A Decentralized Energy Management System

Transcription

1 A Decentralized Energy Management System Ceyhun Eksin, Ali Hooshmand and Ratnesh Sharma Energy Management Department at NEC Laboratories America, Inc., Cupertino, CA, 9514 USA. {ceksin, ahooshmand, Abstract The primary goal of an energy management system (EMS) in power networks is to balance the supply and demand in a cost efficient manner given its operating horizon, and uncertainties in generation due to renewable generators and in demand. This goal is formulated as the economic dispatch problem. A centralized energy management system faces issues in scalability due to introduction of new generator or storage units and in robustness due to failures in some of the entities in the grid including the EMS itself. To alleviate these complexities a versatile decentralized energy management system (d-ems) is developed. The d-ems embeds a decentralized solution to the economic dispatch problem (EDP) based on the alternating direction method of multipliers (ADMM) inside a decentralized implementation of the receding horizon control. The ADMM based algorithm solves the EDP for the scheduling horizon. The receding horizon control allows the system to adapt to changes in the forecasts and network configuration. Decentralized protocols to handle changes to the communication network of devices is provided. These device failure and addition protocols entail network information updates only, thanks to the simple initialization of the ADMM algorithm. I. INTRODUCTION An EMS controls all the devices in a power network with the goal of cost effective performance while matching demand at all times. This goal is translated as the economic dispatch problem (EDP) [1]. A centralized management system that solves the EDP requires information from all devices, is prone to catastrophic failures. In addition, centralized EMS has scalability issue for any future expansion. The EMS is interrupted for the integration of any new device to incorporate its operation cost and device specific constraints to the algorithm that EMS implements. Similarly, when a device is up for maintenance a complete shutdown is required. That is, any failure in the EMS or a device in the network enforces a system-wide operation interruption. These issues can be overcome with a decentralized energy management system. We consider a decentralized solution to the economic dispatch problem (d-edp) which is an ADMM algorithm that operates on the dual of the EDP. The d-edp derivation entails reformulating the dual of the EDP as a consensus problem on the price of power imbalance, that is, each device keeps a local price for power imbalance but is constrained to agree with its neighbors on the local price (Section III-B). This reformulation admits a decentralized solution using the ADMM algorithm. In the d-edp, each device synchronously updates individual power and storage profile variables as well as the local price variable by solving a min-max problem. Then devices exchange their local prices with their neighbors and do an ascent step on the dual variables of the local consensus constraints. It is shown in [] that the algorithm converges to the optimal solution asymptotically when the original problem has strong duality and is convex, and the network is connected. We argue that the asymptotic optimality result carries over to the EDP when the network is connected and cost of each device is convex (Section III-C). Numerical implementations show that convergence to the optimal solution is fast and furthermore a near feasible solution is reached early (Section IV). Here we explore the effects of communication network topology and show that the convergence can degrade with the diameter of the network. The EMS faces uncertainty in supply due to renewables and in demand. As a result the EDP is solved based on predictions of renewables and demand into the time horizon. As time progresses EMS can correct its predictions and make new predictions for the new horizon based on new information revealed. To this purpose, we consider a receding horizon control which makes up for prediction errors by solving the EDP for the whole horizon, applying the first time step of the scheduled optimal actions and then solving the EDP at the next stage based on updated forecasts. In Section V, we present a communication protocol that allows for a decentralized implementation of the receding horizon control. The d-edp coupled with the receding horizon control amounts to a fully decentralized EMS (d-ems). Finally, we consider scalability and robustness. We show that the d-ems can incorporate new devices that register and handle device failures on the fly via a simple update of network information during receding horizon control algorithm execution (Section VI). A. Literature Review Previous efforts to solve EDP in a decentralized manner can be separated into two categories based on whether they are anytime feasible or not. Anytime feasible algorithms assume the start from a feasible point and updates remain feasible matching supply and

