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appliedsciencesArticleTwo-Stage Stochastic Programming SchedulingModel for Hybrid AC/DC Distribution NetworkConsidering Converters and Energy Storage SystemPeng Kang 1, Wei Guo 1, Weigang Huang 1, Zejing Qiu 2,3, Meng Yu 2,3, Feng Zheng 4 andYachao Zhang 4,*1 State Grid Energy Saving Service Co., Ltd., Beijing 100052, China; kangpeng19821125@163.com (P.K.);guowei7359@126.com (W.G.); huangweigang2010@163.com (W.H.)2 Nanrui group company/State Grid Electric Power Research Institute, Nanjing 210000, China;qiuzejing@sgepri.sgcc.com.cn (Z.Q.); yumeng1@sgepri.sgcc.com.cn (M.Y.)3 State Grid Electric Power Research Institute Wuhan Efficiency Evaluation Co., Ltd., Wuhan 430074, China4 School of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350116, China;zf_whu@163.com* Correspondence: yczhang@fzu.edu.cnReceived: 14 November 2019; Accepted: 20 December 2019; Published: 25 December 2019 Abstract: The development of DC distribution network technology has provided a more efficient wayfor renewable energy accommodation and flexible power supply. A two-stage stochastic schedulingmodel for the hybrid AC/DC distribution network is proposed to study the active-reactive powercoordinated optimal dispatch. In this framework, the wind power scenario set is utilized to deal withits uncertainty in real time, which is integrated into the decision-making process at the first stage.The charging/discharging power of ESSs and the transferred active/reactive power by VSCs can beadjusted when wind power uncertainty is observed at the second stage. Moreover, the proposedmodel is transformed into a mixed integer second-order cone programming optimization problemby linearization and second-order cone relaxation techniques to solve. Finally, case studies areimplemented on the modified IEEE 33-node AC/DC distribution system and the simulation resultsdemonstrate the effectiveness of the proposed stochastic scheduling model and solving method.Keywords: hybrid AC/DC distribution network; voltage source converter; active and reactive powercoordination; two-stage stochastic programming1. IntroductionWith the rapid growth of renewable energy and energy storage systems integrated in thedistribution network, the development of a DC distribution network has drawn more and moreattention [1]. Compared to the AC distribution network, the DC distribution network has theadvantages of reducing active power losses, and improving the power quality and utilization rate ofelectric energy [2,3]. As a result, the hybrid AC/DC distribution network will become a new tendencyof intelligent distribution network development in the future [4,5]. By combining the advantages oftwo kinds of distribution networks, the hybrid distribution network can achieve flexible power supplyfor different types of electric load and renewable energy accommodation [6,7].To handle the variations of renewable energy and load demand, a multi-timescale coordinatedstochastic voltage/var control approach was developed and a mixed-integer quadratic programmingmodel was built for the IEEE 33-bus radial AC distribution [8]. The authors of [9] proposed a day-aheadreactive power dispatch method of AC distribution networks by considering power forecast errors ofAppl. Sci. 2020, 10, 181; doi:10.3390/app10010181 www.mdpi.com/journal/applsciAppl. Sci. 2020, 10, 181 2 of 15renewable energy, and a dynamic preliminary-coarse-fine adjustment strategy was designed to achievethe optimal scheduling of distributed generators (DGs) and capacitor banks (CBs).At present, research of the hybrid AC/DC distribution network is in the elementary stage andneeds to expand and deepen. Voltage source converters (VSCs) have been utilized to convert AC linesinto DC lines to improve DG accommodations [10]. To realize the coordinated voltage regulation in ahybrid AC/DC distribution system, a priority-based real-time control strategy was proposed accordingto the voltage control effect of