ADMM-based Distributed Algorithm for Economic Dispatch in Power Systems With Both Packet Drops and Communication Delays

Qing Yang, Gang Chen,, and Ting Wang

Abstract—By virtue of alternating direction method of multipliers (ADMM), Newton-Raphson method, ratio consensus approach and running sum method, two distributed iterative strategies are presented in this paper to address the economic dispatch problem (EDP) in power systems. Different from most of the existing distributed ED approaches which neglect the effects of packet drops or/and time delays, this paper takes into account both packet drops and time delays which frequently occur in communication networks. Moreover, directed and possibly unbalanced graphs are considered in our algorithms, over which many distributed approaches fail to converge. Furthermore, the proposed schemes can address the EDP with local constraints of generators and nonquadratic convex cost functions, not just quadratic ones required in some existing ED approaches. Both theoretical analyses and simulation studies are provided to demonstrate the effectiveness of the proposed schemes.

I. Introduction

AS a key concern in power systems, the economic dispatch problem (EDP) has attracted considerable research interests during the past few decades, which is to achieve optimal power allocation with minimal cost while satisfying both supply-demand balance constraint and local constraints of generators. Since the EDP is essentially an optimization problem, many conventional centralized optimization schemes,such as lambda iteration method [1], [2], genetic algorithm[3], and particle swarm optimization [4], [5], have been applied to solve it. However, these centralized approaches need a single control center to access the system-wide aggregated information, and thus they may be subject to some performance limitations, such as intolerance of a single-point failure,requirement of high-level of connectivity, and lack of privacy protection. Therefore, it is more desirable to design distributed algorithms to overcome these limitations.

Recently, many distributed approaches have been developed to address the EDP in power grids. Especially, a large number of consensus-based approaches (see [6]–[13] and references therein) have been proposed for solving the EDP in a distributed way. In [6], [7], distributed optimal power dispatch schemes are put forward without considering local constraints of generators. Based on incremental cost consensus, the work in [8] presents a distributed algorithm for the EDP with local constraints of generators. In [9], a distributed continuous-time approach is presented to address the EDP, which can obtain the optimal power generation in finite time. The work in [10]further studies the EDP with both transmission loss and communication failure. The work in [11] presents a consensus-based continuous-time approach to address the EDP, where the communication delays are taken into account.However, it is worth noting that the results in [6]–[11] only consider the EDP over undirected graphs rather than directed ones. Actually, directed graphs are more general and practical in network systems because of packet loss, equipment failure and asymmetric bandwidth restriction. There also exist some consensus-based results (see, e.g., [12], [13]) which are able to solve the EDP over directed graphs. However, all these approaches in [6]–[13] require local cost functions to be quadratic, which may fail to solve the EDP with nonquadratic cost functions. In recent years, some distributed approaches(see, e.g., [14]–[20]) have been presented to solve the EDP with general convex cost functions. The work in [19] proposes a distributed iterative scheme with packet drops and the work in [20] takes into account communication delays. However,there are few reports on EDP taking into account both packet drops and communication delays, even though they always occur simultaneously in network systems. Recently, some ADMM-based distributed approaches (see, e.g., [21], [22])have also been proposed to address the EDP in power grids.However, so far, the ADMM-based distributed algorithms for solving the EDP with both packet drops and communication delays have not been well investigated.

Motivated by the above observation and with the aid of the ADMM [23], [24], Newton-Raphson method, the ratio consensus algorithm [25] and running sum method [26], this paper presents two distributed iterative algorithms to address the EDP over directed graphs without/with both packet drops and communication delays. In comparison with existing results on the EDP, the main features of this study are concluded as follows:

1) Although there exist some ADMM-based distributed approaches (see, e.g., [21], [22]) for the EDP, to the best of our knowledge, this paper for the first time presents an ADMM-based distributed algorithm to solve the EDP with both packet drops and communication delays which are ignored in [6]–[22].

2) Different from most of the existing studies (see, e.g.,[6]–[11]) on the EDP which require the communication graphs to be undirected, the proposed schemes can solve the EDP on general digraphs. Since general digraphs are considered, the symmetry property of graphs required in[6]–[11] is no longer necessary, but our algorithms still converge to the optimal solution.

