Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Parallel Computing Based Solution for Reliability-constrained Distribution Network Planning  PDF

  • Yaqi Sun 1,2
  • Wenchuan Wu 1,2 (Fellow, IEEE)
  • Yi Lin 3
  • Hai Huang 3
  • Hao Chen 3
1. State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing, 100084, China; 2. Sichuan Energy Internet Research Institute, Tsinghua University, Chengdu, China; 3. State Grid Fujian Electric Power Co. Ltd., Fuzhou, China

Updated:2024-07-26

DOI:10.35833/MPCE.2023.000760

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Abstract

The main goal of distribution network (DN) expansion planning is essentially to achieve minimal investment constrained by specified reliability requirements. The reliability-constrained distribution network planning (RcDNP) problem can be cast as an instance of mixed-integer linear programming (MILP) which involves ultra-heavy computation burden especially for large-scale DNs. In this paper, we propose a parallel computing based solution method for the RcDNP problem. The RcDNP is decomposed into a backbone grid and several lateral grid problems with coordination. Then, a parallelizable augmented Lagrangian algorithm with acceleration method is developed to solve the coordination planning problems. The lateral grid problems are solved in parallel through coordinating with the backbone grid planning problem. Gauss-Seidel iteration is adopted on the subset of the convex hull of the feasible region constructed by decomposition. Under mild conditions, the optimality and convergence of the proposed method are proven. Numerical tests show that the proposed method can significantly reduce the solution time and make the RcDNP applicable for real-world problems.

I. INTRODUCTION

IN order to improve the reliability of power supply, mesh-designed architecture is commonly adopted in the current urban distribution networks (DNs). The DN operates radially under normal conditions, and redundant lines are used for power rerouting after failures [

1]-[3]. Therefore, the DN with mesh structure has higher reliability [4]. When calculating the reliability index of the mesh-designed DN, it is necessary to consider the fault isolation and load transfer after the fault, which can be achieved through network reconfiguration. Otherwise, the reliability of the DN may be underestimated [5]. Since different customers or lateral grids have different power supply reliability criteria, the DN expansion planning scheme is optimized to achieve the minimum investment cost constrained by specified reliability requirements. Commonly used reliability indicators include customer interruption frequency (CIF), customer interruption duration (CID), system average interruption frequency index (SAIFI), and system average interruption duration index (SAIDI) [6], [7].

DN planning considering reliability has been studied. However, early researches mostly penalize the expectation of power loss in the objective function, which implicitly and approximately reflects the reliability of DNs [

8], [9]. With the improvement of relevant standards, some quantitative indices are used to measure the reliability of DNs [7], [10]. In order to make the planning results meet the specified reliability requirements, the reliability assessment process is required during optimization, which can be achieved by simulation-based methods or analytical methods.

Simulation-based methods often use iterative optimization assessment procedure, i.e., optimization steps are performed with a posterior reliability assessment program [

11]-[15]. In [11], a pool of feasible solutions with diverse expansion plans is first obtained, and the reliability index of each plan is calculated to determine the optimal solution. A comprehensive planning methodology is proposed in [12] considering upgrading the conventional equipment in the DNs. The entire solution process includes the optimal DN reinforcement and the power flow solution process based on Gauss-Seidel iteration, which is used to evaluate the performance of the reinforcement scheme. Based on [11], the choices of customers on reliability have been considered in DN planning model by carrying out Monte Carlo based simulation in the solution process [13]. An integrated method for reliability planning and risk estimation in DNs is proposed in [14], which takes the backup supply or automatic/manual reconfiguration schemes into the consideration. The reliability assessment part of [14] still relies on the time-sequential Monte Carlo based simulation. Decision tree is used in [15] to solve multi-stage network planning problem, in which the switchgear optimization is implemented by simulation software. Tabu search is adopted in [16] to solve the DN planning model considering uncertainty, which requires evaluations to determine the movements for the next search step. Genetic algorithm involving reliability assessment procedure to calculate the fitness function is used for reliability planning stage in [17]. The simulation-based method is intuitive and easy to implement, but suffers from long solution time. The evaluation procedure cannot be embedded in the planning model to obtain a joint solution, while the iterative heuristic method cannot guarantee optimality.

Meanwhile, the analytical solution method of the reliability index has been studied in some literatures. Based on the analytical solution model of reliability, some pioneer works have embedded analytical reliability constraints into the DN planning model. Based on the fault incidence matrix proposed in [

18], the fault incidence matrix is applied in [19] for the joint optimization configuration of sectional switches and tie-lines. Linearized models of different reliability indices are introduced in [20] and then involved in a mixed-integer linear programming (MILP) model of DN planning. Network modeling formulation of reliability indices are derived in [21] and [22] to consider events such as fault isolation and load restoration in DN planning. The multi-level expansion planning problem of the DN is modeled in [23] as an MILP, which has good convergence and can be solved efficiently. However, the post-fault load restoration which can improve the reliability of DNs [5] is not fully considered. For mesh DNs, planning schemes without tie switches for post-fault load restoration may lead to excessive investment.

