Distributed and Asynchronous Operational Optimization of Networked Microgrids
11 Distributed and Asynchronous Operational Optimization ofNetworked Microgrids
Nima Nikmehr,
Student Member, IEEE,
Mikhail A. Bragin,
Member, IEEE,
Peter B. Luh,
Life Fellow, IEEE, and Peng Zhang,
Senior Member, IEEE
Abstract —Smart programmable microgrids (SPM) is an emerg-ing technology for making microgrids more software-defined andless hardware-independent such that converting distributed energyresources (DERs) to networked community microgrids becomesaffordable, autonomic, and secure. As one of the cornerstones ofSPM, this paper pioneers a concept of software-defined operationoptimization for networked microgrids, where operation objectives,grid connection, and DER participation will be defined by softwareand plug-and-play, and can be quickly reconfigured, based onthe development of modularized and tightened models and anovel asynchronous price-based decomposition-and-coordinationmethod. Key contributions include: (1) design the architecture ofthe operational optimization of networked microgrids which canbe readily implemented to ensure the programmability of islandedmicrogrids in solving the distributed optimization models, (2)realize a novel discrete model of droop controller, and (3) introducea powerful distributed and asynchronous method Distributedand Asynchronous Surrogate Lagrangian Relaxation (DA-SLR) toefficiently coordinate microgrids asynchronously. Two case studiesare tested to demonstrate the efficiency of developed DA-SLR, andspecifically, the testing results show the superiority of DA-SLR ascompared to previous methods such as ADMM.
Index Terms —Networked microgrids, Droop control, Distributedoptimization, Software-Defined Networking.
I. I
NTRODUCTION
The smart programmable microgrids (SPMs) is the emergingphenomenon to address the issues associated with the existingdrawbacks of microgrids’ (MGs) structure such as the depen-dence on hardware, challenges in network virtualization, andvulnerability of communication signals to cyber-attacks [1].To provide flexible and easily manageable distributed andasynchronous operational optimization of islanded networkedmicrogrids, software-defined networking (SDN) has been used.Within SDN, the network programmability is enabled throughthe use of logically centralized controllers [2]. The programma-bility of SDN allows the network to efficiently manage thecommunication signals, and to enable the user access to theswitches to manage the network after detecting the failuresowing to the data plane and control plane separation. SinceMGs necessitate the exploitation of SDN in the communica-tion network, the SDN realizes software and plug-and-play-based operation objectives, grid connection, and distributedenergy resources (DER) participation. In [3], an SDN-basedMG framework is designed to manage a self-healing network
This work was supported in part by the National Science Foundation underGrant ECCS-2018492, CNS-2006828 and OIA-2040599.N. Nikmehr and P. Zhang are with the Department of Electrical Engineeringand Computer Science, Stony Brook University, Stony Brook, NY 11794, USA(e-mail: [email protected]).M. A. Bragin and P. B. Luh are with the Department of Electrical andComputer Engineering, University of Connecticut, Storrs, CT06269, USA. and enhance the resilience of the network. To ensure resilientmicrogrids operations, SDN-based communication architectureis developed in [4] to manage the cyber-physical disturbances.Within the SDN infrastructure, discrete operations play an im-portant role and the control signals are sent through the switchesas packets. Further , in programmable MGs, the discrete opera-tion mode can also help the network operators to accuratelyunderstand the network dynamics even in presence of com-munication delays in real applications [5]. Discrete controllerspossess the desired features such as simple programming, cost-efficiency, and digital and analog input and outputs [6]. Discretecontrollers are the prevailing control mode in microgrids toappropriately manage the DERs dispatch, substantially resolvedthe intractable efforts in guaranteeing microgrid stability. In [7],[8], to control the frequency and voltage deviations, distributeddiscrete secondary control are used.Since the computation burden in the centralized operationof networked MGs increases with the increase of the networksize, to coordinate distributed entities, distributed optimizationmethods have been used to improve computational performanceand to resolve data privacy issues of centralized methods [9].Within the distributed methods, the problem is decomposed