active-reactive power adjustment in [11]. Multiple battery energy storagesystems (ESSs) have been considered in distribution networks for real-time voltage regulation in [12].Furthermore, [13] proposed an optimal control management of ESSs to mitigate the fluctuation andintermittence of renewable energy. Taking into consideration prediction errors, the optimal operationof ESSs in a distribution system was developed and solved in a two-level framework to alleviate thenet load uncertainties [14,15]. For the planning problems of ESSs, [16] introduced a new index toquantify wind power fluctuations, and the impact of different ESSs configurations on the solutionswere analyzed. A scenario-based chance-constrained planning approach was developed to handlethe joint planning of multiple technologies of ESSs, and an easy-to-implement variant of Bendersdecomposition algorithm was proposed to solve the chance-constrained optimization problem [17].In order to improve the reliability of electricity supplies with wind power integration, a populationmeta-heuristics algorithm was adopted to determine the minimum capacity of kinetic energy storagein [18].To consider the randomness and uncertainty of wind power output, this paper proposes atwo-stage stochastic scheduling model to realize active-reactive power coordinated economic dispatchfor the hybrid AC/DC distribution network with the integration of capacitor banks and an energystorage system. The first stage is to determine the power setpoints of generator units, switchingnumber of CBs, and charging/discharging state of ESSs, and the second stage is to adjust the transferredactive/reactive power of VSCs and charging/discharging power of ESSs based on the realized windpower scenarios. To sum up, the schematic framework of this study is shown in Figure 1, for which thecorresponding research contents are described in the following sections.Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 16designed to achieve the optimal scheduling of distributed generators (DGs) and capacitor banks(CBs).At present, research of the hybrid AC/DC distribution network is in the elementary stage andneeds to expand and deepen. Voltage source converters (VSCs) have been utilized to convert AC linesinto DC lines to improve DG accommodations [10]. To realize the coordinated voltage regulation ina hybrid AC/DC distribution system, a priority-based real-time control strategy was proposedaccording to the voltage control effect of active-reactive power adjustment in [11]. Multiple batteryenergy storage systems (ESSs) have been considered in distribution networks for real-time voltageregulation in [12]. Furthermore, [13] proposed an optimal control management of ESSs to mitigatethe fluctuation and intermittence of renewable energy. Taking into consideration prediction errors,the optimal operation of ESSs in a distribution system was developed and solved in a two-levelframework to alleviate the net load uncertainties [14,15]. For the planning problems of ESSs, [16]introduced a new index to quantify wind power fluctuations, and the impact of different ESSsconfigurations on the solutions were analyzed. A scenario-based chance-constrained planningapproach was developed to handle the joint planning of multiple technologies of ESSs, and an easyto-implement variant of Benders decomposition algorithm was proposed to solve the chanceconstrained optimization problem [17]. In order to improve the reliability of electricity supplies withwind power integration, a population meta-heuristics algorithm was adopted to determine theminimum capacity of kinetic energy storage in [18].To consider the randomness and uncertainty of wind power output, this paper proposes a twostage stochastic scheduling model to realize active-reactive power coordinated economic dispatch forthe hybrid AC/DC distribution network with the integration of capacitor banks and an energy storagesystem. The first stage is to determine the power setpoints of generator units, switching