Post-fault load restoration is considered in the DN planning model proposed in [

24], but the model is solved by heuristic algorithm. Inspired by [23], a reliability-constrained DN planning (RcDNP) considering post-fault load restoration for mesh DNs is proposed in [25]. The DN expansion planning problem is finally formulated as an MILP. However, the model proposed in [25] does not take the sparse nature of the topological connection relationship of the DN into consideration. Existing studies usually model RcDNP as MILP. As the scale of the DN expands, the number of decision variables in the centralized planning model will explosively grow, making it challenging to handle due to the escalating model complexity. Furthermore, with the proliferation of decision variables, the search space of the MILP problem also expands significantly, leading to reduced solution efficiency. This makes it challenging to find the optimal solution or even a feasible solution within an acceptable timeframe. Planners often need to change the boundary conditions of the problem to create different planning scenarios, rather than forming a single plan at once. This requires repetitive calculations, making the centralized model unsuitable for practical applications.

As an effective mean to improve solution efficiency, parallel computing has been widely applied in power system optimization such as distributed optimal power flow [

26] and distributed reactive power control [27]. However, distributed optimization algorithms adopted in the existing literature such as alternating direction method of multipliers (ADMM) [28] and analytical target cascading (ATC) [29] cannot be used directly in solving the RcDNP problem, since it is an MILP involving a large number of integer variables. A parallel computing method combining branch exchange and dynamic programming for large-scale network layout optimization has been proposed in [30]. Based on [30], the simultaneous optimization of the line layout and type of conductor is further implemented in [31]. But the branch exchange algorithm adopted in network structure optimization still relies on heuristic search and is only suitable for radial DNs. In [32], a genetic algorithm based planning method is proposed considering the sparseness of the rural DN. Reference [33] has established an MILP model for radial DN expansion planning. This model decomposes the original planning problem into several sub-problems and employs a simulated annealing algorithm for the iterative solution. In [34], the load points of the DN are decomposed into different substation supply areas, and subsequently, the static substation planning problem is addressed using an evolutionary algorithm. A parallelizable solution method based on the genetic algorithm is introduced in [35], wherein the genetic algorithm is initially employed for solving the sub-problems in parallel. Subsequently, the fitness function of the integrated solution is decomposed to update the solutions of the sub-problems. Reference [36] divides the planning region into smaller sub-regions during the DN planning process. These smaller sub-regions are independently optimized using heuristic methods. Based on the optimization at the sub-regional level, a global analysis of the planning region is then conducted. However, the heuristic method is not stable and the optimality of the solution cannot be guaranteed theoretically. Parallel accelerated solution method for large-scale mesh DN expansion planning remains to be further studied.

Built upon [

25], this paper presents a parallel computing based solution method for RcDNP to overcome the above difficulties. Firstly, the RcDNP model is reformulated to the backbone grid and sub-areas. Then, a parallelizable augmented Lagrangian algorithm is adopted to solve the model in parallel manner. Furthermore, the Nesterov acceleration method with restart is used to improve the convergence speed.

The main contributions of this paper are summarized as follows.

1) A decomposed RcDNP model is proposed, in which the planning grid is decomposed into the backbone grid and several sub-areas. The number of integer variables of the planning problem roughly increases linearly with the size of the planning DN, while the centralized RcDNP model in [

25] increases quadratically.

2) A parallelizable augmented Lagrangian algorithm with acceleration method is developed to solve the coordination planning problem involving the backbone grid and sub-areas. Numerical tests show that the proposed parallel computing based solution method exhibits a linearly increasing computation time with the growing size of DNs. The optimality and convergence of the algorithm is also proved rigorously.

The remainder of this paper is arranged as follows. The mathematical formulation of the RcDNP model for the backbone grid and sub-area is introduced in Section II. The parallel solution process with acceleration method is discussed in Section III. Numerical tests and results are demonstrated to illustrate the performance of the proposed solution method in Section IV. Conclusions are drawn in Section V.

II. Mathematical Formulation of RcDNP Model

A. Decomposed Planning Model

In the DN, substations are designed to supply power to multiple load concentrated areas. These load concentrated areas are connected to the substation through the backbone grid. This natural sparse structure inspires us to decompose the RcDNP problem into the backbone grid and sub-area planning problems. It can be solved in parallel through coordination. The decomposed planning model is shown in Fig. 1.

Fig. 1  Decomposed planning model.

The proposed model consists of three parts: backbone grid planning module, sub-area planning module, and coordination layer. As shown in Fig. 2, the sub-area is aggregated as an equivalent load node (ELN) i' in the backbone grid planning. For the sub-area planning, the backbone grid is represented by an equivalent distribution source (EDS) i connected a series equivalent outlet branch (EOB) ij with a certain probability of failure.

Fig. 2  Schematic diagram of equivalent decoupled models of backbone grid and sub-area for RcDNP.

The consistency conditions of boundary variables include two aspects: the consistency of the power flow and the consistency of the reliability index.

The consistency of the power flow means that the equivalent load of the backbone grid should be equal to the power flow of corresponding EOB in the sub-area. Besides, corresponding branches in the backbone grid and sub-area share the same capacity while corresponding nodes in the backbone grid and sub-area share the same voltage.

