information about the leader and follower character of each VPL. The controlled system implementation is depicted in Fig. 1. Now, one can propose the following result for balanced- SOC mode (see Definition 1). Theorem 1: Consider a given Pm,i,PM,i>0 for each i-VPL such that condition (3) and (4) are satisfied and any given Kc,i>0 and sampled time Tssuch that 1 Kc,i<Tsfor all i=1,2, .., I. Likewise, the selected Lfulfils Assumption 1. Moreover, for a given parameters Kxr∈(0,1], and K>0, the following holds for any initial condition ξ(0,0)∈Hto system (17)–(20), 1) for all (t,j)∈dom ζ, the solutions ξ(t,j)to (17)–(20) converge to A. 2) All solutions to (17)–(20) eventually evolve continuously in the set A∩{ξ:x=xr,k}. Proof [Proof of item (i)]: The convergence proof cannot follow Lyapunov arguments, because the time to reach the desired attractor can take an arbitrarily long time, being non uniform. Due to this fact, the set Ais unstable (even though globally attractive). Without loss of generality, consider that the initial condition ξ(0,0)flows in C. Indeed, if the cost function is not susceptible to improve and the solution has not flowed a minimum-dwell time, T, the solution will remain evolving in the interior of C. When a condition of Cis violated that means that the solution will be in Dand a jump will occur. After this jump, the system will evolve in C, at least, for a minimum dwell time, i.e., until τ≥T. Note, that Ais attractive, from the map g(ξ). Indeed, the jumps of σensure that for xfar from xr,k,xwill approach slowly to it, and for xclose to xr,kand a fixed σ+,xwill converge to xr,k. Proof of item (ii): In order to prove that xconverges to xr,kduring flows, one needs to ensure that the solution in Cguarantees that the power of each non-saturated VPL, Pns vl,ifor i∈1,2, .., N, will converge to a consensus value P∗ k(Ps vl,σ,Pgen,k)=Pgen,k−γσPcl−αPs vl αsα>0. To this end, let us consider the following function VPns vl =1 2Pns vl TLαnsPns vl +Pns vl −P∗ kPsIILPns vl −P∗ kII, where αns :=α−αsHence, L(αns)represents the Laplacian matrix of the non saturated Pvl,i,Pns vl,i. From the definitions given in (7)–(9), it is hold ˙ Pvl =˙ Pns vl . Moreover, it is easy to see that Pns vl LαnsL(α)Pvl =Pns vl LαnsKcLαnsPns vl . Then, along flows: VcPns vl =−Pns vl LαnsKcLαnsPns vl −2Pns vl LαnsKcLPns vl −P∗ kII −Pns vl −P∗ kIILKcLPns vl −P∗ kII≤0. The objective here is to guarantee a consensus for the nonsaturated powers of the VPLs, Pns vl,i.Kc,iand Tsare selected such that the response time of each Pvl,iis shorter than Ts. Moreover, from the property of the Laplacian matrix, Vc(Pns vl )will be 0, when all Pns vl,iconverge to P∗ k= Pgen,k−γσPcl−αiPs vi αsα, given by the leader nodes, i(defined by i=1). As the level sets of V(Pns vl )are compact, the nonsmooth version of the LaSalle’s invariance principle [24] can be applied, guaranteeing that each Pns vl,iconverges to the consensus P∗ k. This last argument concludes the proof. Theorem 1 proposes a controller, when the microgrid is in balanced-SOC mode. Nevertheless, a microgrid is susceptible of violating condition (10) and (11), requiring to switch to another controller designed for this situation. A. A Non-Balanced SOC Microgrid If condition (10) (and/or condition (11)) are violated, i.e., there is an excess (and/or lack) of generated power compared to the maximum (and/or minimum) demanded power capacity, the variable σis forced to be connected in open loop. That means, the battery will become empty (and/or full) and ˙xcannot converge to zero. Unlike this control loop, the nonsaturated VPLs will be controlled following the consensus algorithm. Hence, one can define this hybrid system: Hsw : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎣ ˙x ˙ Pvl ˙σ ˙τ ⎤ ⎥ ⎥ ⎦ =fsw(ξ), ξ ∈H ⎡ ⎢ ⎢ ⎣ x+ P+ vl σ+ τ+ ⎤ ⎥ ⎥ ⎦ ∈gsw(ξ), ξ ∈∅, (21) fsw(ξ) :=⎡ ⎢ ⎢ ⎣ −η(Pbat)αPvl+γσPcl−Pgen,k Cmax −KcL(α)Pvl −KcLPns vl −P∗ k1I×1 0N,1 rτ T ⎤ ⎥ ⎥ ⎦ , gsw(ξ) :=xP cl σ0,(22) σ=11,N,if 11,NPcl +I i=0PM,i<Pgen,k σ=01,N,if 01,NPcl +I i=0Pm,i>Pgen,k.(23) Theorem 2: Consider a given Pm,i,PM,i>0 for each i-VPL such that condition (10) and/or (11) are not satisfied and any given Kc,i>0 and sampled time Tssuch that Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. ALBEA et al.: ROBUST HYBRID CONTROL FOR DEMAND SIDE MANAGEMENT IN ISLANDED MICROGRIDS 4871 TABLE I “HYLAB”DATA 1 Kc,i<Tsfor all i=1,2, .., I. Likewise, the selected Lfulfils Assumption 1. Then, all solutions to (21)–(23) uniformly asymptotically converge to the set {ξ:x=xr,k}. Proof: Note that (21)–(23) is not a hybrid dynamical system, but a continuous-time dynamical system, because the jump set is empty. In this functioning mode, the system can provide uniform asymptotic stability of {ξ:x=xr,k}. The proof follows the arguments given in item (ii) of Theorem 1, with either γσ=11×Nif condition (10) is not satisfied, or γσ=01×Nif condition (11) is not satisfied. B. Guideline for Parameter Selection In the HDS scheme proposed above there are some parameters to be tuned to adjust the reference, xr,k,(Kx,k), the CPLs (K) and the CVLs (Kc). We summarize here the rules to adjust these parameters. •Kx,r∈(0,1]: This parameter allows to modulate the varying behavior of the reference, xr,k. If the user desires that the reference presents a larger varying behavior, then Kx,r has to be closer to 1. However, if the user prefers a more static behavior of the reference, then Kx,rhas to be closer to 0. •K>0: This parameter manages the number of possible connected CPLs. If the user requires to increase the number of connected CPLs, then Khas to be larger. •Kc>0: This parameter drives the convergence speed of the CVLs to a consensus. Indeed, if the user needs that this convergence is fast, he/she has to chose larger Kc,i. Note that the stability and robustness properties given in the paper are guaranteed for all these possible combinations. V. EXPERIMENTAL RESULTS This section is devoted to validate the proposed control algorithm. The experiments are performed using a detailed dynamic model of a microgrid located at the University of Seville [25]. This microgrid is composed of a lead-acid battery bank, a lithium-ion battery bank, a photovoltaic field, an electrolizer, a fuel cell, hydrogen storage by means of two hydride tanks, N=7 CPLs and I=7 CVLs (emulated by the electronic load) and several DC/DC converters. The specifications of the different devices are described in Table I and Figure 2 shows a picture of the microgrid. The methodology for DSM is tested in this microgrid without using the hydrogen storage elements and with the parameters given in Table II in two different situations, depending on generation: a sunny and Fig. 2. “Hylab” microgrid. TABLE II PARAMETERS FOR DR TABLE III SELECTED PARAMETERS a cloudy day. The control algorithm must behave in a different way depending on the available solar power. The selected parameters for the controlled microgrid appear in Table III. Moreover, we select Kxr=1, which means that the chosen reference follows the dynamic of the sampled Pgen,k and a leader-follower consensus algorithm with 1 leader with the following Laplacian matrix: L(α) =α ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2−10000−1 −12−10000 0−12−1000 00−12−10 0 000−12−10 0000−12−1 −10000−12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ α The microgrid will work with the hybrid control proposed in (17)–(20), when we have a balanced-SOC mode, that means conditions (10) and (11) are fulfilled. On the contrary, the system will be controlled with the algorithm given in (21)–(23), which manages only Pns vl,i, because in this case all (or any) CPLs must be connected to the microgrid. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. 4872 IEEE TRANSACTIONS ON SMART GRID, VOL. 12, NO. 6, NOVEMBER 2021 Fig. 3. Power balance, showing the generated Power, Pgen, in green, the demanded power, Pdin red and the battery power, Pbat in blue, in a sunny day. The shaded area represents the period when condition (10) (blue shadow) and (11) (grey shadow) are not fulfilled. Fig. 4. Evolution of Pgen,k−11,NPcl −I i=0PM,i. The shaded area represents the period when condition (10) is not fulfilled. Fig. 5. Evolution of Pgen,k−01,NPcl −I i=0PM,i. The shaded area represents the period when condition (11) is not fulfilled. After some Pgen(t)evolutions, it is possible to characterize aPgen,kwhich evolves in the following slot 0W≤Pgen,k≤3500 W, such that, Pes gen,av =1750 W and Pes gen,amp =1750 W. A. Sunny Day First, a data base of Pgen(t)in a sunny day is taken, during a complete day (86400 seconds). As mentioned above, the proposed hybrid control will manage the microgrid, whereas the demanded power is larger than the generated one. Hence, Fig. 3 displays the Pgen(t),Pd(σ, t)and Pbat(σ, t)evolutions. It is worth noting that while condition (10) is satisfied, Pd(σ, t) converges to Pgen(t). Note that Figs. 4 and 5 show the validation of condition (10) and condition (11), respectively. If these ones are positive, it means that condition (10) and (11) are not satisfied, which is highlighted with a shaded area. Likewise, during the slot time that condition (10) and (11) are violated, the microgrid is controlled by the algorithm given in (21)–(23). Note that an excess of energy is stocked in the battery, leading toanegativePbat, as shown in the figure. Figure 7 and Fig. 8 show the CPLs and VPLs evolutions, respectively. On one hand, the picture draws the switching of the CPLs, according to the control input, σ. Likewise, note that these CPLs are connected and disconnected into the microgrid, Fig. 6. αscenario. governed by the priority levels, β. During the slot time when condition (10) or (11) are not fulfilled, all loads are supplied or any load is connected, respectively. On the other hand, the non-saturated VPLs evolve in consensus with an αscenario given in Fig. 6, where α4represents that VPL4is disconnected and that VPL3turns to be a connected constant load, during a slot time. Hence, the non-saturated VPLs achieves the discrete variable P∗ ksuch that |Pd−Pg|goes to zero, guaranteeing robustness with respect to plug and play. In particular, when any VPL is disconnected (αi=0), as VPL4with Pvl,4at 34560 s, when any Pvl,iturns to be constant, as VPL3with Pvl,3during the slot time [17280, 69120] s or when any Pvl,i saturates in its upper bound, as occurs for Pvl,7at 17769 s, Pvl,2at 21990 s, Pvl,6and Pvl,5at 22725 s , Pvl,4at 24266 s, and Pvl,4at 24266 s. Note that Assumption 1 is satisfied, i.e., the VPLs evolve in consensus, getting their upper bound, being the leader Pvl,1the last one to saturate. Figure 7 and Fig. 8 show the CPLs and VPLs evolutions, respectively. On one hand, the picture draws the switching of the CPLs, according to the control input, σ. Likewise, note that these CPLs are connected and disconnected into the microgrid, governed by the priority levels, β. During the slot time when condition (10) or (11) are not fulfilled, all loads are supplied or any load is connected, respectively. On the other hand, the non-saturated VPLs evolve in consensus with an αscenario given in Fig. 6, where α4represents that VPL4is disconnected and that VPL3turns to be a connected constant load, during a slot time. Hence, the non-saturated VPLs achieves the discrete variable P∗ ksuch that |Pd−Pg|goes to zero, guaranteeing robustness with respect to plug and play. In particular, when any VPL is disconnected (αi=0), as VPL4with Pvl,4at 34560 s, when any Pvl,iturns to be constant, as VPL3with Pvl,3during the slot time [17280, 69120] s or when any Pvl,i saturates in its upper bound, as occurs for Pvl,7at 17769 s, Pvl,2at 21990 s, Pvl,6and Pvl,5at 22725 s , Pvl,4at 24266 s, and Pvl,4at 24266 s. Note that Assumption 1 is satisfied, i.e., the VPLs evolve in consensus, getting their upper bound, being the leader Pvl,1the last one to saturate. Finally, the SOC evolution is shown in Fig. 9. Note that, in the initial condition, the SOC is in 80%, the CPLs are disconnected and the VPLs are in its minimum values 0 W. Indeed, at this initial condition the microgrid is controlled with Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply. ALBEA et al.: ROBUST HYBRID CONTROL FOR DEMAND SIDE MANAGEMENT IN ISLANDED MICROGRIDS 4873 Fig. 7. Power evolutions of the CPLs in a sunny day. The shaded area in blue and grey represent the period when condition (10) and (11), respectively, are not fulfilled. Fig. 8. Power evolutions of the CVLs in a sunny day. The shaded area in blue and grey represent the period when condition (10) and (11), respectively, are not fulfilled. Fig. 9. SOC evolution in a sunny day. The shaded area in blue and grey represent the period when condition (10) and (11), respectively, are not fulfilled. the hybrid control algorithm given in (21)–(23). When, condition (11) is fulfilled, the CPLs start connecting, reducing the SOC. Then, the SOC evolves following the tendency of Pgen, i.e., if Pgen increases, the SOC also goes up, and reciprocally. Let us highlight that the SOC achieves its maximum rank, when condition (10) is not satisfied any more. B. Cloudy Day Now, the proposed control is tested in an unfavourable situation, which is a cloudy day. In this simulation, one can consider the same scenario as in the sunny day, including the αevolutions, i.e., a disconnection of VPL4and a change of VPL3to a constant power load during a time slot, according to Fig. 6. In Fig. 10, the important oscillations of Pgen are appreciated besides not violating condition (10), as is shown in Fig. 11. Nevertheless, condition (11) is not satisfied during the slot time patched in the figure. The evolution of condition (11) Fig. 10. Power balance, showing the generated Power, Pgen, in green, the demanded power, Pdin red and the battery power, Pbat in blue, in a cloudy day. The shaded area represents the period when condition (11) is not fulfilled. Fig. 11. Evolution of Pgen,k−11,NPcl −I i=0PM,i. Fig. 12. Evolution of Pgen,k−01,NPcl −I i=0PM,i. The shaded area represents the period when condition (11) is not fulfilled. is shown in Fig. 12. It is worth noting that Pdconverges to Pgen even in this strong perturbed situation. Figures 13 and 14 show the CPL and VPL evolutions, respectively. Contrary to a sunny day, there are more switching in the less priority loads. Likewise, the VPLs evolve with strong oscillations, following Pgen,k. In this scenario, there is Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 20,2021 at 15:48:44 UTC from IEEE Xplore. Restrictions apply.