intoseveral subproblems thereby ensuring the privacy of entities,and the avoidance of single-point failures. The Lagrangianrelaxation (LR) method is suitable for distributed coordination.Within the method, after constraints that couple distributedentities are relaxed, the relaxed problem is split into severalsubproblems, which are coordinated by updating Lagrangianmultipliers. To accelerate the convergence of the LR method,Augmented Lagrangian relaxation (ALR) [10] has been used bypenalizing the violations [11]. However, within the method, theproblem is non-separable and nonlinear. To overcome the non-separability issue, the alternate direction method of multipliers(ADMM) was used [12]. While within the ADMM subproblemsare smaller in size and are easier to solve than the relaxedproblem within ALR, the objective function of each subproblemincludes a quadratic penalty entailing the decision variablesfrom other subproblems which leads to communication andprivacy issues. To coordinate distributed entities without spend-ing the time for synchronization, in [13], an asynchronousADMM algorithm was used to allow the coupling variables tobe updated in each subproblem without getting updates fromother subsystems. However, the ADMM does not converge inthe presence of binary variables.To tackle the above issues, we contribute the following: • The architecture of software-defined networking is estab-lished to prepare the implementation and easy managementof distributed operational optimization of programmablemicrogrids in future studies. a r X i v : . [ ee ss . S Y ] F e b SDN ControllerSDNSwitch 1
SDNSwitch 2
SDNSwitch 3 SDNSwitch 4SDNSwitch NHost 0Host 1 Host Host 3Host NCore 0Core 1 Core 2 Core 3Core NCoordinatorMG1 MG2 MG3MG Nλ i λ i P buy , P sell Q buy , Q sell λ i λ i λ i P buy , P sell Q buy , Q sell P buy , P sell Q buy , Q sell P buy , P sell Q buy , Q sell P buy , P sell Q buy , Q sell Fig. 1: Scheme of SDN-enabled operational optimization of networked MGs • Energy management of networked microgrids is formu-lated as a distributed operational optimization problemconsidering the operational limits and power flow con-straints. The droop controllers are discretized for simpleprogrammability and high compatibility in the SDN frame-work due to sending the control signals as packets. • A distributed and asynchronous surrogate Lagrangian re-laxation method [14] is employed to coordinate the inter-connected microgrids. Within DA-SLR, each sub-systemshares data only with the coordinator without sendingdata to neighboring sub-systems or MGs. This sharingpolicy of DA-SLR preserves privacy. Also, compared to theclassical distributed methods, our DA-SLR method ensuresthe convergence in the presence of discrete variables.II. SDN-
ENABLED ENERGY MANAGEMENT OF ISLANDED M ICROGRIDS
A. System model of MGs
Software-defined networking takes the role of the data trans-fer between microgrids and the coordinator. Within the SDNarchitecture, switches are converted to faster and easy forward-ing devices owing to the separation of data and control planes.Besides, the control technique is the centralized operatingsystem [2]. As shown in Fig. 1, the SDN structure includesseveral interconnected switches and only controllers with a wideview of the network are selected to route data transfer [15].In the designed model, a networked structure of islandedmicrogrids is considered. Each microgrid is connected to one ora group of neighbouring MGs. Therefore, the control center ofMGs should collect data from their local controllers to make anaccurate decision about the power exchanging possibility withneighbouring MGs. Each microgrid consists of dispatchable andnondispatchable generators and loads. In this structure, MGcontrol center observes the difference between generation andload, and then make a decision about power exchanging withneighbour MGs. In the presented networked MGs structure,each MG is considered as an autonomous entity. To guaran-tee the operational optimization of networked MGs, all MGentities should be coordinated. Thus, a distributed algorithm isemployed to coordinate the MGs operation. In Fig. 2, a generaldescription for optimal connection between islanded MGs isshown. According to Fig. 1, the shared data with the coordinator
WT PV MTCHPFC LoadESS WT PV MTCHPFC LoadESSOptimal energy dispatch Data Optimal energy dispatch DataControl centerof MG1 Control centerof MGn . . .
MG1 MGn
Coordinator
Fig. 2: Networked MG-based distribution grid are the amount of purchased and sold real and reactive powerswhich are determined after MGs optimization.
B. Optimization model of islanded MGs
Objective function.