3) Unlike the works in [6]–[13] requiring the cost functions to be quadratic, general convex cost functions are considered in this paper. In addition, different from the works in [6], [7]and [16] only considering the coupled equality constraint,both the coupled equality constraint and local constraints of generators are taken into account in this paper.

This paper is organized as follows. Some preliminaries are given in Section II and the ED problem is formulated in Section III. In Section IV, two distributed iterative algorithms for the EDP over digraphs are presented and analyzed.Simulation studies are given in Section V. Finally, concluding remarks are drawn in Section VI.

II. Preliminaries

A. Communication Network Model

The communication network in smart grids is modeled as a general directed graph (digraph)G=(V,E,Q) withV={1,2,...,n} being the set of nodes,E?V×Vbeing the edge set, andQ∈Rn×nbeing the weighted adjacency matrix ofG. The notation (i,j)∈Erepresents there exists a directed communication link from nodeito nodej. Meanwhile,jis said to be an out-neighbour of nodeiandiis called an inneighbour of nodej. Although each node always has access to its own information, for the sake of notational convenience, it is assumed that (i,i)Efor alli∈V. A digraph is said to be strongly connected if for any pair of nodes (i,j), there exists a directed path from nodeito nodej, i.e., a sequence of directed edges of the form (i,i1),(i1,i2),...,(il,j) withir∈V,r=1,...,l. Letdenote the set of outneighbors of nodei. The cardinality ofis denoted aswhich is called the out-degree of nodei. Similarly,letdenote the set of in-neighbors of nodeiand letb e the in-degree of nodei. A digraph is called a balanced digraph, ifA nonnegative matrixQ∈Rn×nis column stochastic if each column sums to 1. The matrixQis primitive if there exists an integerm>0 such that each entry ofQmis positive.

B. Standard ADMM Algorithm

Consider the following minimization problem:

whereandare the decision variables;andare two constant matrices andb∈Rqis a constant vector.

According to [23], the standard ADMM for solving the problem (1) consists of the following iterations:

whereLρ(x,y,z) is the augmented Lagrangian function of the problem (1), defined by

withz∈Rqbeing the Lagrange multiplier associated with the equality constraint (1b) and ρ being a positive penalty parameter.

The convergence properties of the ADMM iterations (2) are described in the following lemma.

Lemma 1 [23], [24]:Assume that ρ>0 and the following conditions hold:

2) The unaugmented Lagrangian functionLhas a saddle point;

3) The matricesS1andS2have full column ranks.

Then the ADMM algorithm (2) is asymptotically convergent to the optimal solution (x?,y?) and the optimal Lagrange multiplierz?of the problem (1), withand

III. Problem Description

The EDP investigated in this paper is to achieve optimal power scheduling through local information interaction and coordination among agents (generators) in the network while meeting the power balance constraint and local constraints of generators. To be specific, the EDP is formulated as the following constrained optimization problem:

wherewithxibeing the power generation of generatori;fi(xi):R →R+and Θiare respectively the local cost function and local constraints of generatori;drepresents the total load demand. The equality (3b) is the supplydemand balance constraint. As a general problem description,all the local inequality constraints are described in the form of(3c). In other words, the set Θiis determined by the local inequality constraints of generators.

For the convenience of later analyses, we impose two assumptions on the problem (3) and one assumption on the communication graphG.

Assumption 1:For alli∈{1,···,n}, the local cost functionfi(xi):R →R+is convex, closed, proper, and twice continuously differentiable. Moreover, the local constraint set Θiis closed and convex.

Assumption 2 (Slater’s Condition):There existsfor alli∈{1,...,n} such thatwhereint(·)represents the set of interiors of a set.

Remark 1:Assumption 1 implies that the problem (3) is a convex optimization problem and Assumption 2 is used to guarantee the existence of feasible solutions of the problem(3). In addition, according to Assumption 1, the local cost function does not have to be quadratic, which generalizes the results in [6]–[13].

Assumption 3:The digraphG=(V,E,Q) is strongly connected and the weighted adjacency matrixQis primitive and column stochastic. Moreover, each nodeihas access to its own out-degree

Remark 2:According to Assumption 3, the communication graphs considered in this paper are general directed graphs rather than the undirected ones considered in [6]–[11]. Thus the symmetric property of communication graphs required in[6]–[11] is not needed here.

