ADMM-Based Distributed OPF Problem Meets Stochastic Communication Delay

An increasing number of distributed generators will penetrate into the distribution power system in future smart grid, thus a centralized control strategy cannot effectively optimize the power loss problem in real-time. This paper examines the idea of a fully distributed optimal power flow (OPF) approach, based on the alternating direction multiplier method, to optimize the power loss. The objectives are not only to effectively obtain the minimization of power loss, but also to analyze the effect of communication time-delay on optimization performance. Both synchronous and asynchronous iterative algorithms are proposed to solve the OPF problem. In addition, four different strategies are proposed to improve convergence speed when delay occurs. The proposed weighted autoregressive strategy can reduce the fluctuation effectively. In comparison with synchronous algorithm, simulation results show that the asynchronous algorithm has a better optimization result.


I. INTRODUCTION
T HE large penetrations of DGs in distribution grid could benefit to the power system, for example, increasing energy diversity, improving reliability and reducing environmental pollution.Meanwhile, these DGs can provide a large quantity of ancillary services that are of great interest to the optimization of the grid [1], [2].Reactive power services play an important role, i.e., satisfy the requirement of reactive power load, control bus voltage in a system wide, decrease the network loss, relieve the transmission block and provide sufficient reserve to ensure the security of system in emergency.The development of smart grid requires Information and Communication Technologies (ICT) architecture to maximise the systems potential.In order to realize the autonomous control in real-time, a distributed optimization strategy will combine the ICT to implement the control and optimization.
A great mass of techniques have been investigated for control and optimization in electrical power system [3].The allocation of reactive power had been designed to solve OPF problems since the 1960s [4] and a number of optimization techniques had been proposed in recent decades [5]- [8].However, these centralized approaches, based on offline techniques, required an accurate network structure and operation conditions in advance.In addition, optimization procedure required a highly reliable communication process and also produced a large communication and computation delay.For these reasons, these approaches are not able to do real-time monitoring and control.
To overcome this shortcoming, a variety of distributed strategies which could achieve real-time optimization had been proposed.The concept of consensus fashion in a distributed system can be found in [9].The partial distributed algorithms, which separated the distribution network into several areas, were studied in [10]- [13].A fast decomposition optimization scheme which used interior point OPF in reference [10] was verified to be an effective decomposition scheme.Reference [11] introduced a distributed semi-definite programming (SDP) relaxation technique which guaranteed the faster convergence speed to obtain solution.A primal and a dual algorithm was propsoed in [12] to coordinate the subproblems decomposed from the convexified OPF, the computation time could be improved dramatically.Reference [13] discusses three ADMM-based distributed optimization algorithms for solving the dynamic DC-OPF problem with DR.The numerical case studies present solution accuracy, convergence performance and communication requirement of the three proposed distributed algorithms.However, these distributed approaches just divided the network into several areas.The problems in centralized approaches still existed because of the distance between two local controllers and requirement of reliable communication.In contrast, distributed algorithms were investigated to work out the distributed OPF problem.An example of full distributed approach in [14] utilized a distributed scheme to control the reactive power and a genetic algorithm (GA) was adopted to optimize objectives.However, the proposed optimization process of GA was time-consuming with respect to the requirements of real-time control [15].Reference [16] capitalized the ADMM to allocate the reactive power to minimize the system power loss which showed a better convergence speed compared to the dual-ascent approach.And [17] also provided the ADMM method to solve the consensus optimization problem, the results demonstrated an expedited convergence speed.In addition, asynchronous ADMM algorithm in [18] based on the distributed convex optimization problem is provided to simulate the convergence results.The simulations confirm the convergence of proposed asynchronous algorithm.These proposed distributed strategies still required a certain limited information exchange.However, they did not analyze the potential effect of time-delay.When the communication delay is considered, some algorithms may not obtain the optimal target and sometimes cannot converge properly.