Pi'd,t=PijNO,t    i'ΨDe,iΨSe,jΨi (1)
Qi'd,t=QijNO,t    i'ΨDe,iΨSe,jΨi (2)
Si'j'C,t=SijC,t      i'ΨDe,iΨSe,j'Ψi',jΨi (3)
Ui'NO,t=Uiss,t    i'ΨDe,iΨSe (4)

The consistency of the reliability index means that the parameter of the EOB in the sub-area and the reliability index of the ELN of the backbone grid should meet the following constraints.

λijt=CIFi'e,t    i'ΨDe,iΨSe,jΨi (5)
τijt=CIDi'e,t/CIFi'e,t    i'ΨDe,iΨSe,jΨi (6)

The EDS i is assumed to be completely reliable, and the influence of the backbone grid failures on the sub-area is reflected in the EOB.

B. Objective Function

The objective function for the RcDNP model is the total investment, which consists of investment cost, maintenance cost, and reliability cost.

min f(x)=t=1T[δtIctI+δtO(ctM+ωEENSt)] (7)

The investment cost ctI and the maintenance cost ctM at stage t are defined as:

ctI=ijΨBaΛCCCa,Ilija,t+TrΨTaΛTCTa,ImTra,t+sΨSSCSs,Inst (8)
ctM=etTijΨBaΛCCCa,Mτ=1tlija,τgCa,τ+lija,0gCa,0+         etTTrΨTaΛTCTa,Mτ=1tmTra,tgTa,τ+mTra,0gTa,0+        sΨSSCSs,Mt=1tnst+ns0 (9)

C. Constraints

The constraints in the model include the following.

1) Operation Constraints

Operation constraints are classified into normal conditions and fault conditions.

-M(1-sijxy,t)Ujxy,t-Uixy,t+2(rijtPijxy,t+xijtQijxy,t)    ijΨB  (10)
Ujxy,t-Uixy,t+2(rijtPijxy,t+xijtQijxy,t)M(1-sijxy,t)    ijΨB  (11)
jΨiPijxy,t+Pixy,t+Pi,gxy,t=0    iΨN (12)
jΨiQijxy,t+Qixy,t+Qi,gxy,t=0    iΨN (13)
U̲Uixy,tU¯    iΨN (14)
Uixy,t=Uiss,t    iΨSS (15)
PTrxy,t=Pijxy,tQTrxy,t=Qijxy,t    TrΨT,ijΨTrout (16)
-Msijxy,tPijxy,tMsijxy,t-SijC,tPijxy,tSijC,t-Msijxy,tQijxy,tMsijxy,t-SijC,tQijxy,tSijC,t    ijΨB (17)
-2SijC,tPijxy,t±Qijxy,t2SijC,t    ijΨB (18)
PTrxy,tSTrC,tQTrxy,tSTrC,t    TrΨT (19)
-2STrC,tPTrxy,t±QTrxy,t2STrC,t    TrΨT (20)
sijxy,tlijt    ijΨB (21)

Here xyΨB{NO} represents the branch fault set and normal operation. The linearized power flow constraints, power balance constraints, voltage constraints, and capacity constraints are given in (10)-(20). Constraint (21) indicates that the connecting status is equal to zero when there is no branch constructed.

2) Fault Indicator Variable Constraints

sxyxy,t=0 (22)
hif,t+hxyf,t-1pixy,t    fΨF,iΨN (23)
pixy,tqixy,t    iΨN (24)
Pixy,t=Pit(1-qixy,t)Qixy,t=Qit(1-qixy,t)    iΨN (25)
ijsijxy,t=i1-qixy,t)    ijΨB,iΨN,iΨSS (26)
-M(1-sijNO,t)+hif,thijf,thif,t+M(1-sijNO,t)    ijΨB,fΨF (27)
-M(1-sijNO,t)+hjf,thijf,thjf,t+M(1-sijNO,t)    ijΨB,fΨF (28)
hijf,t=sijNO,t    if line ij is outlet branch of feeder f (29)
hijf,tsijNO,t    ijΨB,fΨF (30)
fhif,t1    iΨN (31)
fhijf,t1    ijΨB (32)
ijsijNO,t=fihif,t    ijΨB,fΨF,iΨN,iΨSS (33)

Here xyΨB represents the failure scenario. Constraint (22) indicates that branch xy is outage and isolated in the scenario where branch xy fails. Constraint (23) determines that the affected nodes by the outage of branch xy must share the same feeder affiliation variable with the one of branch xy. Constraint (24) indicates the nodes not affected by the fault cannot loss power supply due to network reconfiguration. Constraint (25) indicates that if the node can restore power supply after the post-fault network reconfiguration, its load level returns to the normal state; otherwise, it remains in the outage state. Constraints (27)-(30) are for the feeder affiliation variables related to the network topology in normal state, where (27) and (28) show that feeder affiliation variables can be propagated if branch ij is connected under normal operation conditions. Constraints (26) and (31)-(33) are radial operation constraints under normal and fault conditions [

25].

Regarding the problem of backbone grid planning, the following constraints need to be attached.