number ofCBs, and charging/discharging state of ESSs, and the second stage is to adjust the transferredactive/reactive power of VSCs and charging/discharging power of ESSs based on the realized windpower scenarios. To sum up, the schematic framework of this study is shown in Figure 1, for whichthe corresponding research contents are described in the following sections.Determine the decision variablesfor the first and second stagePower flow modelingAC power flow modelHybrid AC/DC distribution system modelingDC power flow modelVoltage sourceconverter modelComponent modelingOperation constraintsof capacitor banksOperation constraints ofenergy storage systemsHistory data for wind powerWind power uncertainty modelingScenario generation by LHS and CDSimultaneous backward reductionThe reduced scenario set for wind powerEstablish the two-stage stochasticprogramming modelThe linearization and second-ordercone relaxation techniquesa mixed integer second-ordercone programming problemTwo-stage stochastic optimization dispatchObtain the solution by CPLEXFigure 1. Figure 1.The schematic framework of this study. The schematic framework of this study.Appl. Sci. 2020, 10, 181 3 of 15The contributions of this paper can be summarized as follows:(1) The capacitor bank, energy storage system, and voltage source converter are coordinated asthe active adjustment measures simultaneously in the hybrid AC/DC distribution system. By thismeans, a comprehensive and practical framework considering active and reactive powers coordinationdispatch is established.(2) A two-stage stochastic programming model is established to achieve active-reactive powercoordinated economic dispatch for the hybrid distribution system, in which wind power uncertainty isdescribed as several wind power scenarios obtained by scenario generation and reduction techniques.As a result, the decision-making at the first stage not only depends on the day-ahead wind powerpredictions but is also affected by wind power uncertainty in the real-time scheduling stage. In otherwords, the realized wind power scenario set in real time is integrated into the decision process for thetwo-stage scheduling problem.(3) The proposed stochastic programming model is a nonlinear optimization problem, whichis transformed into a mixed integer second-order cone programming problem by the linearizationand second-order cone relaxation techniques for solving. Moreover, simulation results about themodified IEEE 33-node distribution system demonstrate the effectiveness of the proposed stochasticscheduling model.The remainder of this paper is organized as follows. The power flow models for the AC distributionnetwork, DC distribution network, and voltage source converter are established in Section 2. Thetwo-stage stochastic scheduling model for the hybrid AC/DC distribution system is proposed inSection 3. Case studies and simulation results about the modified IEEE 33-node distribution systemare shown in Section 4. Finally, conclusions are drawn in Section 5.2. Power Flow Model for Hybrid AC/DC Distribution Network2.1. AC Distribution Network ModelThe DistFlow equations are used to describe the power flows in AC distribution system [19]: PACi,t = XPACk,t – X[PAC l,t – (IlAC ,t )2rl] 8i 2 NAC,(1)k(i,:)2LACl(:,i)2LAC QAC i,t =Xk(i,:)2LAC QAC k,t – Xl(:,i)2LAC[QAC l,t – (IlAC ,t )2xl] 8i 2 NAC, (2)(UiAC ,t )2 = (UAC j,t )2 – 2(rkPAC k,t + xkQAC k,t ) + (IkAC ,t )2(r2 k + x2 k) 8k(j, i) 2 LAC, (3)(IkAC ,t )2(UAC j,t )2 = PAC k,t 2 +QAC k,t 2 8k(j, i) 2 LAC, (4)PAC k,t 2 +QAC k,t 2 ≤ (SAC k,max)2 8k(j, i) 2 LAC, (5)UACi,min ≤ UiAC ,t ≤ UiAC ,max 8i 2 NAC, (6)where LAC represents the set of AC branches; NAC represents the set of AC nodes; t is the schedulingtime; k(i,;) represents branch k starting with node i and l(:,i) represents the branch l ending with node i;rl and xl are the resistance and reactance of branch l; PAC i,t and QAC i,t are the injection active and reactivepower at node i; IkAC ,t , PAC k,t , and QAC k,t