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.
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 [
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 [
Simulation-based methods often use iterative optimization assessment procedure, i.e., optimization steps are performed with a posterior reliability assessment program [
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 [
Post-fault load restoration is considered in the DN planning model proposed in [
As an effective mean to improve solution efficiency, parallel computing has been widely applied in power system optimization such as distributed optimal power flow [
Built upon [
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 [
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.
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 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 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.
(1) |
(2) |
(3) |
(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.
(5) |
(6) |
The EDS is assumed to be completely reliable, and the influence of the backbone grid failures on the sub-area is reflected in the EOB.
The objective function for the RcDNP model is the total investment, which consists of investment cost, maintenance cost, and reliability cost.
(7) |
The investment cost and the maintenance cost at stage t are defined as:
(8) |
(9) |
The constraints in the model include the following.
Operation constraints are classified into normal conditions and fault conditions.
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
Here 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.
(22) |
(23) |
(24) |
(25) |
(26) |
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
(33) |
Here 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 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 [
Regarding the problem of backbone grid planning, the following constraints need to be attached.
(34) |
(35) |
(36) |
(37) |
(38) |
(39) |
(40) |
(41) |
(42) |
(43) |
(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).
This section introduces a parallelizable augmented Lagrangian algorithm [
The algorithm presented in [
(50) |
where is the objective function; 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:
(51) |
With the substitution of variables in (49), constraint (6) can be written as:
(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:
(53) |
In order to make the structure of the model clearer, we further define and as:
(54) |
(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).
(56) |
The coupling vectors of backbone grid and sub-area are confined to the regions constructed by coordination constraints. Thus, a decomposable RcDNP model is derived in the form of problem (50).
(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 and as:
(58) |
(59) |
Initialize parameters , , , , , , where and are dual functions in the planning problem of the backbone grid and the
2) Step 2: solve initial value of the sub-problem. Solve the problem of the backbone grid:
(60) |
Solve the planning model of each sub-area:
(61) |
Construct the subset of the convex hull of the feasible regions and using the linear combination of above solution, where is the solution of the backbone grid planning problem, and is the solution of the
The coordination layer calculates coordination variables by solving the following optimization problem.
(62) |
where is the set of coordination variables for the backbone grid and all sub-area planning problems.
3) Step 3: update , .
4) Step 4: execute Gauss-Seidel iteration. Update variables: , , , , , , where is the value of dual function in the
Solve the optimization model for the backbone grid:
(63) |
Solve the optimization model of each sub-area:
(64) |
Solve the model (P3) to update the coordination variables: .
5) Step 5: if , , return to Step 4. Otherwise, perform the following steps: , where and are defined as the dual functions of the original problems:
(65) |
(66) |
where the variables with “~” and “^” are temporary variables.
Solve the optimization model of the backbone grid:
(67) |
Solve the optimization model of each sub-area:
(68) |
Add the vertex to the subset of the convex hull of the feasible region: ,, , , ,, where and are the cutting plane functions used to approximate the dual function (65) and (66), which are defined as:
(69) |
(70) |
The algorithm converges if the gaps and are small enough.
6) Step 6: check the convergence criterion. If , the algorithm converges. Otherwise, perform the following steps.
Calculate . If , the Nesterov acceleration method with restart is adopted to update and [
7) Step 7: update : . If , the algorithm stops. Otherwise, return to Step 3.
The flowchart depicting the solution process is presented in

Fig. 3 Flowchart depicting solution process.
In Section III-A, the RcDNP model is reformed to adapt to the algorithm in [
(71) |
In [
In order to analyze the rate of convergence, the dual function of the convex relaxation (71) is introduced as:
(72) |
Assume that the original problem has an optimal dual solution . According to Proposition 2 in [
(73) |
Considering that is non-decreasing, we have:
(74) |
For the case where parameters and both vary but satisfy , 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 ; ③ an “accelerated” step, in which . According to Theorem 3 in [
(75) |
where is the number of acceleration steps performed.