To apply the ADMM, a reformulation of (3) is considered.Define two convex sets:andLeth1andh2be two indicator functions for the sets Γ1and Γ2, respectively defined by

Then, the problem (3) can be rewritten as

wherex,y∈Rnare decision variables.

The objective of this paper is to present distributed iterative strategies to solve the ED problem (3) (or (4)) over directed graphs even in the presence of both packet drops and communication delays.

IV. ADMM-based Distributed Algorithms on Directed Graphs

In this section, two ADMM-based distributed algorithms are proposed to solve the problem (4) over directed graphs, one for reliable communication networks and the other for unreliable communication networks with both packet drops and communication delays.

A. Distributed Algorithm with Reliable Communication Networks

By applying the ADMM [23] to the problem (4), one has

whereLρ(x,y,z) is the augmented Lagrangian function for the problem (4) and is given by

Note that the update ofz, i.e., (5c) is decentralized and thus we only need to solve the subproblems (5a) and (5b) in a distributed way to obtain the updated values of variablesxandy.

First, consider the subproblem (5a). By combining (5a) with(6), one has

where the second and third equalities have used the fact thatandare both constants with respect to the subproblem (5a), and the last equality follows from the definition ofh1(x).

Note that the problem (7) is equivalent to the following constraint problem:

wheredi,i=1,...,n, is regarded as a virtual local demand satisfyingFrom (10), it can be obtained that

Furthermore, in light of (11), ζ ?△ζ can be computed by

It is worth noting that a control center is needed to gather system-wide information if ζ ?△ζ is computed directly by(12). To avoid the usage of any control center, in the following, we devote to calculate ζ ?△ζ in a distributed fashion. For each nodei, define two auxiliary variablesui[k]andwi[k], respectively initialized as

Consider the ratio consensus algorithm [25] used to solve the average consensus problem on digraphs, in which the state variables are updated according to

whereand zeros otherwise.The initial conditions of iterations (14) are given as:φ[0]=[φ1[0],...,φn[0]]Tand ψ[0]=1n. The convergence of the consensus algorithm (14) is given in the following lemma.

Lemma 2 [25]:Let {φi[k],ψi[k]} be the sequence generated by (14). Then under Assumption 3, the average consensus value can be asymptotically obtained via

Now, let φi[0]=ui[0] for alli∈{1,...,n}. By performing the consensus algorithm in (14) and in light of Lemma 2, the average consensus value for the variablesui[k] can be asymptotically obtained via (15), i.e.,Similarly, by letting φi[0]=wi[0], the average consensus value for the variableswi[k]is also asymptotically obtained via (15), which is denoted byBy combining these with (12), at each node ζ ?△ζ can be calculated by

in a distributed fashion. Moreover, by substituting (16) into the first equality in (10), △xican be calculated via

To this end, the optimal solution, to the problem (8) can be computed by

In other words, according to (16)–(18), the optimal solutionx?to the problem (8), i.e., the solutionxt+1to the subproblem(5a) is computed in a distributed way.

In the following, combining (5b) with (6) yields

wherePΓ2represents the projection operator onto the convex set Γ2. To be specific, letand the solution to the subproblem (5b) can be given as:

Based on the above discussion, the proposed distributed algorithm to solve the EDP (3) or (4) on digraphs with reliable communication networks is summarized in Algorithm 1. The convergence of Algorithm 1 is described in the following theorem.

Algorithm 1 ADMM-based distributed algorithm for the EDP (3)on digraphs with reliable communication networks

Theorem 1:If Assumptions 1–3 and ρ>0 hold, then the sequence {(xt,yt,zt)} generated by Algorithm 1 converges to(x?,y?,z?) in a distributed manner, where (x?,y?) andz?are,respectively, the optimal solution and the optimal Lagrange multiplier of the problem (4), with the primal residualand dual residualconverging to zero, i.e.,and

Proof:First, we show that for each node, the iteration valuesandin (5) can be calculated in a distributed manner via Algorithm 1. According to Assumption 3, the digraphGis strongly connected and its weighted adjacent matrixQis primitive and column stochastic. Carry out the consensus iterations in (14) for the variablesui[k] andwi[k]respectively. Then based on Lemma 2, we can obtain the average consensus valuesu?andw?in a distributed way.Furthermore, in light of (16)–(18), the optimal solutionxt+1to the subproblem (5a) is calculated in a distributed manner.Finally, the optimal solutionyt+1to the subproblem (5b) can be distributively obtained according to (19) andzt+1in (5c)can be computed by (20) in a decentralized way. In a word,according to Algorithm 1,andin (5) can be calculated in a distributed fashion.