The ADMM approach, which is an augmented Lagrangianbased algorithm, is a popular choice due to the robust and fast convergence results both in theory and practice [19].In addition, ADMM is well suited to distributed optimization and in particular to large-scale power system.However, when time-delay is considered in the ADMM algorithm, the synchronization will become a critical problem.Although the synchronization of all nodes enables the algorithm to be effectively controlled, the communication time in a synchronous algorithm will be limited due to the slowest node's communication transmission.A few papers [20]- [22] have introduced communication time-delay system into the distributed ADMM algorithm, which was based on a master-slave strategy.Reference [20] presents an asynchronous distributed algorithm by using ADMM to study convergence rate and accuracy of consensus algorithm.The simulation results considering timedelay show possible numerical instabilities and sensitivities of the convergence rate on different strategies.An asynchronous algorithmic framework based on ADMM is proposed to perform convergence with information exchange in [21].The experimental studies demonstrate convergence performance for both synchronous and asynchronous schemes with communication delay.And [22] studied the linear convergence conditions of asynchronous ADMM method.The results reveal the impact of different parameters, network delay and network size on the convergence performance.The proposed algorithms in these papers could still be treated as a centralized control approach, because the master required to collect information from all other independent workers.Only when all the information was received by the master, the master would broadcast other nodes to proceed the next step.
The references above do not discuss a fully distributed OPF problem with stochastic communication delay in a distribution network.The majority of recent papers also have not investigated this particular problem with both synchronous and asynchronous time-delay models.Although The asynchronous model means that each node could operate its own iterative step without considering the time synchronization problem.This paper proposes a feasible fully distributed approach to optimize the allocation of reactive power.The novelty of this paper is to analyze the effect of communication time-delay on the performance of both synchronous and asynchronous algorithms.The main contributions of this work are as follows: • A distributed approach with a limited coordination of neighbouring nodes is presented to solve the OPF problem.An improved iterative step of ADMM algorithm is used to solve the OPF problem in a distributed way.Compared with the widely-used master-slave model, the distributed model can reduce the requirements of reliability of communication system and reduce the total optimization time.Both synchronous and asynchronous algorithms that consider communication timedelay model are discussed in this paper.The simulation results prove that a small probability of communication delay still has a significant influence on the results of asynchronous distributed ADMM algorithm.Comparing with the synchronous algorithm, the proposed asynchronous algorithm considering time-delay has a better convergence speed and optimization results during the same wall clock time period.
• As the fluctuation in experimental results with time-delay, we utilized four strategies, such as, skipping strategy (SS), previous value strategy (PVS), autoregressive strategy (ARS) and weighted AR strategy (WARS), to solve the OPF problem when time-delay issue is applied in both synchronous and asynchronous scenarios.The proposed WARS can effectively reduce the fluctuation and improve the convergence results for both synchronous and asynchronous algorithms with different delay probabilities.
The rest of the paper is structured as follows: in Section II, the distribution network model with communication is described, then the centralized power loss problem formula is presented.Section III provides the synchronous and asynchronous algorithms with end-to-end communication timedelay model, and four different iterative strategies are proposed to assess the convergence results.The simulation results are shown in Section IV.Section V draws conclusions and discusses future work.