If there are enough acceleration steps, we will have . 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 [
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 [
The backbone grid consists of eight nodes and each sub-area consists of 20 nodes. Branch and node parameters partly come from [
The results and solution time of the proposed method are compared with the centralized method [

Fig. 4 Extended planning results of two methods for backbone grid and two sub-areas.
Method | Required 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.
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 [
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 Extended planning results of two methods for backbone grid and three sub-areas.
Method | Total 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 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 [
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 Extended planning results of proposed method for backbone grid and six sub-areas.
Total cost (k$) | Required SAIDI | SAIDI | Solution time (s) |
---|---|---|---|
40429 | 1.9 | 1.8481 | 5358 |
2.6 | 2.5951 | ||
2.2 | 1.9973 | ||
2.8 | 2.7961 | ||
2.4 | 2.3895 | ||
2.9 | 2.7823 |
To further verify the scalability of the proposed method, a 1495-node test system consisting of a backbone grid and 10 sub-areas is used to test the proposed method. The backbone grid is a modified 85-node DN and the sub-area is a modified 141-node DN, the information of which can be found in MATPOWER [
Total cost (k$) | Required SAIDI | SAIDI | Solution 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 |
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.
System | Number of binary variables | |
---|---|---|
Centralized method | Proposed 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 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.
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 |
—— | Set of alternative types of conductors | |
—— | Set of alternative types of transformers | |
—— | Set of branches | |
—— | Set of feeders | |
—— | Set of nodes | |
—— | Set of substation nodes | |
—— | Set of transformers | |
—— | Set of transformer outlet branches | |
—— | Set of nodes connected to node i | |
—— | Set of equivalent load nodes in backbone grid | |
—— | Set of equivalent source nodes in sub-area | |
—— | Subsets of convex hull and | |
—— | Unit vector whose elements are equal to 0 except the | |
, | —— | Aging vectors for conductors and transformers |
—— | Vectors of Lagrangian multipliers | |
—— | Vectors of decision variables of backbone grid and the | |
—— | Feasible sets of backbone grid and the | |
—— | Feasible set | |
—— | Vectors of decision variables and coordination variables | |
—— | Vectors of coordination variables describing boundary conditions for backbone grid and the | |
—— | Variables in the | |
B. | —— | Parameters |
—— | Durations of switching-only interruptions and repair-and-switching interruptions associated with failure of branch connecting nodes x and y | |
—— | Cost coefficient of energy not supplied | |
—— | Number of year up to stage t | |
—— | Present value factor for investment costs at stage t | |
—— | Present value factor for maintenance costs at stage t | |
—— | System average interruption duration index (SAIDI) requirement at stage t | |
—— | Penalty value | |
—— | Investment cost for alternative conductor | |
—— | Maintenance cost for alternative conductor | |
—— | Investment cost for alternative transformer | |
—— | Maintenance cost for alternative transformer | |
—— | Investment cost for substation at node | |
—— | Maintenance cost for substation at node | |
—— | Distance between node i and node j | |
—— | Parameter equal to 1 when type a conductor exists at branch ij originally | |
—— | Parameter equal to 1 when type a transformer exists at transformer Tr originally | |
—— | A sufficiently large positive constant | |
—— | The maximum number of transformers in substation | |
—— | Number of subproblems | |
—— | Parameter equal to 1 when a substation exists at node s originally | |
—— | Number of sub-areas | |