Pit=PiNO,t    iΨN,iΨDePid,t      iΨN,iΨDe (34)
Qit=QiNO,t    iΨN,iΨDeQid,t      iΨN,iΨDe (35)

3) Equipment Selection Constraints

lijt=etTaΛCτ=1tlija,τgCa,τ+lija,0gCa,0    ijΨB (36)
SijC,t=etTaΛCSCaτ=1tlija,τgCa,τ+lija,0gCa,0    ijΨB (37)
rijt=dijetTaΛCrCaτ=1tlija,τgCa,τ+lija,0gCa,0    ijΨB (38)
xijt=dijetTaΛCxCaτ=1tlija,τgCa,τ+lija,0gCa,0    ijΨB (39)
λijt=dijetTaΛCλCaτ=1tlija,τgCa,τ+lija,0gCa,0    ijΨB (40)
mTrt=etTaΛTτ=1tmTra,τgTa,τ+mTra,0gTa,0    TrΨT (41)
STrC,t=etTaΛTSTraτ=1tmTra,τgTa,τ+mTra,0gTa,0    TrΨT (42)
τ=1tnsτ+ns01    sΨSS (43)
τ=1tTrΨTmTrτNsτ=1tnsτ+ns0    sΨSS (44)

Constraints (36)-(43) indicate that an available equipment must already exist or be constructed at the planning stage. Logic constraint between the installation of transformers and the existence of substation is demonstrated as constraint (44).

4) Calculation of Reliability Index

CIDit=xyΨBλxytτxySWpixy,t+xyΨBλxyt(τxyRP-τxySW)qixy,t    iΨN (45)
SAIDIt=iΨNNCitCIDit/iΨNNCit (46)
SAIDItεSAIDIt (47)

Equations (45) and (46) are the commonly used reliability indices, and (47) is the reliability constraint expressed by SAIDI. For the sub-area planning problem, since the failure rate and repair time of the EOB are also variables, (45) should be rewritten as:

CIDit=xΨSeyΨxλxytτxySWpixy,t+xΨSeyΨxλxyt(τxyRP-τxySW)qixy,t+xΨSeyΨxnixy,tiΨN (48)
nixy,tuxyt-(1-pixy,t)Mnixy,t-pixy,tMnixy,tuxyt+(1-pixy,t)Mnixy,tpixy,tM    xΨSe,yΨx (49)

III. Parallel Solution Process with Acceleration Method

This section introduces a parallelizable augmented Lagrangian algorithm [

37], which is applicable for the split-variable reformulation of mixed-integer optimization problems, and is adopted to solve the coordination planning problem of the backbone grid and sub-areas. Independent planning of the backbone grid and sub-areas is achieved by solving the sub-problems in parallel. The global coordination is achieved by the iteration between the coordination layer and sub-problems.

A. Decomposable RcDNP Model

The algorithm presented in [

37] is adapted to solve problems with the following form:

minxi,zi,ii=1nfi(xi):Qixi=zi,xiXi,i, (z1T,z2T,..,znT)TZ (50)

where fi is the objective function; Qi is a matrix; and Z is the set of coordination variables. The model presented in the previous reference is a centralized model, which needs to be reorganized to adapt to the decomposition.

Combining the backbone grid and sub-areas, we can obtain the decomposed RcDNP model of the entire system:

min  f(xb,xs)=fb(xb)+n=1nsubfs,n(xs,n)s.t.  (10)-(47) for backbone grid       (10)-(33), (36)-(44), (46)-(49) for sub-areas       (1)-(6) (51)

With the substitution of variables in (49), constraint (6) can be written as:

uijt=CIDi'e,t    i'ΨDe,iΨSe, jΨi (52)

To further organize the model into a form suitable for the parallelizable augmented Lagrangian algorithm, the decomposable RcDNP model can be written as the compact matrix form:

minf(xb,xs)=fb(xb)+n=1nsubfs,n(xs,n)s.t.  xbXb       xs,nXs,n        Qbxb=Qs,nxs,n       n=1,2,...,nsub (53)

In order to make the structure of the model clearer, we further define zb and zs,n as:

Qbxb=zb=[CIFi'e,t,CIDi'e,t,Pi'd,t,Qi'd,t,Si'j'C,t,Ui'NO,t]Ti'ΨDe,j'Ψi' (54)
Qs,nxs,n=zs,n=[λijt,uijt,PijNO,t,QijNO,t,SijC,t,Uiss,t]TiΨS,ne,jΨi,n=1,2,...,nsub (55)

Finally, the set Z is used to describe the coupling relationship between the coordination variables of different models, which is restricted by coordination constraints (1)-(5) and (52).

Z=[zbT,zs,1T,...,zs,nsubT]T(1)-(5), (52) (56)

The coupling vectors of backbone grid zb and sub-area zs,n are confined to the regions constructed by coordination constraints. Thus, a decomposable RcDNP model is derived in the form of problem (50).

minx,z f(xb,xs)=fb(xb)+n=1nsubfs,n(xs,n)s.t.  Qbxb=zb       xbXb       Qs,nxs,n=zs,n       xs,nXs,n       [zbT,zs,1T,...,zs,nsubT]TZ (57)

Therefore, the parallelizable augmented Lagrangian algorithm can be adopted to solve problem (57). The detailed steps are described as follows.