II. NETWORK MODEL AND PROBLEM FORMULATION
Fig. 1: One-line main feeder with a lateral branching network.
Fig. 1 shows one-line main feeder including a lateral branch distribution network with the description of the notations and also displays communication links for the information exchange.It is a bi-direction communication between neighbouring nodes.We define ν and β as the set of nodes and branches, respectively.(i,j) ∈ β means nodes i,j are neighbour nodes.
For a ladder network, the distribution flow equations can be written as [23] subject to ∀ i ∈ ν where P i and Q i are the active and reactive power flow from node i to i + 1, respectively.V i is the voltage level at node i. r i and x i are the branch resistance and reactance from node i to i + 1, respectively.p i+1 and q i+1 are the active and reactive power at node i + 1, respectively.R i = (r 2 i + x 2 i ) . Both p i+1 and q i+1 consist of local consumption minus generation, for example To meet the requirements of real-time OPF, it is assumed that the active power output in this paper is constant during the optimization procedure.We assume that p iL , p iG and q iL are fixed and only DG's reactive power term q iG can be regulated within limits.For initial node 1, terminated nodes n and m, V 1 is constant and The OPF means that when the structural parameters and load conditions of the system are given, the available control variables (such as generator output power, adjustable transformer taps, etc.) are adjusted to find that all operational constraints can be met, and a certain performance indicator of the system (such as power generation cost or network loss) reaches the optimal distribution of power flow.Our target is to minimize the global optimization objective which is system total active power loss by controlling the reactive power variables of DGs in this paper.The global objective function within multiple nominal operational constraints can be written where (3a) is the total power loss function.
. (3b) is the power balance constraints, s i is the maximum apparent power capacity of inverters of DG.For nodes without DG, s i = 0.The maximum value can be determined by the maximum value of converter voltage and maximum current rating of converter [24].(3c) is the voltage level constraints.V min i , V max i are the minimum and maximum voltage magnitude of node i, respectively.We set V min i = 0.95, V max i = 1.05 according to the American National Standard ANSI C84.1-2011.(3d) is the voltage constraints between two adjacent nodes.(3e) is the transmission line capacity constraints, S max i−1 denotes the maximum apparent power of branch transmission line from node i − 1 to i.
Since consensus ADMM is assured to converge only in convex scenarios, and OPF is a non-convex problem, the distributed solution is in general not guaranteed to converge.It is necessary to derive the linear approximant of the power flow equations (3).Under normal circumstances, the amount of voltage variation from node to node is relatively small compared to the rated voltage level.Hence, we assume that is a constant value, and also approximate (3a) and (3d) to a linear constraint (4a) and (4d), respectively.The optimization function can be approximated as a linear quadratic function in this paper [25], [26].
In [25] and [26], the simulation results show that the linear power flow equations are well justified for a wide range of distribution networks.This observation is powerful because the linear power flow equations in (4) are convex.Convexity means that the optimization solution can be achieved efficiently and each node can communicate with a central authority.In this paper, we focus on the distributed optimization algorithm to solve the problem, which only communicate information between neighbouring nodes in the network.

III. DISTRIBUTED OPTIMIZATION PROBLEM WITH COMMUNICATION MODEL
To solve (4) in a distributed way, a distributed ADMM algorithm will be adopted to minimize the power loss function [27].In addition, the communication delay problem will be considered in the algorithm.

A. Distributed ADMM Model
In the ADMM algorithm, the global objective function will be rewritten to a distributed consensus problem.Each node has its own local objective function and local constraints associated with neighbouring nodes' parameters.As (4) is the global expression, the local variables for each local function are taken into account.Let are inequality constraints and equality constraints which are equal to constraints (4b)-(4e).Partial local variables can be recalculated to obtain the global variables after each iteration.The augmented Lagrangian expression can be defined as follows ( where the detail of the individual augmented Lagrangian formula L i (X i ) for node i ∈ ν can be given as where are the local variables of reactive power flow from node i − 1 to node i and node i to node i + 1, respectively.Q i−1 and Q i are the global variables of reactive power flow from node i − 1 to node i and node i to node i + 1, respectively.y i , y + i are the Lagrangian multipliers for node i.And ρ is the penalty factor.Then the improved ADMM recursive algorithm of node i derived from ( 6) for kth iterative step can be written as follows: where 1 ζi is the iterative factor for node i.For initial node 1, terminal nodes n and mr in Fig. 1, we consider the 7) are completely independent and each node can update by local variables with limited information exchange between neighbouring nodes.The minimization step (7a) is a convex objective function with linear constraints and can be solved by evaluating the corresponding KKT conditions [28].However, the addition of a communication system will inevitably impact the algorithm, for example message delay or loss.To analyze the influence of the communication time-delay model on the performance of the iterative results, we will develop a stochastic end-to-end time-delay model primarily.