In this subsection, the optimizationmodel consisting of objective function and technical constrainsis presented for the networked microgrids.The objective function is described as: OF = (cid:88) t ∈ T (cid:88) i ∈ I [ C g ( P g,MTt,i ) + C g ( P g,F Ct,i ) + C g ( P g,CHPt,i )]+ (cid:88) t ∈ T (cid:88) b ∈ B [ C dch ( P BAT,dcht,b ) − C ch ( P BAT,cht,b )]+ (cid:88) m ∈ M (cid:88) t ∈ T [ ζ IL · P ILt,m + ζ IL · Q ILt,m ] (1)where, C g ( P g,MT ) = α gMT · P g,MT (2a) C g ( P g,F C ) = α gF C · P g,F C (2b) C g ( P g,CHP ) = α gCHP · P g,CHP (2c)are generation costs of micro turbine (MT), fuel cell (FC) andcombined heat and power (CHP), respectively, with P g,MT , P g,F C and P g,CHP being their corresponding power generationlevels and α gMT , α gF C and α gCHP being their generation prices.The costs C ch and C dch of charging and discharging level ofbatteries are calculated as follows: C ch ( P BAT,ch ) = α BATch · P BAT,ch (3a) C dch ( P BAT,dch ) = α BATdch · P BAT,dch (3b)where, P BAT,ch and P BAT,dch are charging and dischargingpower of batteries, respectively and α BATch and α BATdch are thecharging and discharging prices. Decision variables P ILt,m and Q ILt,m are real and reactive power load shedding in microgrid m at hour t . Also, ζ IL is load shedding price and m is usedto describe microgrid index. In the above, sets I , B , M and T are used to denote sets of generation units, batteries, microgridsand time periods, respectively.As shown in (1), the first summation is the total generationcost, the second summation is the total battery charging and discharging cost and the third summation is the total real andreactive powers interruption cost. Constraints.
The objective function (1) is minimized subjectto the following technical constraints. • Generation Capacity.
The constraints restricting real and reactive power generationlevels are defined as follows: u gt,i · P g ≤ P gt,i ≤ u gt,i · P g , ∀ t ∈ T, i ∈ I (4a) u gt,i · Q g ≤ Q gt,i ≤ u gt,i · Q g , ∀ t ∈ T, i ∈ I (4b)where, u gt,i defines commitment status of i th DG at hour t , P gt,i and Q gt,i are real and reactive power generation by unit i at hour t . P g , P g , Q g , and Q g are minimum generated real power,maximum generated real power, minimum generated reactivepower, and maximum generated reactive power, respectively. • Charge/Discharge Power Limits.
The charging and discharging status of batteries should followthe charging limits as below: ≤ P BAT,cht,b ≤ u cht,b · P ch , ∀ t ∈ T, b ∈ B (4c) ≤ P BAT,dcht,b ≤ (1 − u cht,b ) · P dch , ∀ t ∈ T, b ∈ B (4d)where, u cht,b is a binary variable showing the status of b th batterywhich can be either charged ( u cht,b = 1) or discharged ( u cht,b = 0) in each time slot. Besides, P ch , P dch are maximum allowedamount of battery charging and discharging, respectively. • Power Flow Limits.
Each distribution line between two nodes can distribute realand reactive power within a range between zero and P flow ,and Q flow as follows: ≤ P flowt,n − k ≤ P flow , ∀ t ∈ T, ( n, k ) ∈ N (4e) ≤ Q flowt,n − k ≤ Q flow , ∀ t ∈ T, ( n, k ) ∈ N (4f) • Power Interruption Constraints.
The interrupted real and reactive power levels cannot exceedthe limits P IL and Q IL : ≤ P ILt,m ≤ P IL , ∀ t ∈ T, m ∈ M (4g) ≤ Q ILt,m ≤ Q IL , ∀ t ∈ T, m ∈ M (4h) • Voltage and Frequency Restrictions.
Voltage and frequency restrictions are described as follows: | V n | ≤ | V t,n | ≤ | V n | , ∀ t ∈ T, n ∈ N (4i) . ≤ f t ≤ . , ∀ t ∈ T (4j)where, the node voltage is restricted between | V n | and | V n | . • Droop Control.
Droop control is deployed to enhance the ability of MGs inensuring the real and reactive power balances. In this regard,the f − P and V − Q characteristics of a droop control can bedescribed as follows [16]: f = f ref − m p · ( P gi − P gi,ref ) (5a) | V | = | V ref | − m q · ( Q gi − Q gi,ref ) (5b) where, f and f ref are fluctuated frequency and referencefrequency, which is set to 60 Hz. Similarly, | V | and | V ref | are bus voltage magnitude and nominal voltage magnitude,respectively. Also, P gi and Q gi are real and reactive power ofDGs. Droop control coefficients for frequency and voltage basedcontroller are m p and m q , respectively. The limits dealing with m p and m q are described as following: m p ≤ m p ≤ m p (6a) m q ≤ m q ≤ m q (6b)where, m p and m p are minimum and maximum values offrequency-based droop controller respectively. Also, minimumand maximum values of voltage-based droop controller aredescribed by m q and m q respectively. • Power flow.