1) Step 1: initialize. Define the augmented Lagrangian functions Lb and Ls,n as:

Lb(xb,wb,zb)=f(xb)+wbTQbxb+ρ2Qbxb-zb2 (58)
Ls,n(xs,n,ws,n,zs,n)=f(xs,n)+ws,nTQs,nxs,n+ρ2Qs,nxs,n-zs,n2 (59)

Initialize parameters ϕb̲=ϕs,n̲=-, ρ>0, ε>0, k=0, T=0, γ(0,1), where ϕb̲ and ϕs,n̲ are dual functions in the planning problem of the backbone grid and the nth sub-area, respectively; ε is the criterion to stop iteration; and γ is the parameter of the serious step condition.

2) Step 2: solve initial value of the sub-problem. Solve the problem of the backbone grid:

(P1)  min fb(xb)=t=1T[δtIct,bI+δtO(ct,bM+ωEENSbt)]s.t.  Pi'd̲Pi'd,tPi'd¯    i'ΨDe      Qi'd̲Qi'd,tQi'd¯    i'ΨDe      CIFi'e̲CIFi'e,tCIFi'e¯    i'ΨDe      CIDi'e̲CIDi'e,tCIDi'e¯    i'ΨDe       (10)-(47) (60)

Solve the planning model of each sub-area:

(P2)  min  fs,n(xs,n)=t=1T[δtIct,s,nI+δtO(ct,s,nM+ωEENSs,nt)]s.t.  PijNO̲PijNO,tPijNO¯     iΨS,ne,jΨi        QijNO̲QijNO,tQijNO¯    iΨS,ne,jΨi        λij̲λijtλij¯    iΨS,ne,jΨi        uij̲uijtuij¯    iΨS,ne,jΨi         (10)-(33), (36)-(44), (46)-(49) (61)

Construct the subset of the convex hull of the feasible regions Db={xb0} and Ds,n={xs,n0} using the linear combination of above solution, where xb0 is the solution of the backbone grid planning problem, and xs,n0 is the solution of the nth sub-area planning problem.

The coordination layer calculates coordination variables by solving the following optimization problem.

(P3)  z0=argminzb,zs,nQbxb0-zb2+n=1nsubQs,nxs,n0-zs,n2:zb,zs,nZ (62)

where z0={zb0,zs,10,...,zs,nsub0} is the set of coordination variables for the backbone grid and all sub-area planning problems.

3) Step 3: update k=k+1, T=1.

4) Step 4: execute Gauss-Seidel iteration. Update variables: xbk=xbk-1, xs,nk=xs,nk-1, zk=zk-1, wbk=wbk-1, ws,nk=ws,nk-1, ϕk̲=ϕk-1̲, where ϕk̲ is the value of dual function in the kth iteration.

Solve the optimization model for the backbone grid:

(P4)  xbk=argminxb{{Lb(xb,wbk,zbk):xbDb} (63)

Solve the optimization model of each sub-area:

(P5)  xs,nk=argminxs,n{Ls,n(xs,n,ws,nk,zs,nk):xs,nDs,n} (64)

Solve the model (P3) to update the coordination variables: zk=argminzb,zs,nQbxbk-zb2+n=1nsubQs,nxs,nk-zs,n2:zb,zs,nZ.

5) Step 5: if TTmax, T=T+1, return to Step 4. Otherwise, perform the following steps: ϕ˜b=ϕb̲(wbk+ρ(Qbxbk-zbk)), ϕ˜s,n=ϕs,n̲(ws,nk+ρ(Qs,nxs,nk-zs,nk)), where ϕb̲(wb) and ϕs,n̲(ws,n) are defined as the dual functions of the original problems:

ϕb̲(wb)=minxb{fb(xb)+wbTQbxb:xbXb} (65)
ϕs,n̲(ws,n)=minxs,n{fs,n(xs,n)+ws,nTQs,nxs,n:xs,nXs,n} (66)

where the variables with “~” and “^” are temporary variables.

Solve the optimization model of the backbone grid:

(P6)  x^b=argminxb{ϕb̲(wbk+ρ(Qbxbk-zbk))} (67)

Solve the optimization model of each sub-area:

(P7)  x^s,n=argminxs,n{ϕs,n̲(ws,nk+ρ(Qs,nxs,nk-zs,nk))} (68)

Add the vertex to the subset of the convex hull of the feasible region: Db=conv(Db{x^b}),Ds,n=conv(Ds,n{x^s,n}), εbk=ϕ^b(wbk,xbk,zbk)-ϕbk̲, εs,nk=ϕ^s,n(ws,nk,xs,nk,zs,nk)-ϕs,nk̲, Δϕbk=ϕ˜b-ϕbk̲,Δϕs,nk=ϕ˜s,n-ϕs,nk̲, where ϕ^b(wb,xb,zb) and ϕ^s,n(ws,n,xs,n,zs,n) are the cutting plane functions used to approximate the dual function (65) and (66), which are defined as:

ϕ^b(wb,xb,zb)=Lb(xb,wb,zb)+ρ2Qbxb-zb2 (69)
ϕ^s,n(ws,n,xs,n,zs,n)=Ls,n(xs,n,ws,n,zs,n)+ρ2Qs,nxs,n-zs,n2 (70)

The algorithm converges if the gaps εbk and εs,nk are small enough.