are the phase current, active, and reactive power at branch k; UAC j,tand UACi,t are the phase voltage at node j and i; UiAC ,min and UiAC ,max are the minimum and maximumvoltage at node i; and SAC k,max is the maximum load flow of AC branch k.Appl. Sci. 2020, 10, 181 4 of 152.2. DC Distribution Network ModelThe DC distribution system can be described as:PDCi,t = Xk(i,:)2LDCPDCk,t – Xl(:,i)2LDC[PDC l,t – (IlDC ,t )2rl] 8i 2 NDC, (7)(UiDC ,t )2 = (UDC j,t )2 – 2rkPDC k,t + r2 k(IkDC ,t )2 8k(j, i) 2 LDC,(8)(IkDC ,t )2(UDC j,t )2 = PDC k,t 2 8k(j, i) 2 LDC,(9)UDCi,min ≤ UiDC ,t ≤ UiDC ,max 8i 2 NDC,(10) – SDCk,max ≤ PDC k,t ≤ SDC k,max 8k(j, i) 2 LDC, (11)where LDC represents the set of DC branches; NDC represents the set of DC nodes; PDC i,t is the injectionactive power at node i; IkDC ,t and PDC k,t are the phase current and active power at branch k; UiDC ,min andUDCi,max are the minimum and maximum voltage at DC node i; and SDC k,max is the maximum load flow ofDC branch k.2.3. Voltage Source Converter ModelThe schematic diagram of the voltage source converter (VSC) is shown in Figure 2.Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 162.2. DC Distribution Network ModelThe DC distribution system can be described as:DC DCDC DC DC DC 2, , , , DC( ,:) (:, )i t k t l t l t l [ ( ) ]k i l iP P P I r i∈ ∈= – – ∀ ∈   , (7)DC 2 DC 2 DC 2 DC 2( ) ( ) 2 ( ) ( , ) U U r P r I k j i i t j t k k t k k t , , , , DC = – + ∀ ∈  , (8)DC 2 DC 2 DC 2( ) ( ) ( ) ( , ) I U P k j i k t j t k t , , , DC = ∀ ∈  , (9)DC DC DCU U U i i i t i ,min , ,max DC ≤ ≤ ∀ ∈  , (10)DC DC DC– ≤ ≤ ∀ ∈ S P S k j i k k t k ,max , ,max DC ( , )  , (11)where DC represents the set of DC branches; DC represents the set of DC nodes; Pi t DC , is theinjection active power at node i; Ik t DC , and Pk t DC , are the phase current and active power at branch k;DCUi,min and UiDC ,max are the minimum and maximum voltage at DC node i; and SkDC ,max is themaximum load flow of DC branch k.2.3. Voltage Source Converter ModelThe schematic diagram of the voltage source converter (VSC) is shown in Figure 2.Figure 2. Schematic diagram of the voltage source converter.As shown in Figure 1, riVSC and xiVSC represent the equivalent resistance and reactance of theVSC connected with node j in the AC network. The branch from node j to AC side node i of the VSCcan be regarded as the AC branch in the AC network. The VSC model can be described as [20,21]:VSC(AC) AC AC 2 VSCP P I r i i t k t k t i , , , SVC = – ∀ ∈ ( )  , (12)SVC(AC) AC AC 2 VSCQ Q I x i i t k t k t i , , , SVC = – ∀ ∈ ( )  , (13)VSC(AC) 2 AC 2 VSC AC VSC AC AC 2 VSC 2 VSC 2, , , , ,( ) ( ) 2( ) ( ) [( ) ( ) ] U U r P x Q I r x i t j t i k t i k t k t i i = – + + + , (14)Figure 2. Schematic diagram of the voltage source converter.As shown in Figure 1, rVSC i and xVSC i represent the equivalent resistance and reactance of the VSCconnected with node j in the AC network. The branch from node j to AC side node i of the VSC can beregarded as the AC branch in the AC network. The VSC model can be described as [20,21]: PVSC(AC)i,t = PAC k,t – (IkAC ,t )2rVSC i 8i 2 NSVC,(12) QSVC i,t (AC) = QAC k,t – (IkAC ,t )2xVSC i 8i 2 NSVC, (13)(UiVSC ,t (AC))2 = (UAC j,t )2 – 2(rVSC i PAC k,t + xVSC i QAC k,t ) + (IkAC ,t )2[(rVSC i )2 + (xVSC i )2], (14)3PVSC(AC)i,t = PVSC i,t (DC) 8i 2 NSVC, (15)UVSC(AC)i,t =p33 µMi,tUiVSC ,t (DC) 8i 2 NSVC, (16)where PVSC(AC)i,t and QSVC i,t (AC) are the input active and reactive power at the AC side of the VSC whilePVSC(DC)i,t is the output active power at the DC side of the VSC; UiVSC ,t (AC) and UiVSC ,t (DC) are the voltageAppl. Sci. 2020, 10, 181 5 of 15at the AC and DC sides of VSC i; µ is the utilization rate of the DC voltage; and Mi,t is the modulationdegree of SVC i within the range of 0 to 1.2.4. Second-Order Cone Relaxation for the Distribution System ModelFor the above model of the hybrid AC/DC distribution network, and the nonlinear terms existingin Equations (1)–(5), (7)–(9), and (12)–(14), the following variables are defined to eliminate the quadraticterms of voltage and current for the AC/DC power flow.