—— | Number of customers at node i at stage t | |
—— | Resistance unit of type a conductor | |
—— | Capacity of type a conductor | |
—— | Capacity of type a transformer | |
—— | Inductance unit of type a conductor | |
—— | Lower and upper bounds of variable | |
C. | —— | Continuous Variables |
—— | Failure rate of branch ij at stage t | |
—— | Repair time of branch ij at stage t | |
—— | Customer interruption frequency (CIF) of node i at stage t | |
—— | Customer interruption duration (CID) of node i at stage t | |
—— | Temporarily estimated CIF of equivalent load node (ELN) i at stage t | |
—— | Temporarily estimated CID of ELN i at stage t | |
—— | Expected energy not supplied at stage t | |
—— | Variable in the range of and equal to 1 when node i is supplied by feeder f under normal operation condition at stage t | |
—— | Variable in the range of and equal to 1 when branch ij is supplied by feeder f under normal operation condition at stage t | |
—— | Substitution variable of product term for equivalent outlet branch of equivalent distribution source in sub-area at stage t | |
—— | Resistance of branch ij at stage t | |
—— | Active and reactive load demands of node i at stage t | |
—— | Equivalent active and reactive loads of sub-area at node i at stage t | |
—— | Active and reactive demands at node i after post-fault network reconfiguration due to a fault at branch xy ( represents normal operation condition) at stage t | |
—— | Active and reactive outputs of distributed generation at node i after post-fault network reconfiguration due to a fault in branch xy ( represents normal operation condition) at stage 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 ( represents normal operation condition) at stage t | |
—— | Active and reactive power flows through transformer Tr after post-fault network reconfiguration due to a fault at branch xy ( represents normal operation condition) at stage t | |
—— | Capacity of branch ij at stage t | |
—— | Capacity of transformer Tr at stage t | |
—— | SAIDI at stage t | |
—— | Substitution variable for product of failure rate and repair time of equivalent outlet branch (EOB) xy at stage t | |
—— | Square of voltage at reference node i at stage t | |
—— | Square of voltage at node i due to a fault at branch xy ( represents normal operation condition) at stage t | |
—— | Inductance of branch ij at stage t | |
D. | —— | Binary Variables |
—— | Binary variable equal to 1 when there exists a conductor at branch ij, otherwise equal to 0 at stage t | |
—— | Binary variable equal to 1 when alternative conductor a at branch ij is installed, otherwise equal to 0 at stage t | |
—— | Binary variable equal to 1 when transformer Tr exists, otherwise equal to 0 at stage t | |
—— | Binary variable equal to 1 when alternative transformer a at candidate position Tr is installed, otherwise equal to 0 at stage t | |
—— | Binary variable equal to 1 when substation at node s is built, otherwise equal to 0 at stage t | |
—— | Binary variable equal to 1 when node i is affected by outage due to a fault in branch xy at stage 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 | |
—— | Binary variable equal to 1 when branch ij is connected after network reconfiguration due to a fault in branch xy ( represents normal operation condition) at stage t |
References
C. Gandioli, M.-C. Alvarez-Hérault, P. Tixador et al., “Innovative distribution networks planning integrating superconducting fault current limiters,” IEEE Transactions on Applied Superconductivity, vol. 23, no. 3, pp. 5603904-5603904, Jun. 2013. [Baidu Scholar]
D. S. Kumar, D. Srinivasan, A. Sharma et al., “Adaptive directional overcurrent relaying scheme for meshed distribution networks,” IET Generation, Transmission & Distribution, vol. 12, no. 13, pp. 3212-3220, Jul. 2018. [Baidu Scholar]
M. Gholami, J. Moshtagh, and N. Ghadernejad, “Service restoration in distribution networks using combination of two heuristic methods considering load shedding,” Journal of Modern Power Systems and Clean Energy, vol. 3, no. 4, pp. 556-564, Dec. 2015. [Baidu Scholar]