6) Step 6: check the convergence criterion. If εbk+n=1nsubεs,nkε, the algorithm converges. Otherwise, perform the following steps.

Calculate ηk=Δϕbk/εbk+n=1nsubΔϕs,nk/εs,nk. If ηkγ, the Nesterov acceleration method with restart is adopted to update wbk and ws,nk [

38].

7) Step 7: update ρ: 1ρ=minmax2ρ(1-ηk),110ρ,1ρmax, 10ρ,1ρmin. If k>kmax, the algorithm stops. Otherwise, return to Step 3.

The flowchart depicting the solution process is presented in Fig. 3.

Fig. 3  Flowchart depicting solution process.

B. Optimality and Convergence

In Section III-A, the RcDNP model is reformed to adapt to the algorithm in [

37]. Since the objective function f(x) is linear, the solution of (57) can be obtained by optimizing on the convex hull conv(X) of the original domain, i.e.,

minx,z{f(x):Qx=z,xconv(X),zZ} (71)

In [

37], based on the conclusion of Proposition 3 and Lemma 1, Proposition 4 proves that the sequence {(xk,zk)} generated by Step 4 in the main loop has limit points, which is optimal for (71). Thus, it is also the optimal solution of the original problem (57).

In order to analyze the rate of convergence, the dual function of the convex relaxation (71) is introduced as:

ϕC(w)=minx,z{f(x)+wT(Qx-z),xconv(X),zZ} (72)

Assume that the original problem has an optimal dual solution w*. According to Proposition 2 in [

37], the sum of the gap between ϕC(w*) and ϕk̲ in all iterations is limited. For the case that parameter ρ is fixed, the following conclusion holds:

k=1N(ϕC(w*)-ϕk̲)< (73)

Considering that ϕk̲ is non-decreasing, we have:

ϕC(w*)-ϕk̲=o(1/k) (74)

For the case where parameters ρ and γ both vary but satisfy ρk(1/γk-1)=c, the rate of convergence can be quadratic. It is worth noting that the actual convergence rate may not reach the theoretical level since the serious step condition may not be guaranteed in each iteration.

After adopting Nesterov acceleration method with restart, the parallelizable augmented Lagrangian algorithm will perform one of the following three operations when updating the dual variable: ① a “restart”; ② a “nonaccelerated” step immediately after a “restart”, in which the acceleration factor αk=1; ③ an “accelerated” step, in which αk>1. According to Theorem 3 in [

38], the residual ck decreases by at least a factor of δ in an accelerated step. Therefore, after k iterations, we can obtain:

ckc0δk^ (75)

where k^ is the number of acceleration steps performed.

If there are enough acceleration steps, we will have ck0. And after the last acceleration step, the dual variable updating will be equivalent to the original algorithm, for which the convergence is known. The numerical test in [

38] proves the advantages of the restart. Research works on restarted variants of other acceleration methods also share similar results [39].

IV. Numerical Tests

The RcDNP model and proposed solution method is tested on the planning of backbone grid and different numbers of sub-areas. System data can be accessed from [

40]. All numerical tests are carried out on a laptop computer with an Intel Core i7-10875H CPU and 16 GB RAM. The MILP model is solved by Gurobi (version 9.5.0).

A. Backbone Grid and Two Sub-areas

The backbone grid consists of eight nodes and each sub-area consists of 20 nodes. Branch and node parameters partly come from [

25]. The topology of the test system can be found in [41].

The results and solution time of the proposed method are compared with the centralized method [

25]. The extended planning results of the two methods for the backbone grid and two sub-areas are shown in Fig. 4. Reliability evaluation [5] is used to verify the results to ensure that the results meet the reliability requirements. The reliability indices, construction costs, and solution time of the two methods are listed in the Table I.

Fig. 4  Extended planning results of two methods for backbone grid and two sub-areas.

TABLE I  Reliability Indices, Construction Costs, and Solution Time of Two Methods
MethodRequired SAIDI (area 1)Evaluated SAIDI (area 1)Required SAIDI (area 2)Evaluated SAIDI (area 2)Total cost (k$)Solution time (s)
Centralized 1.5 1.4997 2 1.978 1007 1335
Proposed 1.5 1.4997 2 1.978 1007 929

It can be observed that the constructed branches and the operation mode planned by the proposed method are the same as those planned by the centralized method. However, due to the parallel solution architecture, the proposed method solves the problem faster than the centralized method.

B. Backbone Grid and Three Sub-areas

Another numerical test is conducted on a larger-scale case with a backbone grid and three sub-areas. The detailed setting of this case can be found in [

41].

The planning results of the centralized planning method and the proposed method for the backbone grid and three sub-areas are shown in Fig. 5. We have compared the proposed method with traditional centralized method and parallel solving method based on genetic algorithm. The construction costs and solution time of different methods are listed in Table II.

Fig. 5  Extended planning results of two methods for backbone grid and three sub-areas.