(IkAC ,t )2 = eIkAC ,t , (IkDC ,t )2 = eIkDC ,t , (17) (UiAC ,t )2= UeiDC ,t .(18) 2 = UeiAC ,t , (UiDC ,t )Then, Equations (4), (5), and (9) can be converted to second-order cone constraints as follows [22,23]:kh 2PAC k,t 2QAC k,t eIkAC ,t – UeAC j,t iTk2 ≤ eIkAC ,t + UeAC j,t , (19)kh PAC k,t QAC k,t iTk2 ≤ SAC k,max, (20)kh 2PDC k,t eIkDC ,t – UeDC j,t iTk2 ≤ eIkDC ,t + UeDC j,t . (21)By this means, the hybrid AC/DC distribution system model is transformed as a second-ordercone programming problem.3. Two-Stage Stochastic Scheduling Model for the Hybrid AC/DC Distribution NetworkThe two-stage stochastic scheduling model is established to cope with wind power uncertainty.The decision at the first stage is applicable for all the wind power scenarios, and the respectiveadjustment decision for each wind power scenario is made at the second stage.3.1. Scenario Generation and ReductionThe wind power prediction error is regarded as a random variable, which obeys a normaldistribution, then the wind power output in real time can be expressed by [24]:PeWt = PˆW t + DPeW t , DPeW t ∼ G(0, δ2 t ), (22)where DPeWt is the wind power prediction error at time t; PˆW t is the day-ahead wind powerprediction value; δt is the standard deviation of the prediction error; and G represents the normaldistribution function.The scenario generation and reduction techniques are implemented to handle wind poweruncertainty. Since the Monte Carlo simulation method based on simple random sampling suffers froma higher computational burden, the Latin hypercube sampling (LHS) is adopted to obtain the originalwind power scenario set, which can achieve good coverage from the entire distribution of the randomvariable [25].The procedures of LHS can be described as follows:Step 1: Assuming the number of samples for the random variable, DPeW t , is N, the interval [0, 1]can be equally divided into N intervals, and a uniformly distributed random number within the rangeof [(k-1)/N, k/N] is generated for k = 1 to N;Step 2: Disorder the order of the above n random numbers obtained by Step 1;Appl. Sci. 2020, 10, 181 6 of 15Step 3: The n random numbers by Step 2 are regarded as the probability values of samples, and thevalue of the random variable, DPeW t , can be obtained according to the inverse function of the cumulativedistribution function (CDF) for DPeW t :DP(k)t = F- t 1(k -N0.5), k = 1, · · · , N, (23)where Ft(·) represents the CDF of random variable DPeW t , and DPt(k) is the ith sample value of DPeW t .Cholesky decomposition (CD) is introduced to alleviate the undesired correlations betweenrandom variables at different time points [26].The sample matrix for DPeW t by LHS can be expressed as: DP =………2DP2· · · DPt 2666666666666666664DP(1) 1DP(1)DP1· · · DP (2) t(N)(2) (N)…DP(1)T DPT(2) · · · DPt(N)3777777777777777775T×N. (24)The procedures of CD are as follows:Step 1: A T × N ordering matrix, L, is generated, and the element at the tth row and ith columnrepresents the position of the sample, DPt(k), in the matrix, DP, to be arranged;Step 2: The elements of each row in DP are arranged according to the ordering matrix, L;Step 3: The corresponding correlation matrix of L noted as ρL can be calculated and decomposedby the Cholesky decomposition: ρL = DDT.Step 4: A T × N matrix, G, can be obtained by:G = D-1L.(25)(26) Step 5: The elements of each row in the matrix, L, are arranged according to the order of elementsin the corresponding row in G;Step 6: The elements of each row in the matrix, DP, are arranged according to the updatedordering matrix, L.Then, the simultaneous backward reduction method is utilized to decrease the number of windpower scenarios [27]. For the matrix, DP, obtained by the LHS and CD techniques, the original scenariok noted as DP(k) is a vector composed of the elements of the kth row in DP, for which the occurrenceprobability is π(k). The Kantorovich distance of scenarios DP(k) and DP(j) can be defined as:dk,j = kDP(k) – DP(j)k2. (27)The scenario number in the reduced scenario set is assigned as Ns. The procedures of thesimultaneous backward reduction method can be described as follows:Step 1: Delete the scenario, k, which satisfies the following condition:π(k)π(j)mink,jdk,j = minm2f1,··· ,Ngπ(m)(n2f1,··· min ,Ng,n,mπ(n)dn,m). (28)Step 2: The scenario number is set by N = N – 1, and the scenario, k*, closest to the deletedscenario, k, is chosen according to: dk∗,k = mins,kπ(k)π(s)ds,k.