For the power flow study, the linearized DistFlow model isdeployed. The model has been utilized and justified in dis-tribution systems and microgrids studies [17]- [18]. An exactapproximation of AC power flow is considered in linearizedDistFlow model [19]: P flowt,n − k = (cid:88) ( κ :( k,κ )) ∈ N P flowt,k − κ − ( P gt,k − P Lt,k ) , ∀ t ∈ T, n, k, κ ∈ N (7a) Q flowt,n − k = (cid:88) ( κ :( k,κ )) ∈ N Q flowt,k − κ − ( Q gt,k − Q Lt,k ) , ∀ t ∈ T, n, k, κ ∈ N (7b) | V n,t | = | V k,t | − r n,k · P flowt,n − k + χ n,k · Q flowt,n − k | V |∀ t ∈ T, n, k ∈ N (7c)where, the bus index is described by n and k and the set N denotes a set of nodes. The real and reactive power flow betweensending node n and receiving node k at hour t are defined by P flowt,n − k and Q flowt,n − k , respectively. Also, real and reactive loadsare denoted by P Lt,k and Q Lt,k , respectively, and | V n,t | and | V | are voltage magnitudes at bus n and the point of commoncoupling in an MG, respectively. The resistance and reactancebetween lines n and k are determined by r n,k and χ n,k . • Real power balance.
The real power balance ensures the balance between the realpower consumption and real power generation and exchangingreal power through the distribution lines which is formulated asfollows: (cid:88) i ∈ I ( s gn · P gt,i ) + s W Tn · P g,W Tt + s P Vn · P g,P Vt + 1 m p ( f ref,m − f t,m ) + s BATn ( P BAT,dcht − P BAT,cht )+ P ILn,t + ( P flowt,k − n − (cid:88) ( k :( n,k )) ∈ N P flowt,n − k )++ ( (cid:88) m ∈ M ( s T rann − m · P buyt,n − m − s T rann − m · P sellt,n − m )) = P Lt,n , ∀ t ∈ T, ( n, k ) ∈ N, m ∈ M (8)where, P g,W Tt and P g,P Vt are real power generation by WT andPV panel. Also, s gn , s W Tn , s P Vn , s BATn and s T rann − m are connection indicator of a generator, WT, PV, battery, and transacted powerof node n to a node in m th microgrid, respectively. • Reactive power balance.
The reactive power balance denotes the balance between thereactive power generation and loads considering the reactivepower flow and voltage-based droop control impact as follows: (cid:88) i ∈ I ( s gn · Q gt,i ) + s W Tn · Q g,W Tt + s P Vn · Q g,P Vt + 1 m q ( | V | refn − | V n , t | ) + ( Q flowt,k − n − (cid:88) ( k :( n,k )) ∈ N Q flowt,n − k )+ ( (cid:88) m ∈ M ( s T rann − m · Q buyt,n − m − s T rann − m · Q sellt,n − m )) + Q ILn,t = Q Lt,n , ∀ t ∈ T, ( n, k ) ∈ N, m ∈ M (9)where, Q g,W Tt and Q g,P Vt are reactive power generation by WTand PV panel. • Interface Power Exchange Limits.
The amount of exchanging power by each MG is limited asfollows: ≤ P buyt,m − w ≤ u buyt,m − w · P buy , ∀ t ∈ T, { m, w } ∈ M (10) ≤ P sellt,m − w ≤ (1 − u buyt,m − w ) · P sell , ∀ t ∈ T, { m, w } ∈ M (11) ≤ Q buyt,m − w ≤ u buyt,m − w · Q buy , ∀ t ∈ T, { m, w } ∈ M (12) ≤ Q sellt,m − w ≤ (1 − u buyt,m − w ) · Q sell , ∀ t ∈ T, { m, w } ∈ M (13)where, P buyt,m − w and P sellt,m − w are purchased/sold power levelsby m th microgrid at hour t from/to w th microgrid. Similarly, Q buyt,m − w and Q sellt,m − w are purchased/sold reactive power levels,respectively. Also, u buyt,m − w is the binary variables denoting thestatus of purchasing power by m th MG from w th MG. Themaximum amount of purchased and sold real and reactive pow-ers are illustrated by P buy , P sell , Q buy and Q sell , respectively. • Interface Power Flow Constraints.