2 demand at all times [3] [5]. All of these algorithms are synchronous and gradient based, that is, the change in power in each update is a linear function of the graph Laplacian multiplied by the gradient. These algorithms require initial feasibility to remain feasible. Hence, when network is time varying they require another algorithm that determines a feasible starting solution. In addition it is not clear how the algorithms can be modified to include storage units. Among these algorithms only [3] can handle changes in the network. The not anytime feasible approaches are divided into consensus gradient algorithms [6] [8] and primal-dual subgradient algorithms [9]. The proposed consensus gradient based algorithms work for cost functions that have quadratic form. Except for [6] they require consensus iterations to converge at each iteration before updating the power in the next time step. It is not clear how to incorporate storage unit constraints to these consensus based algorithms. Finally, while the subgradient based algorithms can handle asynchronous updates, they are known to have slow convergence which is problematic since the algorithms are not anytime feasible. More recently, an ADMM based algorithm called proximal message passing is proposed in [1]. The model that is considered in [1] incorporates optimal power flow (OPF) equations for AC and DC devices. The algorithm operates on the primal OPF problem. This requires that the local power imbalance at each iteration is known by the devices in the same net where a net is a lossless energy carrier responsible for maintaining power balance among terminals that are connected to it. Consequently, the communication network needs to align with the transmission lines and the approach relies on entities called nets on transmission lines, e.g., power routers, that transmit power imbalance related information to the devices it connects. This makes the decentralized solution prone to failures upon failure of a net device. While the proposed d-edp is not anytime feasible, through numerical experiments we show that feasibility is achieved fast, i.e., an order of magnitude faster than convergence to optimality. Furthermore, the d-ems can handle failures and additions by simply restarting the algorithm after updating the total number of devices and each device s neighborhood set. The d-edp uses a synchronous decentralized consensus based ADMM algorithm which has been shown to converge to the optimal operating points for convex optimization problems with strong duality []. Furthermore, ADMM algorithms empirically have been shown to have faster convergence than subgradient algorithms [11], [1]. For strongly convex functions the decentralized consensus ADMM has linear convergence rate [13], whereas convergence rate of subgradient algorithms is sublinear. Moreover, the proposed d-edp incorporates storage units that have dynamic state of charge. Finally, the communication network does not have to align with a power network and each node in the network corresponds to a device. II. ECONOMIC DISPATCH PROBLEM In energy systems, EDP considers cost optimal power dispatch decisions to match the load profile. We use d(h) to denote the predicted demand at time h H and the demand profile is defined as d := [d(1),..., d(h)]. We assume that the load profile d is known or d represents the predicted load profile. The energy system is composed of devices N that can generate, store or generate and store power. We use G and S to denote the set of generator and storage units, respectively. The set of all devices in the system is then the union of generator and storage units, N := G S. A generator unit i G has the ability to inject p i (h) amounts of power to the system at time h H not to exceed its generation capacity p max i (h). The generation profile p i := {p i (h)} h H results in monetary cost of C i (p i ) for the system where C i (p i ) is some increasing function that maps load profile to positive reals R +. A storage unit i S can charge/discharge its battery by s i (h) amounts of power not to exceed its maximum charge/discharge amount s max i (h). When s i (h) >, we say that the storage unit charges its battery otherwise we say that it discharges its battery. The battery s state of charge at time h H is denoted by q i (h) and is modeled by the following difference equation, q i (h) = q i (h 1) + αs i (h) (1) where α is the coefficient converting kw units into Ah. The state of charge cannot exceed qi max amounts at any point in time due to the specifications of the battery, that is, q i (h) qi max for any h H. The initial state of charge level q i () is assumed to be given. We use Ω Gi for the set of feasibility constraints of power generation of device i N, that is, p i Ω Gi. Similarly, we use Ω Si to denote the set of feasibility constraints for the storage profile of i N, s i := {s i (h)} h H Ω Si. Given the constraints regarding device specifications the EDP chooses generation p := {p i } and storage s := {s i } profiles that will match supply and demand while minimizing cost, C i (p i ) () s.t. min p,s p i s i = d (3) p i Ω Gi, s i Ω Si for all i N. (4) The supply demand matching constraint (3) couples the decision variables of the devices. We denote the