(29) Appl. Sci. 2020, 10, 1817 of 15Step 3: The occurrence probability of scenario k* is set as:π(k∗) = π(k∗) + π(k).(30)Step 4: If N > Ns, return to Step 1; otherwise, the scenario reduction process is finished.As a result, the reduced scenario set for wind power can be obtained as:P = nPW t,s, s = 1, 2, · · · , Nso.(31) The parameters, N and Ns, are set to 1000 and 10, respectively. The standard deviation of thewind power prediction error at time point t is set to 20% of PˆW t . The original and reduced wind powerscenario sets are shown in Figure 3.Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 16Step 2: The scenario number is set by N = N – 1, and the scenario, k*, closest to the deleted scenario,k, is chosen according to:*( ) ( ),,min k sd d k k = s k ≠ π π s k . (29)Step 3: The occurrence probability of scenario k* is set as:* *( ) ( ) ( ) k k kπ π π = + . (30)Step 4: If N > Ns, return to Step 1; otherwise, the scenario reduction process is finished.As a result, the reduced scenario set for wind power can be obtained as: = = { } P s N t s s W, , 1, 2, ,  . (31)The parameters, N and Ns, are set to 1000 and 10, respectively. The standard deviation of thewind power prediction error at time point t is set to 20% of PˆtW . The original and reduced windpower scenario sets are shown in Figure 3.Figure 3. The original and reduced wind power scenario sets.3.2. Objective FunctionThe day-ahead scheduling cost includes the generating cost of gas-fired units in the first stageand the power purchasing cost, penalty cost of power fluctuation from the substation, network losscost, load shedding, and wind curtailment cost:AC DCAC DCAC DCSubCur Cur, , , , ,Cur Cur, , , ,Loss AC 2 DC 2, , , ,Sub Sub, ,min ( 3 )( 3 )( 3( ) ( ) )(G G Li t s i t s i t st i s t i iWs i t s i t ss t i is k t s k k t s ks t k ks t i t sic P c L Lc W Wc I r I rc Pππππ∈ ∈∈ ∈∈ ∈∈Ω+ + ++ ++ +             Ramp Sub Sub| |) i t s i t s , , , 1,s tc P P–  + –, (32)where cG is the unit generating cost of gas-fired units; cL is the load shedding price; cW is the windcurtailment price; cLoss is the network loss price; ctSub is the electricity price of the main grid at timet; cRamp is the penalty price for power fluctuation; ΩSub is the node set of substation; and πs is theoccurrence probability of the wind power scenario, s.Figure 3. The original and reduced wind power scenario sets.3.2. Objective FunctionThe day-ahead scheduling cost includes the generating cost of gas-fired units in the first stage andthe power purchasing cost, penalty cost of power fluctuation from the substation, network loss cost,load shedding, and wind curtailment cost:min PtPicGPGi,t + PsπsPtcL( Pi2NAC3LCuri,t,s + Pi2NDCLCuri,t,s )+PsπsPtcW( Pi2NAC 3WCuri,t,s + PWCuri,t,s )+ i2NDCPsπsPtcLoss( Pk2LAC3(IkAC ,t,s)2rk + Pk2LDC(IkDC ,t,s)2rk)+PsπsPtP i2WSub (cSub t PSub i,t,s + cRampjPSub i,t,s – PSub i,t-1,sj)(32)where cG is the unit generating cost of gas-fired units; cL is the load shedding price; cW is the windcurtailment price; cLoss is the network loss price; cSub t is the electricity price of the main grid at timet; cRamp is the penalty price for power fluctuation; WSub is the node set of substation; and πs is theoccurrence probability of the wind power scenario, s.3.3. Constraints for the Two-Stage Stochastic Programming3.3.1. Operation Constraints of the Capacitor BankQCB i,t = yCB i,t QCB step, (33)Appl. Sci. 2020, 10, 181 8 of 150 ≤ yCB i,t ≤ Nmax CB , (34)XtyCB i,t+1 – yCB i,t ≤ yCB max, (35)where yCB i,t is the operation number of CB; Nmax CB is the maximum operation number of CB; QCB step is thecapacity of single group CB; and yCB max is the maximum adjustable number of CB in one day.It can be seen that the operation number of CB is determined in the first stage, which is applicablefor all wind power scenarios in the second stage. In addition, the nonlinear constraint (Equation (35))can be linearized as follows [28]:8>>>:yCB i,t+1 – yCB i,t ≤ Nmax CB bCB tyCB i,t – yCB i,t+1 ≤ Nmax CB bCB tPtbCBt ≤ yCB max, (36)where bCBt is the binary variable indicating whether the operation number of CB has changed.3.3.2. Operation Constraint of the Energy Storage Systemych i,t + ydis i,t ≤ 1, (37)0 ≤ Pdisi,t,s ≤ ydis i,t PESS max, (38)0 ≤ Pchi,t,s ≤ ych i,t PESS max, (39)EESSi,t+1,s = EESS i,t,s + ηchPch i,t,s – ηdisPdis i,t,s, (40)0.2EESSmax ≤ EESS i,t,s ≤ 0.9EESS max, (41)Xtych i,t+1 – ych i,t ≤ yESS max, (42)Xtydis i,t+1 – ydis i,t ≤ yESS max, (43)where ych i,t and ydis i,t represent the charging and discharging status of the energy storage system; ηch andηdis are the charging and discharging efficiency; PESS max is the maximum charging/discharging power;EESSmax is the maximum storage capacity; EESS i,t,s is the storage capacity of ESS at time t under scenario s;Pdisi,t,s and Pch i,t,s represent the discharging and charging power output of ESS at time t under scenario s;and yESS max is the maximum charging/discharging number in one day.It can be seen that the charging/discharging state of ESS noted as ych i,t and ydis i,t is not affectedby wind power fluctuations under different scenarios, which remains unchanged during the wholescheduling period. On the other hand, the charging and discharging power output under wind powerscenario s are defined as Pchi,t,s and Pdis i,t,s, respectively, which indicates that the charging/dischargingpower of ESS depends on the possible wind power scenarios. As a result, the charging/dischargingstate of ESS is regarded as the decision variables in the first stage, which is determined before thepossible wind power scenarios happened. Moreover, the charging/discharging power output of ESSis considered as the decision variable in the second stage, which is used to alleviate the wind powerfluctuation in real time.Appl. Sci. 2020, 10, 181 9 of 153.3.3. Relevant Constraints of Two StagesFor the two-stage scheduling model, the relevant constraints of the first and second stages aredescribed as follows:PACi,t,s = PSub i,t,s + PW i,t,s + PG i,t + Pdis i,t,s – Pch i,t,s – PL i,t 8i 2 NAC, (44)QAC i,t,s = QSub i,t,s + QW i,t,s + QG i,t + QCB i,t – QL i,t 8i 2 NAC, (45)PDCi,t = PW i,t,s + PVSC i,t,s (DC) – PL i,t 8i 2 NDC, (46)where PSubi,t,s and QSub i,t,s represent the active and reactive power from the substation connected withnode i at time t under scenario s; PW i,t,s and QW i,t,s represent the active and reactive power from the windgenerator under scenario s; PG i,t and QG i,t represent the active and reactive power output by the gas-firedunits; and PL i,t and QL i,t represent the active and reactive power load at node i.In this study, the gas consumption plan by gas-fired units is determined before wind poweruncertainty is observed and will not adjust in real time. Consequently, the power output of gas-firedunits cannot be changed under different wind power scenarios at the second stage.Furthermore, the hybrid AC/DC distribution network model in Section 2 is established for thecertain wind power scenario. In other words, all the variables in Equations (1)–(21) should be changedto the corresponding variables for different wind power scenarios. Taking the DC distribution networkas an example, UiDC ,t , IkDC ,t , and PDC k,t should be replaced by UiDC ,t,s, IkDC ,t,s, and PDC k,t,s, respectively. Additionally,the power flow model for the DC distribution network under scenario s can be expressed as follows:8

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