F. R. Islam, K. Prakash, K. A. Mamun et al., “Aromatic network: a novel structure for power distribution system,” IEEE Access, vol. 5, pp. 25236-25257, Oct. 2017. [Baidu Scholar]
Z. Li, W. Wu, B. Zhang et al., “Analytical reliability assessment method for complex distribution networks considering post-fault network reconfiguration,” IEEE Transactions on Power Systems, vol. 35, no. 2, pp. 1457-1467, Mar. 2020. [Baidu Scholar]
H. L. Willis, Power Distribution Planning Reference Book, 2nd ed. New York: Marcel Dekker, 2004. [Baidu Scholar]
IEEE Guide for Electric Power Distribution Reliability Indices, IEEE Standard 1366-2003, May 2004. [Baidu Scholar]
Y. Tang, “Power distribution system planning with reliability modeling and optimization,” IEEE Transactions on Power Systems, vol. 11, no. 1, pp. 181-189, Feb. 1996. [Baidu Scholar]
R. E. Brown, S. Gupta, R. D. Christie et al., “Automated primary distribution system design: reliability and cost optimization,” IEEE Transactions on Power Delivery, vol. 12, no. 2, pp. 1017-1022, Apr. 1997. [Baidu Scholar]
A. A. Chowdhury and D. O. Koval, Power Distribution System Reliability: Practical Methods and Applications. Hoboken: Wiley, 2009. [Baidu Scholar]
R. C. Lotero and J. Contreras, “Distribution system planning with reliability,” IEEE Transactions on Power Delivery, vol. 26, no. 4, pp. 2552-2562, Oct. 2011. [Baidu Scholar]
I. Ziari, G. Ledwich, A. Ghosh et al., “Optimal distribution network reinforcement considering load growth, line loss, and reliability,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 587-597, May 2013. [Baidu Scholar]
S. M. Mazhari, H. Monsef, and R. Romero, “A multi-objective distribution system expansion planning incorporating customer choices on reliability,” IEEE Transactions on Power Systems, vol. 31, no. 2, pp. 1330-1340, Mar. 2016. [Baidu Scholar]
I. Hernando-Gil, I.-S. Ilie, and S. Z. Djokic, “Reliability planning of active distribution systems incorporating regulator requirements and network-reliability equivalents,” IET Generation, Transmission & Distribution, vol. 10, no. 1, pp. 93-106, Jan. 2016. [Baidu Scholar]
N. N. Mansor and V. Levi, “Integrated planning of distribution networks considering utility planning concepts,” IEEE Transactions on Power Systems, vol. 32, no. 6, pp. 4656-4672, Nov. 2017. [Baidu Scholar]
I. J. Ramirez-Rosado and J. A. Dominguez-Navarro, “Possibilistic model based on fuzzy sets for the multiobjective optimal planning of electric power distribution networks,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1801-1810, Nov. 2004. [Baidu Scholar]
N. N. Mansor and V. Levi, “Operational planning of distribution networks based on utility planning concepts,” IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 2114-2127, May 2019. [Baidu Scholar]
C. Wang, T. Zhang, F. Luo et al., “Fault incidence matrix based reliability evaluation method for complex distribution system,” IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 6736-6745, Nov. 2018. [Baidu Scholar]
T. Zhang, C. Wang, F. Luo et al., “Optimal design of the sectional switch and tie line for the distribution network based on the fault incidence matrix,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 4869-4879, Nov. 2019. [Baidu Scholar]
M. Jooshaki, A. Abbaspour, M. Fotuhi-Firuzabad et al., “MILP model of electricity distribution system expansion planning considering incentive reliability regulations,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 4300-4316, Nov. 2019. [Baidu Scholar]
R. H. Fletcher and K. Strunz, “Optimal distribution system horizon planning - part I: formulation,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 791-799, May 2007. [Baidu Scholar]
R. H. Fletcher and K. Strunz, “Optimal distribution system horizon planning - part II: application,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 862-870, May 2007. [Baidu Scholar]
G. Muñoz-Delgado, J. Contreras, and J. M. Arroyo, “Distribution network expansion planning with an explicit formulation for reliability assessment,” IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 2583-2596, May 2018. [Baidu Scholar]