In the following, we show that if ρ>0, the sequence{(xt,yt,zt)}generated by the ADMM iterations (5) is asymptotically convergent to (x?,y?,z?) with (x?,y?) being the optimal solution to the problem (4). First, note that the cost functionf(x)+h1(x)+h2(y) in the problem (4) is proper,closed, and convex becausef(x) is convex from Assumption 1 and the indicator functionsh1(x),h2(y) of the convex sets are all proper, closed, and convex. Moreover, the equality constraint in (4b) is affine. From Assumptions 1 and 2, the constraint set Θ1×···×Θnis a closed convex set and the Slater’s Condition holds. Thus, the strong duality property holds [27] and the optimal solution to the problem (4) exists.In light of the saddle point theorem [28], the existence of the optimal solution (x?,y?) to the problem (4) implies that there exists a dual optimal solutionz?such that (x?,y?,z?) is a saddle point of the Lagrangian function of the problem (4).Furthermore, note thatS1andS2in the problem (1)respectively correspond toIand ?Iin the problem (4) which have full column ranks. Therefore, the three conditions required in Lemma 1 are all satisfied. Thus, the remainder of the convergence proof is similar to that in [24] only by replacingf1(x),f2(y),S1,S2andbin (1) withf(x)+h1(x),h2(y),I, ?Iand 0 in the problem (4), respectively.

Remark 3:In Algorithm 1, the outer loop iterations are ADMM iterations and the inner loop iterations are average consensus iterations. In addition, it is worth pointing out that the inner loop iterations in Algorithm 1 are only asymptotically convergent to the average consensus values.Therefore, the stopping criteria (threshold or number of iterations) for the inner loop iterations are needed in actual implementation.

B. Distributed Algorithm With Both Packet Drops and Time Delays

While both packet drops and bounded communication delays are considered, an ADMM-based distributed algorithm for solving the problem (4) on digraphs is presented in this subsection.

First, some notations about packet drops and time delays are introduced. For a digraphG=(V,E,Q), letpji[k] be the indicator variables to denote whether there exist packet drops over (j,i)∈Eor not, which is defined by:pji[k]=0 if there exist packet drops over (j,i)∈Eat time instantkandpji[k]=1 otherwise. Let the integer τji[k]≥0 represent the delay of a message sent from nodejto nodeiat time instantkand it is required thatfor allk≥0, whereThe following assumption on packet drops and time delays can be found in [25], [26].

To solve the EDP (4) over digraphs with both packet drops and time delays, the running-sum method [26] and the ratio consensus algorithm with time delays [25] are employed here to compute the average consensus values. In particular, each nodei,i∈V, is assigned with six variables. Besides the variables φi[k] and ψi[k], nodeialso maintains the variablesand θji[k] forj∈Ni+∪{i}, whereri[k] andsi[k] are respectively the running sums of the variables φi[k]and ψi[k]; πji[k] and θji[k] keep track of the running sumri[k]andsi[k] respectively. Letri[0]=0,si[0]=0 for alli∈Vand πji[0]=0 , θji[0]=0 for alland alli∈V. Then each nodei,i∈V, computes the running sumsri[k+1] andsi[k+1]by

whereand zeros otherwise. Based on Assumption 4, the tracking variables πji[k+1] and θji[k+1] are computed as

According to (22), πjiand θjiremain unchanged if there exist packet drops on link (j,i)∈E. Moreover, each nodeialways knows the running sum of itself, i.e.,πii[k+1]=ri[k+1] and θii[k+1]=si[k+1] for alli∈V. Let τjibe the delay on link (j,i)∈E. The variables φi[k] andψi[k]of each node are updated according to:

where φ[0]=[φ1[0],...,φn[0]]Tand ψ[0]=1n. Note that φi[k] and ψi[k] are set as zero ifk<0. Then the convergence of the iterations (21)–(23) is described in the following lemma.