B. ADMM Algorithm With Time Delay Model
Communication Delay Model: From a measurement point of view, the end-to-end delay over a fixed path consists of three main components: 1) determined delay, 2) internet traffic caused stochastic delay, 3) route processing caused stochastic delay [29].The independent sum of the deterministic delay and the stochastic delay has probability density function (PDF) as [20] where p + q = 1 and ϕ is the deterministic delay density that can be approximat-ed by a Gaussian density ϕ with mean σ and standard deviation µ. µ is larger than the minimum deterministic delay.ϕ 2 (t) = λe −λt is the stochastic delay density that assumes to follow the exponential distribution by one alternating renewal process with mean length of the closure periods λ −1 .
In order to calculate the time-delay probability of each communication for all nodes, the PDF (8) will be recalculated to derive the Cumulative Distribution Function (CDF) of time delay model as [20] where η = 1 2 λ 2 σ 2 + µλ − λt and erf is the error function.The relative parameters can be set as µ = 5.3ms, σ = 0.078, p = 0.580 and λ = 1.39 [30].According to equation ( 9), the probabilities of different time-delay for each communication procedure can be added into the ADMM algorithm to analyze the performance.
and Lagrange multipliers y i (k + 1), y + i (k + 1) can be obtained by (7b)-(7d); (Step 4) node i sends updated global variable Q i (k + 1) and V i (k + 1) to node i + 1. Node i will simultaneously receive the updated variables from node i − 1.Then the iterative process will be repeated until the result satisfies a certain error condition.The above communication mechanism shows that each node will exchange the information with backward and forward neighbouring nodes independently in every iteration.It is easy to find that the mechanism has a synchronization issue.In this paper, both synchronous and asynchronous algorithms are proposed to analyze the performance of ADMM algorithm.The detail of both algorithms will be discussed below.
The synchronous distributed ADMM algorithm is outlined in Algorithm 1.We assume that the computation time t dm (t 1 to t 2 ) and t du (t 3 to t 4 ) are the constant time in Fig. 2. The red and pink lines are backward and forward links, which means each node only sends message to backward or forward neighbouring nodes.Fig. 2(a) is the synchronous algorithm timing diagram, where the maximum communication time is locked by the threshold time t ds (assume t 2 to t 3 equals t 4 to t 5 ).According to equation ( 9), we set 6.3363 ms (10% probability of time-delay) as the t ds .Each node will be equipped with an external GPS to keep each step have the same clock.Specifically, after the step of calculation, only when communication time of backward or forward link reaches t ds , each node will start the next step.It is a complete iteration period for the synchronous algorithm.
Algorithm transmit Q i (k + 1) to forward nodes 13: receive Q i−1 (k + 1) from backward nodes 14: wait t ds 15: k=k+1 16: untill satisfied the defined minimum error 17: end process An asynchronous algorithm is presented in Fig. 2(b).Each communication also has a bounded maximum transmission time t da , which is set to 9.6146 ms (0.1% delay probability).In case of a delay, the message may lead the algorithm to remain in the status of receiving the message.This algorithm does not need to wait until all nodes have received the updated variables before beginning the next step.Every node can execute the update steps to obtain the new local variables independently and immediately when it receives the message.Hence, each node will not have idle status and more iterations can be obtained.However, the communication message comes from neighbouring nodes, it is possible that node i has not received any message from neighbouring nodes (perhaps the message is still in transit due to late transmission) when time reaches the t da .Then the previous data will be used to update the variables in order to keep the algorithm running without extra waiting time.The outline of asynchronous algorithm is listed in Algorithm 2.
During the same time interval in Fig. 2, all nodes in synchronous ADMM algorithm just finish one iterative period, but some nodes in asynchronous ADMM algorithm almost completes one and a half iterative period.It is obviously that asynchronous algorithm can shorten the iteration period and increase the number of iterations.Consequently, the asynchronous algorithm can speed up the convergence rate faster than the synchronous algorithm to obtain a high theoretical accuracy.We assume that all node can always receive the message within threshold time for no-delay case.If some nodes do not receive the message within threshold time, the message will be considered as lost.Then other measures will be taken to continue the next step.