3 optimal power and storage profiles to the above equation with p and s, respectively. Next, we provide a fully decentralized algorithm based on ADMM that converges to the optimal power generation and storage values. III. DECENTRALIZED EDP We first present a general overview of the ADMM algorithm see [14] for a more detailed explanation and then restructure the EDP such that the application of the ADMM algorithm emits a decentralized solution. A. ADMM Algorithm Define the variables x X R n, z Z R m. The generic form of the problems that the ADMM algorithm provides decentralized solutions to contain objective functions f( ) : X R and h( ) : Z R and linear equality constraints, min f(x) + h(z) (5) x X,z Z s.t. Ax + Bz = c where A R k n, B R k m and c R k. Note that the objective of the above optimization problem has the form that is separable with respect to its variables while its constraint couples the variables. The ADMM operates on the augmented Lagrangian defined as L ρ (x, z, λ) :=f(x) + h(z) + λ T (Ax + Bz c) + ρ Ax + Bz c (6) where λ R k is the price associated with violation of the equality constraint and ρ > is a penalty parameter that penalizes infeasibility of (5). The algorithm consists of a coordinate descent in the primal variables in an alternating manner followed by an ascent step in the price variable, x t+1 := argmin L ρ (x, z t, λ t ) (7) x z t+1 := argmin L ρ (x t+1, z, λ t ) (8) z λ t+1 := λ t + ρ(ax t+1 + Bz t+1 c) (9) The minimization of the augmented Lagrangian with respect to x at iteration t + 1 requires values of other variables from iteration t, namely, primal variable z t and price variable λ t in (7). The minimization of the augmented Lagrangian with respect to z at iteration t+1 requires the updated primal variable x t+1 and the price variable λ t in (8). The order of primal updates can be interchanged, that is, we can update z first and then x; however, the second variable that is updated still requires the updated value of the first primal variable. The dual ascent step at iteration t + 1 in (9) uses the updated primal variables x t+1 and z t+1 to ascent with step size equal to the penalty parameter ρ. While the form of the EDP in ()-(4) belongs to the type of problems that ADMM is designed for, the ADMM algorithm presented above is not a fully decentralized update. It is possible to see this from the discussion above where each primal variable update requires previously updated primal variables. This means that devices need to receive the most recent updates from all of the devices that updated before them in order to update their primal variables. Furthermore the dual ascent step (9) requires a centralized coordinator that has access to network-wide updated primal variables. In the next section we introduce a communication network and present a decentralized solution to EDP utilizing the dual consensus ADMM (DC-ADMM) presented in []. B. d-edp using ADMM Consider a connected network with set of nodes corresponding to devices in the grid N and an edge set E where pair of nodes (i, j) belong to E if i can send information to and receive information from j, i j, that is, E := {(i, j) : i j, i N, j N }. The neighborhood set of i is the set of agents from which agent i can receive information from N i := {(j, i) E : j N }. We adopt the convention that device i is not the neighbor of itself, that is, i / N i. We relax the coupling equality constraint (3) of the EDP problem with the price variables λ to obtain the following Lagrangian L({p i }, {s i }, λ) = ( C i (p i ) + λ T (p i s i ) λ T d/n ). (1) The dual function for the relaxed EDP is obtained by maximizing the negative of the above Lagrangian with respect to the primal variables which we can do separately for each device, g(λ):= max p i Ω Gi s i Ω Si ( C i (p i )+λ T (p i s i ) λ T d/n ). (11) We define the local dual function resulting from maximization of local variables of i as g i (λ) := max pi Ω Gi,s i Ω Si C i (p i ) λ T (p i s i ) and rewrite the dual function above as a sum of local dual functions, g(λ) := g i (λ) + λ T d/n. (1) The minimization of (1) with respect to λ yields the optimal price variables and the optimal primal variables when the original EDP problem in () has zero duality gap. However, λ is a global variable associated with the equality constraint in () and solution to (1) requires information from all devices. In order to solve the dual problem above in a decentralized manner we introduce