The following interface power flow constraints are consideredto ensure that the power bought by microgrid m from microgrid w equals to the power sold by microgrid w to microgrid m : P buyt,m − w = P sellt,w − m , ∀ t ∈ T, { m, w } ∈ M (14) Q buyt,m − w = Q sellt,w − m , ∀ t ∈ T, { m, w } ∈ M (15)These constraints are coupling with respect to the microgrids. C. Problem linearization
The optimization solvers such as CPLEX and Gurobi cannotsolve nonlinear problems due to cross products of variables.In (8), frequency f t,m is a continuous variable while m p isdiscrete. Similarly, in (9), voltage magnitude | V n,t | and m q arecontinuous and discrete variables, respectively. Therefore, thedroop control terms make the problem constraints nonlinear,which results in a nonlinear optimization problem [20]. Never-theless, the presented terms are linearized as described below. Generic Linearization Procedure.
The linerization proce-dure will be explained by using an expression z = A · d , where A is a continuous variable and d is a binary variable. If A hasbounds [¯ A, ¯ A ] , then the exact form of the linearized inequalitiesare as following for expression d [20]: min { , ¯ A } ≤ z ≤ ¯ A (16a) ¯ A · d ≤ z ≤ ¯ A · d (16b) A − (1 − d ) · ¯ A ≤ z ≤ A − (1 − d ) · ¯ A (16c) z ≤ A + (1 − d ) · ¯ A (16d)Therefore, products of binary variable and continuous vari-ables is linearized following the procedure (16a)-(16d) throughthe introduction of new continuous variables ( z ). However, inthe designed optimization model, d is a discrete variable (integernon-negative variable). If d j is bounded by positive integer D ,so that d j ∈ { , , ..., D } , we can introduce binary variables w , w , ..., w D and add the following constraints: D (cid:88) l =0 w lj = 1 , ∀ j ∈ J (17a) d j = D (cid:88) l =0 l · w lj , ∀ j ∈ J (17b)Therefore: z j = D (cid:88) l =0 l · w lj · A, ∀ j ∈ J (17c)Now, we can linearize each of the products of w lj · A based oninequalities (16a)-(16d). In f − P droop control, m p is a discretevalue, while frequency ( f ) is a continuous variable. Since m p is discrete, therefore, k p = m p is also discrete. Therefore, w p,j = k p,j · f m , ∀ j ∈ J (18)Following (16a)-(16d), to resolve the non-linearity difficulty,we replace (18) by the following constraints: z p,j = D (cid:88) l =0 l · w p,lj · f m , ∀ j ∈ J (19)Similarly, by introducing k q = m q , the nonlinear terms w q,j = k q,j · | V n | , ∀ j ∈ J (20)are linearized as follows: z q,j = D (cid:88) l =0 l · w q,lj · | V n | , ∀ j ∈ J (21)III. D ISTRIBUTED AND A SYNCHRONOUS S URROGATE L AGRANGIAN R ELAXATION M ODEL FOR N ETWORKED MG S A. Distributed model of Surrogate Lagrangian Relaxation
In the networked microgrids structure, assume there are m MGs conndected to several neighbouring MGs. To coordinatethe MGs, Lagrangian multipliers are introduced first to relaxconstraints (14) and (15) that couple MGs. The Lagrangian function then becomes: L ( P, Q, λ ) = (cid:88) t ∈ T (cid:88) i ∈ I [ C g ( P g,MTt,i ) + C g ( P g,FCt,i ) + C g ( P g,CHPt,i )]+ (cid:88) t ∈ T (cid:88) b ∈ B [ C dch ( P BAT,dcht,b ) − C ch ( P BAT,cht,b )]+ (cid:88) m ∈ M (cid:88) t ∈ T [ ζ IL · P ILt,m + ζ IL · Q ILt,m ]+ (cid:88) t ∈ T, { m,w }∈ M λ p,t,m − w · ( P buyt,m − w − P sellt,w − m )+ (cid:88) t ∈ T, { m,w }∈ M λ q,t,m − w · ( Q buyt,m − w − Q sellt,w − m ) , ∀ t ∈ T, i ∈ I, b ∈ B, (22) where, λ p,t,m − w and λ q,t,m − w are Lagrange multipliers asso-ciated with real and reactive power levels.The relaxed problem is then decomposed into individual sub-problems. The formulation for “buying” MG m is formulatedas: min MG m { (cid:88) i ∈ I [ C g ( P g,MTt,m,i ) + C g ( P g,F Ct,m,i ) + C g ( P g,CHPt,m,i )]+ (cid:88) b ∈ B [ C dch ( P BAT,dcht,m,b ) − C ch ( P BAT,cht,m,b )]+ [ ζ IL · P IL ( t,m ) + ζ IL · Q ILt,m ]+ (cid:88) t ∈ T, { m,w }∈ M λ p,t,m − w · ( P buyt,m − w )+ (cid:88) t ∈ T, { m,w }∈ M λ q,t,m − w · ( Q buyt,m − w ) , ∀ i ∈ I, b ∈ B, m ∈ M (23)Note that decision variables P sellt,w − m and Q sellt,w − m within (22)belong to subproblem w and are thus not included in (23).While subproblem formulation m includes variables P buyt,m − w and Q buyt,m − w indicating power bought, the microgrid can infact sell power, in which case the corresponding optimizedvalues will be negative. The formulation for “selling” MG w is formulated using the same logic with the exception thatvariables P sellt,w − m and Q sellt,w − m will appear within the