C. Convergence Analysis for Asynchronous ADMM Algorithm
The convergence analysis of synchronous ADMM algorithm has been investigated in [17], [19], [27].In this section, the convergence behavior of asynchronous ADMM algorithm will be studied.Some assumptions regarding the problem are made.Let ∇F i be the gradient or subgradient of F i in (3a).And we also define k τ i as a new sequence of node i, which implies that the gradient calculation may use old parameters due to the delayed message or lost message.x i and x i are the global and local variables for each node i, respectively.Then, the following assumptions are listed.Assumption 1: For each node i, the individual function gradient ∇F i is Lipchitz continuous, and there exists a constant K i > 0, such that In addition, X is a closed, convex and compact set.The power loss function F i (x i ) is bounded from below over X .
Assumption 2: The total delays are bounded.For each node i, there exists finite constant T i such that k − k τ i ≤ T i for all k.
Assumption 3: For each node i, the stepsize ρ i is chosen large enough such that: Assumption 1 is the standard in the context of non-convex optimization [31] and is satisfied for most problems of interest.According to assumption 2, when x i is updated in asynchronous algorithm, the parameter used to update the variables should be the latest received parameter.The convergence of the asynchronous algorithm can be verified by the following lemma.Lemma 1: Suppose that assumptions 1, 2 and 3 hold true.
Then we have the following true for asynchronous algorithm.(a) (b) The augmented Lagrangian is lower bounded and satisfies The proof of Lemma 1 can be found in Appendix A. Lemma 1(a) presents that there exists certain finite k < ∞ such that the augmented Lagrangian values are non-increasing after k iterations.Lemmas 1(b) presents that the Lagrangian is lower bounded.Then, we achieve that the augmented Lagrangian function is convergent.
The subsequent theorem proves the final convergence result and other properties.Theorem 1: (a) The iterates generated by the asynchronous algorithm converges if the following is true (b) For each node i at k iterations, certain sequences {{x * i }, {x * i }, {y * i }} converges to the set of stationary solution of (4.5) and satisfies The proof of Theorem 1 can be found in Appendix B. Note that it suffices to show that asynchronous algorithm converges to a stationary solution of (4) which is equivalent to (3).In other words, {x * i } can be the solution of (3).It is emphasized that the asynchronous algorithm may not converge to a globally optimal solution.

D. Delay-robust Strategies
The convergence performance for both algorithms will certainly be affected when time-delay is considered, for example, unstable results, low quality results and slow convergence.Different measures will assess the effect of time-delay in both synchronous and asynchronous algorithms.
1) Strategy I-Skipping Strategy (SS): In this strategy, if the communication message is not delivered within the threshold time, the unreceived node will not carry out current step and skip it.Then this node will wait until the next successful communication.Other nodes will continue to update the variables by (7).The SS not only interrupts the normal update of the algorithm, but also affects the neighbouring nodes' normal update of the algorithm due to the time-delay. 2