4 the local copies of the price variable, that is, i s local copy of λ is λ i. Then we can equivalently represent the minimization of (1), min λ g(λ) in terms of local copies of the price variable given a connected network, min g i (λ i ) λ T i d/n (13) λ 1,...,λ N,{γ ij} s.t. λ i = γ ij, λ j = γ ij for all j N i, i N (14) where γ ij are the local auxiliary variables. Note that in a connected network the solution of the above optimization is equivalent to solving min λ g(λ). Further observe that the above optimization is of the form in (5) which implies we can derive an algorithm using the same arguments as in Section III-A. We first form the augmented Lagrangian for the above problem using the dual variables u ij and v ij for the consensus constraints in (14) with the penalty constant ρ >, L ρ (λ 1,..., λ N, {γ ij, u ij, v ij })= g i (λ i ) λ T i d/n + u ij (λ i γ ij ) + v ij (γ ij λ j ) + ρ λ i γ ij + λ j γ ij (15) Define the set of price variables of all the agents λ := {λ i } and the set of all auxiliary variables as γ = {γ ij } j Ni,. When we apply the ADMM steps in (7)- (9) to the above augmented Lagrangian, we have the following steps at iteration t: λ t+1 = argmin L ρ ( λ, γ t, {u ijt, v ijt }) (16) λ γ t+1 = argmin L ρ ( λ t+1, γ, {u ijt, v ijt }) (17) γ u ijt+1 = u ijt + ρ ( ) λ it+1 γ ijt+1 (18) v ijt+1 = v ijt + ρ ( ) λ jt+1 γ ijt+1 (19) Starting from the auxiliary variable updates (17) and primal variable updates (16) and using their decomposable structure into i N i and N quadratic subproblems, respectively, [1] argue inductively that the set of updates above (16)-(19) simplifies and decouples into the following updates y it+1 = y it + ρ λ it λ jt () λ it+1 = argmin g i (λ i ) λ T i d/n + yjt+1λ T i λ i + ρ λ i λ it + λ jt (1) where we define y it := u ijt + v ijt when initial dual variables are all zero u ij = and v ij =. As noted in [], the minimization in (1) is actually a minmax optimization problem that implicitly includes the maximization of the primal variables p i and s i in the dual function g i ( ). When C i ( ) is convex and strong duality holds for the EDP ()-(4), they provide the following closed form solution for the min-max problem in (1) using the minimax theorem in [15, Proposition.6.], λ it+1 = 1 N i ( (λ it + λ jt ) + 1 ρ (p it+1 s it+1 1 N d) 1 ρ y it+1), () (p it+1, s it+1 ) = argmin C i (p i ) + ρ p i Ω Gi,s i Ω Si 4 N i 1 (p i s i 1N ) ρ d 1 ρ y it+1 + (λ it + λ jt ). (3) Observe that the price variable update in () requires the updated primal variables of time t + 1 from (3). Hence device i first updates the primal variables and then the price variable. The updates for device i are summarized in Algorithm 1. Algorithm 1 solves the dual EDP in a fully decentralized manner. The initialization consists of setting dual variables y i to zero. Other variables p i, s i, λ i can be arbitrarily set. The algorithm at time t starts by sending local price variables λ it and observing neighbors local price variables λ Nit := {λ jt : j N i }. The algorithm is synchronous in that device i requires prices of all of its neighbors from iteration t. Then in step device i averages the neighbors price variables to update its dual variable (). Along with observed price variables λ Nit, the dual variables y it+1 are used to update the primal power and storage variables following (3) in step 3. Finally in step 4 device i updates its local price variable λ it+1 using all of the observed and updated variables according to (). The algorithm continues by moving Algorithm 1 d-edp updates at device i Require: Initialize primal variables p i, s i, λ i and dual variables y i =. Set t =. Require: Determine stopping condition, e.g., maximum number of steps T N +. while Stopping Condition Not Reached do [1] Transmit λ it and receive λ jt from j N i. [] Compute y it+1 using (). [3] Update primal variables p it+1, s it+1 using (3). [4] Compute prices λ it+1 using (). [5] Set t= t+1. end while

5 the iteration step forward. The derivation above is worth retracing. The primal EDP problem in ()-(4) contains a power balance constraint that couples the variables of all the devices. Therefore, we consider a decentralized solution operating on the relaxation of the power balancing constraint. The relaxed problem in (1), i.e., the dual EDP entails a global price variable λ that is associated with the price of power imbalance. We then write an equivalent representation of the dual EDP as a dual consensus EDP in (13)-(14) where each device carries their local copy of the price. Applying the ADMM algorithm to the dual consensus EDP yields a fully decentralized solution. C. Convergence Properties of the d-edp According to Theorem in [] the iterations of Algorithm 1 asymptotically converge to optimal variables when the problem is convex, the network (N, E) is connected and strong duality holds. We assume that the cost C i (p i ) is convex in p i Ω Gi and the network is connected for the centralized EDP in ()-(4). A sufficient condition for strong duality is Slater s condition which requires that there exists a strictly