Lemma 3:Let {φi[k],ψi[k]} be the sequence generated by iterations (21)–(23). Then under Assumptions 3 and 4, the average consensus value can be asymptotically obtained via

Proof:First, we show that with the aid of the augmented graph representation method used in [25] and [26], the directed graphG=(V,E,Q) with both packet drops and bounded time delays can be converted to an augmented directed graph without packet drop and time delay. “Virtual buffers” and “virtual nodes” are introduced here to model the packet drops and the bounded time delays, respectively. To be specific, for each link (j,i)∈E, we introduce a “virtual buffer”bji, which stores the information that may be lost due to packet drops over (j,i) . The information stored inbjiwill be released to nodeiif there is no packet drop on link(j,i)∈E. That is to say, ifpji[k]=1, bothqjiφj[k](qjiψj[k])and the information stored inbjiwill be sent to nodei; whenpji[k]=0 , no information is received at nodei, but the information from nodejwill be accumulated inbji. On the other hand, for each nodei∈V, we introduce“virtual nodes”:At each time stepk, the virtual nodepossesses the information which is ready to reach nodeiafterlsteps. The weights between the original nodes, “virtual buffers” and “virtual nodes” are shown in Fig. 1.

Fig. 1. “Virtual buffer” b ji and “virtual node” on link ( j,i) and corresponding weights.

Letdenote the augmented graph described above. It is easy to see that the augmented graphG1hasnodes, which are labeled in the following order:noriginal nodes fromG,m“virtual buffers”,n“virtual nodes” with one time delay,n“virtual nodes” with two time delays and so on. Moreover, the augmented graphG1hasm1=|E1| edges. For each “virtual buffer”bji, define two variables φpdiand ψpdi. Let φpd[k] and ψpd[k] be the column vectors with each entry being φpdi[k] and ψpdi[k],respectively, for alli=1,...,m. For each “virtual node”i(l),define two variablesand, and let φ(l)[k] and ψ(l)[k] be the column vectors with each entry beingand

respectively, for alli=1,...,nandl=1,...,τˉ. Moreover, let

and

In addition, the initial values for the variablesandare respectively defined as

whereV={1,...,n} is the set of original nodes.

According to the above discussion, the iterations in(21)–(23) with both packet drops and time delays can be converted to the following compact form without packet drop and time delay:

where

withQ(l)[k] (l=0,1,...,τˉ) being the weighted matrix that depends on both packet drops and time delays and is defined as

Note that for each (j,i)∈E, only one ofQ(0)[k](j,i),is equal topji[k]Q(j,i) and the others are zero.Qpd1[k]={p ji[k]}n×mis a weighted matrix which depends on the weights of the links from “virtual buffers” to original nodes.Qpd2[k]={(1?pji[k])qji}m×nis a weighted matrix which relies on the weights of the links from original nodes to “virtual buffers”.Qpd3[k]={1?pji[k]}m×mis the self-loop weighted matrix of the “virtual buffers”.

As both packet drops and communication delays are considered here, we run the iterations (21)–(23) instead of the iterations (14) at Step 3 in Algorithm 1 and get the ADMMbased distributed algorithm with both packets drops and time delays, which is named Algorithm 2 in the following statement.

Since Algorithm 2 is in some extent similar to Algorithm 1,for brevity, we only need to make the following modifications on the basis of Algorithm 1 to get Algorithm 2: 1) Add the information of packet drops and delays in the “Input” step; 2)Replace the iterations (14) in Step 3 with the iterations(21)–(23).

The following theorem shows the convergence properties of Algorithm 2.

Theorem 2:If Assumptions 1–4 and ρ>0 hold, then the sequence {(xt,yt,zt)} generated by Algorithm 2 converges to(x?,y?,z?)in a distributed manner even when there exist both packet drops and bounded time delays, where (x?,y?) andz?are, respectively, the optimal solution and the optimal Lagrange multiplier of the problem (4), with the primal residualand dual residualconverging to zero, i.e.,and

Proof:We first show that when there exist both packet drops and bounded time delays, the iteration valuesandin (5) still can be calculated in a distributed fashion via Algorithm 2. Consider the variablesui[k] andwi[k] with initial values defined in (13). First, let φi[0]=ui[0] and ψi[0]=1 for alli∈V. Then carry out the iterations (21)–(23)for the variableui[k]. According to Lemma 3, the average consensus valueu?for the variableui[k] can be asymptotically obtained by (24). With a similar method, the average consensus valuew?for the variablewi[k] can also be obtained.That is to say, based on Lemma 3, the average consensus values for the variablesui[k] andwi[k] are both calculated in a distributed fashion even though packet drops and delays appear in the communication network simultaneously. Then according to (16)–(18),is obtained in a distributed way and in light (19)–(20),andcan also be computed in a distributed manner. The remaining proof is similar to that of Theorem 1.