) Strategy II-Previous Value Strategy (PVS):
The PVS replaces the delayed message by using the saved data from the last successful communication.Each node can install a storage device to save the data generated from the previous normal iteration.When the delay occurs, the stored information can be invoked instead of the message that have not be received within the threshold time.The update of other nodes still follows equations (7).This strategy may decrease the convergence rate because the partial iterative processes using previous data can reduce the update of both self-node and neighbouring nodes.It is more suitable for certain scenarios that are prone to transmission delay while the requirement of optimization accuracy is not very stringent.
3) Strategy III-AR Strategy (ARS): The predicted value in advance is utilized instead of the previous saved message in the PVS.Forecast begins when the communication message is received in the last successful communication, which means no extra time is required.Suppose the relationship between past successfully received messages and current unreceived message can be estimated using previous saved data.The AR model of order ω is defined as [32] where ω is the order of the AR model and φ 1 , ..., φ ω are the AR coefficients which are constant, c is also the fixed value and t is the stochastic parameter that we define as white noise, t ∼ T (0, σ 2 ).In addition, a 1 , a 2 , ..., a T are the previous values of time series data used to predict the current value.According to the AR coefficient, the likelihood function of the AR model can be written as where the parameters can be obtained by using the Yule-Walker method [33].The performance of AR model is depended on the order.Decreasing the order against the time series data reduces the estimation performance of the AR model and increasing the order leads to more complex behavior which can even fail to obtain appropriate estimations.The Akaike-Information-Criterion(AIC) is adopted as the evaluation function to balance the order and likelihood [34].
where ω is the selected order of the AR model and the value of ω can be chosen from the order index 1,2,3...,ω.The evaluation function of AIC will be applied to estimate the unreceived message.If the time-delay happens, the algorithm will adopt the predictive values to replace the unreceived variables.Other nodes will follow ( 6) to update the variables.For brevity, we have not included the detail of the AR derivation process.

4) Strategy VI-Weighted AR Strategy (WARS):
The estimated value from AR strategy exists certain errors during the prediction process.Therefore, we propose to add a weighted term into the ADMM algorithm to reduce the error caused by the prediction to improve the convergence results.The ADMM recursive expression in (7) for kth iteration can be rewritten as (19c) where the iw is the set of nodes whose message do not arrive on time.γ i and γ i are the weighted factors.The weighted terms γ i a iw−1 (k) and γ i a iw+1 (k) are added to execute the iterative steps if delay happens.Other nodes will continue to use the (6) to update the variables.The weighted factors can reduce the predictive error and optimize the predicted value to obtain a better convergence result.

E. Connection between WARS and Conventional ADMM
It is necessary to observe the difference to find the connection between WARS and conventional ADMM.Reference [27] presents the traditional ADMM iterative form for consensus minimization problem.
Comparing ( 19) and (20), it is obviously that the traditional ADMM algorithm is a particular case of the WARS by setting All steps are set to use the same coefficient 1 ρ in the conventional ADMM.Contrarily, the WARS assigns the different coefficient 1  ζi instead of the constant coefficient 1 ρ .In addition, the WARS adds different weighted terms γ i a iw+1 (k) and γ i a iw−1 (k) to each node because of the forecast error.
These changes can make the WARS to achieve a better efficiency than the traditional ADMM.To begin with, the traditional ADMM requires information exchange with all nodes to update the global variable Q(k + 1).In contrast, the WARS is able to decrease the communication cost by communicating with limited neighbouring nodes.Furthermore, when delay happens, the unreceived message not only affects the variable update of self-node, but also affects the variable update of neighbouring nodes.Using modified predicted value in proposed WARS can decrease the error as much as possible against the traditional ADMM.In addition, the adopted weighted terms may produce a certain deviation from the original value which can improve the results from a local optimization to a global optimization.Finally, the conventional ADMM can only optimize its convergence speed by adjusting the penalty factor ρ, since other parameters are fixed.Whereas, the WARS has the flexibility by adjusting both weighted terms γ i a iw+1 (k), γ i a iw−1 (k) and coefficient term 1 ζi .Consequently, the WARS not only has the potential to reduce the volatility, but also accelerate the convergence speed to obtain a better optimization result.

IV. SIMULATION RESULTS
In order to verify the ADMM algorithm with communication delay, a 33-bus medium-voltage DN has been used as a case study [35].It contains 33 buses and 32 branches.In addition, four wind DGs (2,12,15,18) and four solar DGs (23,25,27,33) are installed in the network [36].