feasible point. For a set of linear equality and inequality, this condition relaxes to the existence of a feasible point. For the EDP problem in ()-(4) the constraints are all affine. Furthermore, we assume that the energy system is connected to the grid which can satisfy demand at all times. This assumption makes sure that there exists a feasible point {p i, s i } that satisfies power balance constraint (3) and other device specific constraints (4). Consequently, the assumptions of Theorem in [] are all met and the iterations of Algorithm 1 converge to the optimal, that is, p it p i, s it s i. The d-edp is such that the solution yields optimal local prices along with the optimal power and storage profiles, that is, λ it λ. The local optimal prices can then be used to design smart pricing policies such as real time pricing for demand response management [16]. IV. NUMERICAL IMPLEMENTATION OF D-EDP We consider a microgrid with a single battery (B), a diesel generator (DG) and a photo-voltaic () generator and a connection to the grid (G). Including the grid the number of devices is equal to N = 4. The operation cycle is 4 hours with hourly scheduling decisions, that is, H = 4. The battery has a storage capacity of kwh, qb max = kwh. We assume charging or discharging the battery has no cost and roundtrip efficiency is 1%. The diesel generator maximum power is.kw p max DG =.kw. The cost of dispatching p DG(h) at time h is equal to zero when the power dispatched is zero, otherwise it is a linear function of p DG (h) > with slope a =.5$/kW and a positive y-intercept equal to b, that is, C DG (p i (h)) = b1(p i (h) > ) + ap i (h) where 1( ) is an indicator function [17]. Note that the cost function is discontinuous when the intercept is strictly positive, b >. In this case the problem is formulated as a mixed integer programming problem which is not convex, that is, violates the assumptions of convergence in Section III-C. For the numerical analysis we allow the convexity assumption to be violated and set b =.5$. The generator profile p max P V is determined a priori based on collected data. This profile determines the amount available for dispatch at each hour and is shown by the dashed line in Fig. 4. There are no costs for using the power. We assume that the grid can supply all the power that the microgrid can require. The grid electricity cost is set according to a time-of-use rate tariff. In this tariff, the baseline price.3 jumps at time h = 8 to.1 $/kw, that is, price G (h) = (h 8) $/kw. Note that the grid is cheaper than DG until h = 8 and afterwards it is cheaper given the power dispatched is less than 1kW. The demand profile is determined from the real load data of a commercial building over the operation cycle and is shown by the solid line in Fig. 4. All devices know their individual specifications and the forecasted demand profile. The d-ems has been implemented in the Java Agent Development Framework (JADE) which is a platform for decentralized applications in Java [18]. We test the convergence properties of the algorithm in four generic networks, namely line, star, ring and fully connected network as depicted in Fig. 1. Fig. shows the total cost convergence for each communication network setup. The results show that the line network is the slowest to converge and the fully connected network is the first to converge. This indicates that the convergence rate degrades with the increasing diameter of the network. All network structures converge in less than one thousand iterations to a near optimal cost computed by solving the centralized EDP. We further observe in Fig. 3 near feasibility is achieved at least an order of magnitude earlier than convergence to optimal value in less than 5 iterations. The computation time of 15 iterations takes less than one minute. While the network structure has an influence on the convergence rate, a decentralized solution will be more computationally effective than a centralized solution as the number of devices in the network grows. That is, a d-ems will scale better in comparison to a centralized solver [1]. Fig. 4 shows the optimal power and storage profiles of the devices. In the first seven hours the source of total dispatch is the G. Note that the G power exceeds the demand in this period because extra power is used to charge the battery. The reason for charging the battery is the price increase in the G power by hour eight. Note that the demand between the hours h = 8 and h = 1 cannot