TABLE II  Construction Costs and Solution Time of Planning Results of Different Methods
MethodTotal cost (k$)Solution time (s)
Centralized method 1243 6980
Proposed method 1243 1935
Parallel solving method based on genetic algorithm 1355 85758

The effect of the acceleration method is shown in Fig. 6. It can be observed that the convergence speed is significantly improved by introducing the acceleration method. The method with acceleration method converges to the optimal value (1243 k$) at the 30th iteration, while the method without acceleration method only reaches 1255 k$ in the same iteration steps.

Fig. 6  Effects of acceleration method.

The effect of the proposed method on multi-stage planning problems and the model considering the uncertainties of load and distributed generation are shown in [

41].

C. Backbone Grid and Six Sub-areas

We further expand the scale of the case to a 139-node DN containing a backbone grid and six sub-areas. As the scale of the centralized model is too large, it cannot be solved in an acceptable time. There are only the results of the parallel computing method shown in Fig. 7. The results of the proposed method with acceleration for the 139-node DN are listed in Table III. When the centralized method is adopted, the gap of the solution is still 22.02% after seven days, and the total cost of the planning scheme is 51333 k$.

Fig. 7  Extended planning results of proposed method for backbone grid and six sub-areas.

TABLE III  Results of Proposed Method with Acceleration for 139-node DN
Total cost (k$)Required SAIDISAIDISolution time (s)
40429 1.9 1.8481 5358
2.6 2.5951
2.2 1.9973
2.8 2.7961
2.4 2.3895
2.9 2.7823

D. 1495-node Test System

To further verify the scalability of the proposed method, a 1495-node test system consisting of a backbone grid and 10 sub-areas is used to test the proposed method. The backbone grid is a modified 85-node DN and the sub-area is a modified 141-node DN, the information of which can be found in MATPOWER [

42]. The results of the proposed method with the acceleration are shown in Table IV, while the centralized method still cannot obtain the solution after seven days.

TABLE IV  Results of Proposed Method with Acceleration for 1495-node Test System
Total cost (k$)Required SAIDISAIDISolution time (s)
71411 1.5 1.4977 98036
1.4 1.3494
1.7 1.6998
1.6 1.5794
1.8 1.7983
1.5 1.4850
1.4 1.3749
1.7 1.6699
2.0 1.9158
1.9 1.8987

E. Analysis of Solution Efficiency

For MILP problems, the number of binary variables can roughly reflect the size of the problem. The search space of the MILP problem rapidly expands with the increase of binary variables.

Table V illustrates the comparison of the number of binary variables between centralized method and the proposed method for the different systems.

TABLE V  Number of Binary Variables BETWEEN CENTRALIZED METHOD AND PROPOSED METHOD
SystemNumber of binary variables
Centralized methodProposed method
48-node 12620 4736
72-node 24398 6800
139-node 82820 12992
1495-node 7130296 56236

As the system scale increases, the number of binary variables in the centralized method will explosively grow, leading to a significant expansion of the search space. To provide a more intuitive illustration, a comparison graph for the number of binary variables and solution time of the two methods at different numbers of sub-areas is shown in Fig. 8.

Fig. 8  Comparison of centralized method and proposed method in terms of number of binary variables and solution time. (a) Number of binary variables. (b) Solution time.

It can be observed that as the size of the DN increases, the number of binary variables in the centralized method rises rapidly, while the number of binary variables in the parallel computing method increases almost linearly with the problem size. The rapid expansion of the model size not only makes the modeling more challenging, but also makes it more difficult to obtain an optimal or even feasible solution within an acceptable time. From the comparison graph of solution time, it can be observed that when the case increases to four sub-areas, the running time of centralized method has exceeded one day while the proposed method converges within the acceptable time in all the cases. The proposed method demonstrates a significant advantage in terms of efficiency.

V. Conclusion

We propose a parallel computing based solution method for solving the RcDNP problem. A decomposition planning model containing the backbone grid and sub-areas is presented, in which the integer variables increase linearly with the size of networks, while those in the original model increase quadratically. A parallelizable augmented Lagrangian algorithm incorporating Nesterov acceleration method with restart is adopted to solve the RcDNP model. Numerical tests on different systems demonstrate that the proposed method has significant advantages in terms of solution efficiency on the premise of ensuring optimality. The proposed method enables the RcDNP model with the potentiality of real-world application.