A. Observation of General Convergence Speed
To maximize the convergence speed, the applied ADMM formula has two important parameters, the penalty factor ρ and the convergence tolerance ε.Fig. 3 illustrates the convergence speed of synchronous ADMM with different ρ, which is decreased from 1 to 0.00001.The high values of ρ typically have a slow convergence speed and poor iterative results.The best choice of ρ which provide the fastest convergence speed is 0.001 in the early stage of the iteration process.Subsequently, the convergence speed for ρ = 0.001 is slower than the speed for ρ = 0.0001 after 200 iterations.In other different cases, a little higher or smaller value of ρ may result in a slightly better convergence speed.The ADMM only needs 9 iterations to reach the tolerance goal of less than 10% compared to centralized result.However, the error is only less than 5% after 200 iterations and 2.5% after 400 iterations.It is typically that the ADMM algorithm has a faster convergence rate at the beginning of the iteration process.In some special cases, it also should be noted that setting a high tolerance value (e.g., ε = 0.1 ) may results in a large fluctuation.In this paper, we set the tolerance value as ε < 0.05 which could also help to choose the penalty factor ρ = 0.001 with faster convergence rate.

B. Performance of Synchronous algorithm
In this section, the communication delay is applied to synchronous algorithm to test the performance.We divide communication into three different cases, the backward link case, the forward link case and two-way link case.Fig. 4 shows the effect of convergence speed in synchronous SS.The left side of Fig. 4 presents the fluctuated convergence results in three cases under a fixed probability of time-delay.For these three cases, the optimization results always have a larger fluctuation during the iterations.The result in forward link has less fluctuation range than that in backward link.Meanwhile, two-way link case has the largest fluctuation.The right side of Fig. 4 demonstrates the statistical results for these three cases (1000 stochastic time-delay runs for each case).The maximum fluctuation range is 21.33% in backward link case, 11.17% in forward link and 22.58% in two-way link.Overall the results indicate that the fluctuation mainly depends on the communication of step 2 in Fig. 2(a), which means the global reactive power variable Q i (k + 1) would significantly influence the minimization results for both self-node and other neighbour nodes in (5).
Fig. 5(a) shows the optimization results of different synchronous strategies under a fixed time-delay scenario.Only two-way link case is performed as the space limitation.It is clear that the results of PVS dramatically reduce the fluctuation range compared to the results of SS.However, the optimization result after 200 iterations does not converge to a desirable value (near 0.13 MW) and the maximum 8.98% error could be very inefficient.The warm-starting of ADMM algorithm (forecast in advance) can be utilized to improve both the fluctuation in SS and the convergence speed in PVS.In Table I, 1000 stochastic time-delay runs are simulated for each strategy.The smaller variance values indicate the less fluctuation.Table I shows that the synchronous ARS obtains a better result against the PVS.Although the proposed ARS speed up the convergence rate, the result still has a fluctuation because of the forecasting error.Hence, weighted terms are added to improve the estimation error to optimize the results.The peculiarity of WARS shows an increased convergence speed, decreased fluctuation range and even better optimization result than that in the synchronous no-delay case.Comparing with the results 0.1258 MW of synchronous no-delay case, the mean value of WARS can converge to 0.1243 MW which dramatically improve the result.The results in synchronous WARS demonstrate that it is an efficient strategy from a

C. Performance of Asynchronous Algorithm
Fig. 6(a) demonstrates the results of the different asynchronous strategies.The result is based on the wall clock time because the asynchronous algorithm do not require a synchronization device to wait in the process.Although the asynchronous strategy has a larger threshold time tolerance, the result in asynchronous no-delay case has a better optimization result which can achieve 0.1238 MW compared to the 0.1258 MW in synchronous no-delay case at the same time interval.Furthermore, the synchronous strategy requires 1260 ms to achieve 0.1258 MW, but the asynchronous strategy reaches the same result only after 300 ms.The increased convergence speed by the asynchronous algorithm mainly comes from the reduction of the waiting time and the increasing number of the iterations.
Meanwhile, the result of asynchronous SS show that the time-delay still brings a slight influence on the stability of the convergence result with 0.1% delay probability.The mean value of asynchronous WARS in Table I presents that the result can be improved to 0.1227 MW.Furthermore, the fluctuation performance is also improved with such low probability of delay.Consequently, the proposed WARS can significantly improve the optimization results from an asynchronous optimization perspective.