objectivefunction with the negative sign.The MG subproblems are coordinated by Lagrangian mul-tipliers, which are updated based on violation of relaxed con-straints as: λ r +1 p,t,m − w = λ rp,t,m − w + e r · (cid:0) P buy,rt,m − w − P sell,rt,w − m (cid:1) , (24a) λ r +1 q,t,m − w = λ rq,t,m − w + e r · (cid:0) Q buy,rt,m − w − Q sell,rt,w − m (cid:1) , (24b)where, r is the coordinator iteration number, P buy,rt,m − w is themost recent value available of active power at iteration r thatMG m has the intention to buy and e r is the stepsize. In [14],the contraction mapping concept is employed to derive e r inthe following way: e r = γ r e r − || g ( P r − , Q r − ) |||| g ( P r , Q r ) || , < γ r < , r = 1 , , ... (25)where, g ( P r , Q r ) is the vector of constraint violations, γ r is stepsize-setting parameter which can be updated in eachiteration. An exact formula to determine γ r can be found in [14].As mentioned in Section II and as shown in (23), MG subproblems do not involve decision variables from other sub-problems, thus, unlike ADMM, the DA-SLR method does notrequire data exchange among MGs thereby preserving privacy. B. SDN-enabled DA-SLR solution steps
According to Fig. 1, each core is devoted to a microgrid andthe multipliers are asynchronously updated using plug and playproperty of DA-SLR method without waiting for all subproblemsolutions to arrive at the coordinator. MGs share their purchasedand sold power levels with the coordinator, the coordinatorupdates the multipliers and then broadcasts the multipliers toall MGs.
Feasible Cost Search.
Feasible solutions are searched byusing heuristics. Heuristics are operationalized by solving theoriginal problem (1)-(15) with decision variables fixed at themost recent values obtained by solving MG subproblems. Ifthe feasible solution to the original problem is not found, themultipliers are updated for several iterations before feasiblesolutions are searched again.
Dual value of DA-SLR.
Dual values provide the lower boundfor the feasible cost to quantify its quality, and the dual valuesare obtained by minimizing each MG subproblem by using themost recent available values of multipliers.
Stopping criteria.
The stopping criteria is the differencebetween the feasible cost and the dual value: OG = cost feasible − cost dual cost feasible (26)The steps of distributed and asynchronous of surrogate La-grangian relaxation is described as Algorithm 1. Algorithm 1:
Execution of DA-SLR on operationaloptimization of networked MGs
Result:
Exchanging powers ( P buym − w / P sellm − w , Q buym − w / Q sellm − w )between microgrids initialization; M microgrids’ exchanged powers P ,buym − w / P ,sellm − w , Q ,buym − w / Q ,sellm − w , Lagrange multipliers λ , updating stepsize e iteration r ← Coordinator receives MGs subproblem solutions and updatesmultipliers per (24a)-(24b) without waiting for all solutionsto arrive. Coordinator then broadcasts multipliers to all MGs Each idle MG starts solving its subproblem by using thelatest available multipliers and sends its solution to thecoordinator criteria to search to feasible solution are satisfied then search for a feasible solution the feasible solution is obtained then7 obtain the dual value and calculate the gap (26) else go to step ; endelse go to step ; end8 if the stopping criteria is satisfied then the optimization process is finished else go to step ; end IV. S
IMULATION RESULTS
In this section, the efficacy and efficiency of the DA-SLRbased operational optimization method are tested and validatedusing two case studies. In case study 1, a four-MG networkedmicrogrid system based on a modified IEEE 33-bus distributionnetwork [21] is considered. In the second case study, a nine-MGnetworked microgrid system based on a modified IEEE 123-bussystem [22] is tested. In both studies, the range of frequency-based droop coefficients m p is assumed to fall in [0 . , . with a discrete step of m p considered as 0.0018. Similarly, therange of voltage-based droop coefficients m q is set as [0 . , . with a discrete step of 0.0045. A. Test results on a 33-bus networked microgrid system
In Fig. 3, the single line diagram of the 33-bus networkedmicrogrids system with four MGs is shown, and in Table I,DERs information is described. The power base of the systemis set to be 10 MVA. The line resistance and reactance of therest of the network for added buses are set to be 0.006 and 0.01p.u., respectively.