6 DG DG G B G G B DG B Fig. 1. Generic line, star and ring networks of four devices. Each device can only communicate with its neighbors. Power (kwh) G DG B Demand Max Total Cost Iteration Number LINE STAR RING FULL OPT Fig.. Convergence to optimal cost for generic networks. Devices apply the d-edp in Algorithm 1. Line network converges to the optimal value the slowest while the fully connected network is the fastest to converge. All networks converge within 1 iterations. fully be matched by the available generation, p max P V indicated by the dashed line. During these hours and stored power is used together to meet the demand. After hour h = 11 the available is used to charge the battery. The stored power between the hours h = 1 to h = 19 is used to meet demand in the final hours h = to h = 4 together with the DG power. Note here that even though DG power is more expensive than G power Absolute Supply Demand Iteration Number LINE STAR RING FULL Fig. 3. The evolution of the supply and demand infeasibility gap. The x-axis is the iteration number t and the y-axis is the h (p it(h) s it (h)) d(h). All network types achieve feasibility within 5 iterations which implies feasibility is achieved an order of magnitude faster than convergence to optimality Time (Hours) Fig. 4. Optimal power and storage profile for the device network in Fig. 1. The solid line represents the demand profile. The dashed line is the availability over the horizon p max P V. The power dispatch bars are color coded according to the devices (G,, DG and B) as shown in the legend. In the first 7 time slots, grid power is used to charge the battery. The stored power is used for matching demand together with available power in times h = {8, 9} when the grid price is increased. Between time slots 1 h 18 the power is used to supply demand and store power in the battery. The stored power is then discharged when power is not sufficient to meet the demand after h =. Diesel dispatches power after time h = 1. This is due to the fact that storage capacity of the battery device is not enough to meet the demand from h = to h = H H Fig. 5. Receding horizon control for the operating horizon H. At time h = the system solves the EDP in ()-(4) for the optimal power and storage schedules p i and s i. Then the devices apply the first element of the optimal schedule p i (1) and s i (1). In the next time step devices plan for the next H time steps and the process continues. for the amounts required at time h = to h = 4, the algorithm uses DG power to balance power. This implies that the algorithm converges to a suboptimal point which can happen when the problem is not convex. The optimal schedule in Fig. 4 is obtained based on the load profile d and generation p max P V forecasted at hour h =. As time progresses these forecasts might be updated for the remaining horizon. Furthermore there might be changes to the device network as new devices are added or devices leave due to failure or maintenance. We would like the d-ems to be adaptable to changes in predictions and robust to changes in device network. In order to incorporate prediction changes we use the receding horizon control method that allows for forecast updates at each step of the planning horizon. We further propose a communication protocol in the next section that allows for the decentralized implementation of the receding horizon. In Section VI, we provide a decentralized protocol that allows the system to handle device failures and additions on the fly. V. DECENTRALIZED RECEDING HORIZON CONTROL Given a time horizon H a centralized receding horizon control for EDP solves the optimization in () for the

7 whole horizon to obtain power and storage profiles but only uses the first element of the optimal profile for step h =. Before h = 1, the EDP is solved for hours h = 1 to H +1 and the scheduled element for h = 1 is applied and the horizon is propagated by one again. This process is schematically shown in Fig. 5. In order to implement the decentralized receding horizon control, we use the d-edp algorithm for each hour and furthermore we require a communication protocol that synchronizes devices for starting the updates for the next time step. Note that the updates in d-edp require only local forecasts and device specifications. Hence each device can update its forecasts locally. We assume the demand profile is forecasted by a load device and is communicated to all the devices. The proposed communication protocol for device i is detailed in Algorithm. Each device starts the algorithm by updating its local forecasts for the operating horizon. Then each device sends a request to start planning for the horizon in step and waits to receive requests from all their neighbors in step 3. When all neighboring devices send their requests, device i starts the d-edp Algorithm 1 in step 4. Once Algorithm 1 is complete, device i applies the first element of the scheduled power and storage profile in step 5. When the power is dispatched, the device propagates its time in step 6 and goes back to step 1 to start planning for the the current time horizon. Algorithm Decentralized receding horizon at device i Require: Initialize time h =. Require: Initial demand forecast profile d and local forecasts if applicable. loop [1] Update local forecasts for h to H + h. [] Request neighbors to start planning. [3] Wait until all neighbors also send their requests. [4] Do d-edp (Algorithm 1). [5] Apply