D. Performance of Loss communication
The above simulation is based on the practical communication system which is low probability of communication delay.However, in case some nodes' communication devices are out of work, the normal convergence rate will have large fluctuation even if the probability of communication delay is low.The simulation results in Fig. 7 show the convergence results of different strategies both in synchronous and asynchronous algorithms.In this case, we assume that these nodes (node 15, 18, 23, 25) are out of work.Fig. 7(a) presents that  the proposed WARS can effectively reduce the fluctuation of experimental results.In addition, the statistical results in Table II show that the mean value can be reduce to 0.1260 MW and the variance value is the smallest among the synchronous strategies.Comparing with the synchronous algorithm, the asynchronous algorithms can not only improve the mean value from 0.1260 MW to 0.1230 MW, but also demonstrate the lowest fluctuation of experimental results in Fig. 7(b) and Table II.

V. CONCLUSION
The future smart grid system will become more and more granular because of the increased penetration of renewable generators, distributed storage and electric vehicles.The investigation of a distributed ADMM algorithm considered communication delay model was discussed in this paper.Both synchronous and asynchronous algorithms combined with stochastic delay analyzed the convergence results compared to the no-delay algorithm.It shows that time-delay would significantly affect the performance of results in both algorithms.The simulations presented that the proposed WARS can effectively improve the volatility and achieve a better optimization result compared to other strategies.
It should be noted that this paper is a first step to developing the communication model into the distributed problem in power system.Developing a more accurate communication model, proposing a more effective strategy and investigating the cyber-attack would be very meaningful for future work.

APPENDIX A
Proof of Lemma 1(a): According to the update of global variables (20), we can obtain that the following is true Further, from (21), we have Similarly, it also have the following equality for iteration k Therefore, when the new information is delayed or lost for each node i, it follows that According to (24), we have It means that Lemma 1(a) is true when message delayed or loss occurred.If the information arrives, we also have the following results by using triangle inequality.
Subsequently, the desired result in Lemma 1(a) is achieved.

Fig. 2 :
Fig. 2: Illustration of timing diagram.(a) Synchronous algorithm; (b) Asynchronous algorithm.Message transmission mechanism: The aim of this paper is to investigate the ADMM algorithm with stochastic timedelay.The detailed iterative process of ADMM algorithm with communication delay for kth iteration is summarized as follow: (Step 1) each individual augmented Lagrangian function L i (X i ) with linear constraints would be solved by KKT conditions to achieve the new local variables Q

Fig. 3 :
Fig. 3: Illustration of convergence speed of distribution AD-MM algorithm with different values of penalty factor ρ.

Fig. 3
Fig.3also shows the different tolerance values for different iterations.The ADMM only needs 9 iterations to reach the tolerance goal of less than 10% compared to centralized result.However, the error is only less than 5% after 200 iterations and 2.5% after 400 iterations.It is typically that the ADMM algorithm has a faster convergence rate at the beginning of the iteration process.In some special cases, it also should be noted that setting a high tolerance value (e.g., ε = 0.1 ) may results in a large fluctuation.In this paper, we set the tolerance value as ε < 0.05 which could also help to choose the penalty factor ρ = 0.001 with faster convergence rate.

Fig. 4 :
Fig. 4: Comparison of convergence speed and statistic results of three different cases for synchronous SS (1000 runs each).

Fig. 7 :
Fig. 7: The convergence results for loss communication case.

TABLE II :
Statistical results with 1000 runs for each strategies of loss communication case