Nomenclature

Symbol —— Definition
A. —— Sets and Vectors
ΛC —— Set of alternative types of conductors
ΛT —— Set of alternative types of transformers
ΨB —— Set of branches
ΨF —— Set of feeders
ΨN —— Set of nodes
ΨSS —— Set of substation nodes
ΨT —— Set of transformers
ΨTrout —— Set of transformer outlet branches
Ψi —— Set of nodes connected to node i
ΨDe —— Set of equivalent load nodes in backbone grid
ΨSe —— Set of equivalent source nodes in sub-area
Db,Ds,n —— Subsets of convex hull conv(Xb) and conv(Xs,n)
et —— Unit vector whose elements are equal to 0 except the tth element that is equal to 1
gCa,τ, gCa,0,gTa,τ, gTa,0 —— Aging vectors for conductors and transformers
wb,ws,n —— Vectors of Lagrangian multipliers
xb,xs,n —— Vectors of decision variables of backbone grid and the nth sub-area planning model
Xb,Xs,n —— Feasible sets of backbone grid and the nth sub-area planning model
Xi —— Feasible set
xi,zi —— Vectors of decision variables and coordination variables
zb,zs,n —— Vectors of coordination variables describing boundary conditions for backbone grid and the nth sub-area planning model
()k —— Variables in the kth iteration
B. —— Parameters
τxySW,τxyRP —— Durations of switching-only interruptions and repair-and-switching interruptions associated with failure of branch connecting nodes x and y
ω —— Cost coefficient of energy not supplied
τ(t) —— Number of year up to stage t
δtI —— Present value factor for investment costs at stage t
δtO —— Present value factor for maintenance costs at stage t
εSAIDIt —— System average interruption duration index (SAIDI) requirement at stage t
ρ —— Penalty value
CCa,I —— Investment cost for alternative conductor a
CCa,M —— Maintenance cost for alternative conductor a
CTa,I —— Investment cost for alternative transformer a
CTa,M —— Maintenance cost for alternative transformer a
CSs,I —— Investment cost for substation at node s
CSs,M —— Maintenance cost for substation at node s
dij —— Distance between node i and node j
lija,0 —— Parameter equal to 1 when type a conductor exists at branch ij originally
mTra,0 —— Parameter equal to 1 when type a transformer exists at transformer Tr originally
M —— A sufficiently large positive constant
Ns —— The maximum number of transformers in substation
n —— Number of subproblems
ns0 —— Parameter equal to 1 when a substation exists at node s originally
nsub —— Number of sub-areas
NCit —— Number of customers at node i at stage t
rCa —— Resistance unit of type a conductor
SCa —— Capacity of type a conductor
STa —— Capacity of type a transformer
xCa —— Inductance unit of type a conductor
()̲,()¯ —— Lower and upper bounds of variable (·)
C. —— Continuous Variables
λijt —— Failure rate of branch ij at stage t
τijt —— Repair time of branch ij at stage t
CIFit —— Customer interruption frequency (CIF) of node i at stage t
CIDit —— Customer interruption duration (CID) of node i at stage t
CIFie,t —— Temporarily estimated CIF of equivalent load node (ELN) i at stage t
CIDie,t —— Temporarily estimated CID of ELN i at stage t
EENSt —— Expected energy not supplied at stage t
hif,t —— Variable in the range of [0, 1] and equal to 1 when node i is supplied by feeder f under normal operation condition at stage t
hijf,t —— Variable in the range of [0, 1] and equal to 1 when branch ij is supplied by feeder f under normal operation condition at stage t
nixy,t —— Substitution variable of product term λxytτxytpixy,t for equivalent outlet branch of equivalent distribution source in sub-area at stage t
rijt —— Resistance of branch ij at stage t
Pit,Qit —— Active and reactive load demands of node i at stage t
Pid,t,Qid,t —— Equivalent active and reactive loads of sub-area at node i at stage t
Pixy,t, Qixy,t —— Active and reactive demands at node i after post-fault network reconfiguration due to a fault at branch xy (xy=NO represents normal operation condition) at stage t
Pi,gxy,t, Qi,gxy,t —— Active and reactive outputs of distributed generation at node i after post-fault network reconfiguration due to a fault in branch xy (xy=NO represents normal operation condition) at stage t
Pijxy,t, Qijxy,t —— Active and reactive power flows through branch ij (from node i to node j) after post-fault network reconfiguration due to a fault in branch xy (xy=NO represents normal operation condition) at stage t
PTrxy,t, QTrxy,t —— Active and reactive power flows through transformer Tr after post-fault network reconfiguration due to a fault at branch xy (xy=NO represents normal operation condition) at stage t
SijC,t —— Capacity of branch ij at stage t
STrC,t —— Capacity of transformer Tr at stage t
SAIDIt —— SAIDI at stage t
uxyt —— Substitution variable for product of failure rate λxyt and repair time τxyt of equivalent outlet branch (EOB) xy at stage t
Uiss,t —— Square of voltage at reference node i at stage t
Uixy,t —— Square of voltage at node i due to a fault at branch xy (xy=NO represents normal operation condition) at stage t
xijt —— Inductance of branch ij at stage t
D. —— Binary Variables
lijt —— Binary variable equal to 1 when there exists a conductor at branch ij, otherwise equal to 0 at stage t
lija,t —— Binary variable equal to 1 when alternative conductor a at branch ij is installed, otherwise equal to 0 at stage t
mTrt —— Binary variable equal to 1 when transformer Tr exists, otherwise equal to 0 at stage t
mTra,t —— Binary variable equal to 1 when alternative transformer a at candidate position Tr is installed, otherwise equal to 0 at stage t
nst —— Binary variable equal to 1 when substation at node s is built, otherwise equal to 0 at stage t
pixy,t —— Binary variable equal to 1 when node i is affected by outage due to a fault in branch xy at stage t
qixy,t —— Binary variable equal to 1 when node i is still in outage after network reconfiguration following a fault in branch xy at stage t
sijxy,t —— Binary variable equal to 1 when branch ij is connected after network reconfiguration due to a fault in branch xy (xy=NO represents normal operation condition) at stage t

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