A. Bosisio, A. Berizzi, D. Lupis et al., “A tabu-search-based algorithm for distribution network restoration to improve reliability and resiliency,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 1, pp. 302-311, Jan. 2023. [Baidu Scholar]
Z. Li, W. Wu, X. Tai et al., “A reliability-constrained expansion planning model for mesh distribution networks,” IEEE Transactions on Power Systems, vol. 36, no. 2, pp. 948-960, Mar. 2021. [Baidu Scholar]
C. Lin and S. Lin, “Distributed optimal power flow with discrete control variables of large distributed power systems,” IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 1383-1392, Aug. 2008. [Baidu Scholar]
Z. Tang, D. J. Hill, and T. Liu, “Fast distributed reactive power control for voltage regulation in distribution networks,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 802-805, Jan. 2019. [Baidu Scholar]
Y. Wang, L. Wu, and S. Wang, “A fully-decentralized consensus-based [Baidu Scholar]
ADMM approach for DC-OPF with demand response,” IEEE Transactions on Smart Grid, vol. 8, no. 6, pp. 2637-2647, Nov. 2017. [Baidu Scholar]
A. Mohammadi, M. Mehrtash, and A. Kargarian, “Diagonal quadratic approximation for decentralized collaborative [Baidu Scholar]
optimal power flow,” IEEE Transactions on Smart Grid, vol. 10, no. 3, pp. 2358-2370, May 2019. [Baidu Scholar]
J. C. Moreira, E. Miguez, C. Vilacha et al., “Large-scale network layout optimization for radial distribution networks by parallel computing,” IEEE Transactions on Power Delivery, vol. 26, no. 3, pp. 1946-1951, Jul. 2011. [Baidu Scholar]
J. C. Moreira, E. Miguez, C. Vilacha et al., “Large-scale network layout optimization for radial distribution networks by parallel computing: implementation and numerical results,” IEEE Transactions on Power Delivery, vol. 27, no. 3, pp. 1468-1476, Jul. 2012. [Baidu Scholar]
J. R. E. Fletcher, T. Fernando, H. H.-C. Iu et al., “Spatial optimization for the planning of sparse power distribution networks,” IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 6686-6695, Nov. 2018. [Baidu Scholar]
Ž. N. Popović, V. D. Kerleta, and D. S. Popović, “Hybrid simulated annealing and mixed integer linear programming algorithm for optimal planning of radial distribution networks with distributed generation,” Electric Power Systems Research, vol. 108, pp. 211-222, Mar. 2014, [Baidu Scholar]
S. M. Mazhari, H. Monsef, and R. Romero, “A hybrid heuristic and evolutionary algorithm for distribution substation planning,” IEEE Systems Journal, vol. 9, no. 4, pp. 1396-1408, Dec. 2015. [Baidu Scholar]
S. Heidari and M. Fotuhi-Firuzabad, “Integrated planning for distribution automation and network capacity expansion,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 4279-4288, Jul. 2019. [Baidu Scholar]
A. Navarro and H. Rudnick, “Large-scale distribution planning - part II: macro-optimization with Voronoi’s diagram and tabu search,” IEEE Transactions on Power Systems, vol. 24, no. 2, pp. 752-758, May 2009. [Baidu Scholar]
B. Dandurand, N. Boland, J. Christiansen et al., “A parallelizable augmented lagrangian method applied to large-scale non-convex-constrained optimization problems,” Mathematical Programming, vol. 175, no. 1-2, pp. 503-536, May 2019. [Baidu Scholar]
T. Goldstein, B. O’Donoghue, S. Setzer et al., “Fast alternating direction optimization methods,” SIAM Journal on Imaging Sciences, vol. 7, no. 3, pp. 1588-1623, Jan. 2014. [Baidu Scholar]
B. O’Donoghue and E. Candès, “Adaptive restart for accelerated gradient Schemes,” Foundations of Computational Mathematics, vol. 15, no. 3, pp. 715-732, Jun. 2015. [Baidu Scholar]
Y. Sun, W. Wu, Y. Lin et al. (2023, Oct.). Data for case studies. [Online]. Available: https://www.jianguoyun.com/p/DXWXhKoQ5ZijCRjX2p4FIAA [Baidu Scholar]
Y. Sun, W. Wu, Y. Lin et al. (2023, Oct.). Supplemental file for parallel computing based solution for reliability-constrained distribution network planning. [Online]. Available: https://www.jianguoyun.com/p/DVEJNn0Q5ZijCRif4p4FIAA [Baidu Scholar]
R. D. Zimmerman and C. E. Murillo-Sanchez. (2020, Dec.). MATPOWER (Version 7.1). [Online]. Available: https://matpower.org [Baidu Scholar]