Remark 4:Different from the results in [6]–[10], [12]–[18],and [21] only concerning reliable communication networks and the results only considering communication delays [11],[20] or packet drops [19], this work takes into account both packet drops and communication delays. Though the result in[25] can deal with communication delays while the work in[26] can handle packet drops, they may fail to solve the issues considering both packet drops and communication delays. In fact, the coexistence of packet drops and communication delays may bring difficulties in analyses and execution of the algorithm. However, these difficulties are conquered here by adopting the consensus iterations (21)–(23), in the convergence proof of which a new augmented graph representation has been constructed to transform the problem with both packet drops and time delays into the one without packet drop and delay.

Remark 5:Note that Algorithm 2 is more general and practical than Algorithm 1 since it takes into account both packet drops and time delays which usually occur simultaneously in communication networks. But on the other hand, the coexistence of packet drops and time delays may make an effect on the convergence rate of Algorithm 2. That is to say, compared with Algorithm 1, Algorithm 2 needs more iteration steps to get the optimal solution of the problem (3).

Remark 6:For the case of time-varying delays, if each node in the network has knowledge of the upper boundof the time delays and all the time-varying delaysare set asthen the case with time-varying delays is equivalent to the time-invariant delay case. Thus, for the case with bounded time-varying delays, the average consensus values for the variablesui[k] andwi[k] can also be obtained in a distributed manner via Algorithm 2 and the result in Theorem 2 still holds.

V. Simulation Study

In this section, several case studies are provided to verify the effectiveness of the proposed strategies for solving the EDP over directed graphs without/with packet drops and communication delays.

Case Study 1: The simulation results with Algorithm 1

In this case study, the proposed algorithm with reliable communication networks is verified on a power system with three generators and the local cost functionfi(xi) of generatoriis given as

where the γiand βiare the cost coefficients given in Table I.The local constraint set Θi(i= 1,2,3) and the initial conditions are also provided in Table I. Let ρ=1 and the total demandd=90 MW. The local virtual demands are all set as30 MW. The directed communication graph is shown in Fig. 2,where each node chooses its weight and the weights of its outgoing links to beIt is easy to verify that the weighted adjacency matrixQcorresponding to this digraph is primitive and column stochastic. The threshold values ?1and ?2in Algorithm 1 are both set as 0.001. The stopping criteria for the inner loop iterations of Algorithm 1 are given as: for alli, |φi[k+1]/ψi[k+1]?φi[k]/ψi[k]|<ε1with φi[0]=ui[0],and |φi[k+1]/ψi[k+1]?φi[k]/ψi[k]|<ε2with φi[0]=wi[0],

TABLE I Parameters and Initial Conditions of Three Generators

Fig. 2. Communication graph of three generators.

where ε1>0,ε2>0 are the threshold values which are both set as 0 .001 here.

By implementing Algorithm 1, the simulation results are shown in Figs. 3(a)–(f) and the number of iterations needed to reach the threshold is given in Table II. As can be seen, the incremental cost (IC) of each generator reaches consensus asymptotically and the consensus value is 27.722 $/MWh. The optimal solution to the problem (3) isx?=[33.038,36.962,20.000]TMW, which satisfies the local constraint of each generator and the supply-demand balance constraint.

Case Study 2: The simulation results with Algorithm 2

In this subsection, both packet drops and fixed time delays are considered in the communication network shown in Fig. 2 and the other conditions are the same as those in Case Study 1. The packet drop probabilities on communication links are set as:pd12=0.7,pd21=0.5,pd23=0.4,pd31=0.3. The delay profile is given as: τ12=τ31=τ23=1, τ21=2. Note that the maximum time delay is=2. By running Algorithm 2,the simulation results are illustrated in Figs. 4(a)–(f) and the number of iterations required to reach the threshold is provided in Table II. It can be seen that though there exist both packet drops and fixed time delays, the optimal incremental cost and the optimal solution still can be obtained and are the same as those in Case Study 1. Thus, Algorithm 2 is robust to both packet drops and fixed time delays.