the first element of p it (h) and s it (h). [6] Advance time h = h + 1. end loop VI. ADAPTABILITY TO NETWORK CHANGES Changes to the network can occur when a device leaves or a new device joins. A device removal implies that the node associated with the device and all the edges to and from that node are removed. When a device is added, we assume that the new device connects with at least one existing device. The communication protocols to handle removal and addition differ as removal of a node can be more critical than an addition. When a device is removed all the devices interrupt their operations and update their network information, that is, their neighborhood list N i and the total number of DG1 DG Fig. 6. A six device communication network composed of two storage devices, two diesel generators, one and one Grid. There are two scenarios for the evolution of the communication network. In the first the communication network remains the same and in the second the storage device B is removed from the network at h =. devices N. Then they restart Algorithm from step 1. When a device is added, devices update their network information at the beginning of the next time step, that is, after advancing time in step 6. This means that the new device is admitted to the system in the next time step. The new device updates its network information upon connection to the network and starts from step 1 and waits at step 3 for his neighbors to send requests. For a centralized solution to the EDP a device removal or addition implies reconfiguring the optimization problem by removing or adding new variables, constraints and cost functions. Reformulating the optimization problem can be cumbersome when the network is changing frequently. The steps of the decentralized receding horizon method described in Algorithm require that the devices only need their neighbor list and the total number of devices in the system. A. Device failure numerical example We consider a microgrid containing two batteries (B, B1), two diesel generators (DG, DG1), one generator () and the grid (G) see Fig. 6. The specifications of each device and the load profile are the same as in Section IV. We compare the results of the d-ems between two scenarios. In the first scenario, all of the devices run without any failures. In the second storage device B fails during step 4 in Algorithm at time h =. Fig. 7(a) depicts the final generation and storage profiles of the no failure scenario whereas Fig. 7(b) shows the profiles of the scenario with failure. We observe that the system adapts to failure by increasing the state of charge levels of the remaining battery. Note that when there is no failure the generation in 1 h 19 are used to charge batteries which are then discharged at times h 4 to match demand. Observe that when there is failure, the storage capacity of the remaining battery is not sufficient to match all of the demand for h 4. As a result, the system uses diesel generators to match demand. G B B1

8 Power (kwh) Power (kwh) G DG + DG1 B1 B Demand Max Time (Hours) (a) G DG + DG1 B1 B Demand Max Time (Hours) Fig. 7. Optimal power and storage profile for the device network in Fig. 6 for the two scenarios. In (a), the network remains the same over the horizon. In (b), the storage device B fails and the system operates with single storage device. Compared to (a), in (b) the available storage capacity qb1 max limits the system s ability to balance demand for h using solely stored renewable power. As a result, diesel generator power is needed for balancing power. (b) VII. CONCLUSION We proposed a decentralized energy management system (d-ems) that solves an economic dispatch problem with generator and storage units for the planning horizon in a fully decentralized manner using an ADMM algorithm in which each device iteratively makes decisions based on its specifications, demand profile prediction and neighbors price variables. Based on existing results we argued that the proposed algorithm converges to the optimal under the assumptions that the network is connected and each device has convex cost functions. We provided numerical experiments that associates the convergence rate with the structure of the network. The proposed d-ems further implements a receding horizon control via a communication protocol that allows for updates to the demand profile forecasts and to the device specifications of EDP as time progresses. Furthermore, we incorporated a protocol to the d-ems that handles changes to the device network on the fly. We showed that the d-edp algorithm only requires the number of neighboring devices and the total number of devices to reinitialize with respect to the changes in the device network. Finally, we provided a numerical example where a storage device fails and the system adapts by utilizing the remaining storage device for the remainder of the horizon. REFERENCES [1] A. Hooshmand, B. 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