11 12
16 17 1819 20 21 22
23 24 25
26 27 28 29 30 31 32
35 36
48 49 50
52 53 54 PV MT WT FC ES CHP
CHP WT ES PV MTCHP
ES PV
CHP WT FC MT ES PVFC
MTWT
CHP
MG1 MG3 MG4MG2
Dispatchable DER Non-dispatchable DEREnergy storage Switch (open)Node
Fig. 3: Modified IEEE 33-bus distribution system with four MGsTABLE I: DERs information in networked islanded MGs structure
DERtype Bus No. Max. realpower (kW) Max. reactivepower (kVAR)PV , , ,
53 200 80 WT , , ,
40 150 60 MT , , ,
51 400 200 FC , ,
49 300 − CHP , , , ,
44 400 300
As demonstrated in Fig. 4, the operation cost of the systemobtained by the new method decreases fast thereby reaching thegap of less than 0.2% within 20 iterations.
Fig. 4: The DA-SLR method in tracking the feasible solution in modified IEEE33-bus system
Performance of the new method is compared against that ofADMM in Table II.
TABLE II: Comparison of feasible costs using DA-SLR and ADMM
Method Objective function ( $ )MG1 MG2 MG3 MG4 TotalDA-SLR ADMM
The DA-SLR and ADMM methods take 16 and 102 secondsper iteration to solve MGs’ subproblem, respectively. This com-parison clearly shows the advantage of DA-SLR as compared tothe ADMM. As reviewed in Introduction, ADMM diverges inthe presence of discrete variables; and as demonstrated in TableII, DA-SLR obtains the total feasible cost of $13,562, whichis $18,240 less (or 57.3%) than that obtained by using ADMMafter 20 iterations. In this case study, each iteration includes atotal of four subproblems, and after finishing the optimizationprocess of those subproblems, the next iteration starts.The contribution of active and reactive power levels withineach DER obtained by DA-SLR are shown in Fig. 5 and Fig.6, respectively.
Fig. 5: Generated real power by DERs in each microgrid obtained by DA-SLRmethod
Fig. 6: Generated reactive power by DERs in each microgrid obtained by DA-SLR method
According to Fig. 5 and Fig. 6, CHP and MT units havea significant contribution in providing the real and reactiveloads in each MG. Since dispatchable DERs are equipped withdroop controllers, MT, FC, and CHP generators contribute tofrequency-based droop control, in which frequency variationscan cause changes in generation level of DERs, while MT andCHP generators are used for voltage-based droop control toadjust the reactive power in a predetermined range. In Fig. 7, thecontribution of droop control of DERs in each MG is illustrated.
Fig. 7: Contribution of droop control on real and reactive power
In Fig. 7, generally, the contribution of frequency-based droopcontrol in generating real power for a DER is higher than thecontribution of voltage-based droop control in reactive powergeneration for the same DER. The reason is that the maximumreal power generation capacity is more than the maximumcapacity of DERs in reactive power generation (see Table I).Precisely, the real and reactive power balances are decidingfactors of the contribution of droop controllers in the power generation of DERs. The frequency fluctuation of each DER iscaused by frequency-based droop control is shown in Fig. 8.
Fig. 8: Frequency fluctuation due to droop control performance in each DER
Comparing Fig. 7 and Fig. 8 shows that the DER decreasesits frequency to enhance its droop control contribution in realpower generation and vice versa. Take the CHP unit in MG1as an example. As shown in Fig. 7, CHP contribution usingfrequency-based droop control is in the maximum level afterhour , which results in the lowest frequency level comparingto previous hours according to Fig. 8. Also, as the frequencyand voltage droop coefficients are considered discrete variables,the optimized values for droop coefficients is shown in Figs. 9and 10. Fig. 9: Optimized values for real power droop control coefficients obtained byDA-SLR method
A droop controller prefers to adjust the frequency and voltagedroop coefficients in the higher or lower levels or step to keepthe power balance guaranteed. Comparing the Fig. 7 and Fig.9 for frequency-based droop control shows that when the real power generation of a DER increases the frequency-based droopcontrol coefficient increases compared to the hours that thegenerated real power is lower. Similarly, the reactive powergeneration level of a DER in Fig. 6 has a direct relationship withthe contribution of voltage-based droop control in Fig. 7 therebythe droop control coefficient dealing with reactive power will bechanged to guarantee the reactive power balance. Take MT(24)of MG1 as an example, where a direct relationship betweenthe reactive power generation and corresponding droop controlcoefficient is observable.