Fig. 3. The simulation results of Case Study 1. (a) Primal and dual residuals (MW); (b) Auxiliary variables ui ; (c) Auxiliary variables wi; (d) Incremental cost ($/MWh); (e) Power outputs (MW); (f) Total outputs vs. demand(MW).

TABLE II Comparison of Different Communication Conditions and Number of Nodes

Case Study 3: Test on IEEE 118-bus system with Algorithm 1

In this case study, the IEEE 118-bus system with 14 generators is utilized to verify the scalability of Algorithm 1 and quadratic cost functions considered in [6]–[13] are employed here. The generator parameters are adopted from[29]. The local virtual demands are all set as 1200/14 MW.That is to say, the total demandd=1200 MW. The local constraint of generatoriis set as Θi=[50,200] MW,i=1,...,14 . We set ρ=1,MW,and the stopping criteria are the same as those in Case Study 1. The directed communication graph is illustrated in Fig. 5.

Fig. 4. The simulation results of Case Study 2. (a) Primal and dual residuals (MW); (b) Auxiliary variables ui ; (c) Auxiliary variables wi; (d) Incremental cost ($/MWh); (e) Power outputs (MW); (f) Total generation vs. demand (MW).

Fig. 5. Communication graph of fourteen generators.

By implementing Algorithm 1, the simulation results are shown in Figs. 6(a)–(f). The number of iterations needed to reach the threshold is given in Table II. It can be seen that the consensus value of incremental costs is 347.2 $/MWh and the optimal solution is obtained as:x*= [158.2,133.8, 50, 50, 50,51.6, 50, 50,102.2,102.2,102,200, 50, 50]TMW which meets the local constraint of each generator and supply-demand balance constraint. Hence, Algorithm 1 is still effective for a large network.

Case Study 4: Test on IEEE 118-bus system with Algorithm 2

In this subsection, both packet drops and time-varying delays are considered for the network shown in Fig. 5 and the other conditions are the same as those in Case Study 3. The packet drop probability on each communication link is randomly selected from the set [0.1,0.9]. The upper bound of the time-varying delays is set as=2. In particular, at each time step, the delays on communication links vary randomly in the set {1,2}. According to Remark 6 and by running Algorithm 2, the simulation results are shown in Figs. 7(a)–(f)and the number of iterations needed to reach the threshold is provided in Table II. As can be seen, the optimal incremental cost and optimal solution are the same as those in Case Study 3. Therefore, Algorithm 2 is effective for a large network with both packet drops and bounded time-varying delays.

Fig. 6. The simulation results of Case Study 3. (a) Primal and dual residuals (MW); (b) Auxiliary variables ui; (c) Average variables wi; (d) Incremental cost ($/MWh); (e) Power outputs (MW); (f) Total outputs vs. demand(MW).

Remark 7:According to Table II, it can be concluded that the convergence rate of Algorithm 1 is affected by the number of nodes. The more nodes, the more iteration steps are required. The convergence rate of Algorithm 2 is affected not only by the number of nodes but also by the packet drops and communication delays. That is to say, even if the number of nodes is the same, Algorithm 2 needs more iteration steps than Algorithm 1 to get the optimal solution, which confirms the statement in Remark 5. Moreover, from Table II, it can be seen that the iteration number of the outer loop is only associated with the number of nodes, not affected by communication conditions, but the iteration number of the inner loop is affected by both factors.

VI. Conclusions

In this paper, two ADMM-based distributed strategies have been proposed to address the EDP with general convex cost functions over possibly unbalanced digraphs. The communication networks without/with both packet drops and bounded time delays were considered and the ratio consensus method was employed to solve the EDP in a distributed fashion. It has been proved that the ADMM iterations in both the reliable communication networks case and the unreliable communication networks case (with both packet drops and time delays) are convergent to the optimal solution of the ED problem in a distributed way. Simulation results have demonstrated the effectiveness of the proposed schemes.

Fig. 7. The simulation results of Case Study 4. (a) Primal and dual residuals (MW); (b) Auxiliary variables ui ; (c) Auxiliary variables wi; (d) Incremental cost ($/MWh); (e) Power outputs (MW); (f) Total generation vs. demand (MW).