Fig. 10: Optimized values for reactive power droop control coefficients obtainedby DA-SLR method
Moreover, to show the impact of droop controller contributionin this study, the applied technical limit on droop controlcoefficients is enhanced from 20 percent to 30 percent of thetotal generation of each DER in the next scenario. In Table III,this impact is demonstrated.
TABLE III: Impact of droop control contribution increment on the operationcost using DA-SLR
EntityMG1 MG2 MG3 MG4 Total
Objectivefunction ($)
Reduction(%) . . . . . In Table III, the importance of droop control coefficients andconsequently the optimized values are shown. In the optimiza-tion process, changing droop control coefficients limits can havea profound impact on the operation cost of microgrids.
B. Test results on a 123-bus networked microgrids system
To demonstrate the scalability of DA-SLR, a larger network isconsidered. Fig. 11 shows the single line diagram of a 123-busnetworked microgrids system with nine interconnected MGs.
134 5 62 7 81211 141020 1922 2118 35374013533 323127 26 252829 30 25048 47 49 50 5144 45 4642 4341 36 38 396665 64636260 160 67575859 545352 55 5613341516 17 9695 94 93152 92 90 8891 89 87 86 808182 838478 8572 73 74 7577 79300 111 110108 109 107112 113 114105 106101 102 103 104 45010097 9968 69 70 71197151150 61 610
924 23 251 195 451149 350 WT PVMT
FC ES
CHP
MG1
CHP MT ES WT MG2
FCCHPMT WT ES PV MG3 PV CHP MT FC WTES
MG4 WT CHP FC PV ES MG8 WT CHP MT ES MG5 PV CHP FC WTES MG7
WTPV CHP
MT FC ES MG6 WT FC MT CHP ES PV MG9
21 48 6634 Dispatchable DER
Non-dispatchable DER
Energy storage
Switch (open)
Fig. 11:
Modified IEEE 123-bus distribution system with nine MGs
The results obtained by the designed DA-SLR method areanalyzed upon approaching the feasible cost (see Fig. 12).The gap reaches 0.3% and 0.01% after 10 and 20 iterationsrespectively. In Table IV, the operation cost of MGs is analyzedusing three methods, DA-SLR, sequential SLR, and ADMM.
Fig. 12: Tracking the feasible solution using SLR and DA-SLR in modifiedIEEE 123-bus systemTABLE IV: Impact of droop control contribution increment on the operationcost using DA-SLR
Method Objective function ($)MG1 MG2 MG3 MG4 MG5
DA-SLR
ADMM − − Objective function ($)MG6 MG7 MG8 MG9 Total
DA-SLR
ADMM − − − According to the results of Table IV, the DA-SLR has abetter performance of optimization compared to sequential SLRand ADMM. It is seen that the ADMM method suffers fromdivergence problems in solving some subproblems.The DA-SLR and ADMM spend 35 and 210 seconds periteration to solve MGs’ subproblems, respectively. This com-parison shows the definite advantage of DA-SLR as comparedto the ADMM.
V. C
ONCLUSION
In this paper, the distributed and asynchronous surrogateLagrangian relaxation method is used to coordinate networkedmicrogrids to schedule the islanded MGs in a distributed man-ner. The new method efficiently handles binary decisions andexcellent performance, as well as scalability, is demonstrated.The method paves the way to facilitate software-defined net-working in enabling efficient coordination of distributed entitiesasynchronously. It was shown that the DA-SLR method canachieve acceptable convergence in fewer iterations and guaran-tee a minimum amount of operation cost. The classical methodADMM, in contrast, could not efficiently manage a distributedproblem when the number of microgrids or subproblems aregrowing. Moreover, discrete variables have no impact on theconvergence of the DA-SLR method, while classical methodsface difficulties in solving problems with discrete decisionvariables. The next step is to discuss the potential of the DA-SLR method on stability issues of operational optimization ofprogrammable microgrids.R
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