Open AccessArticle

Zonal Planning for a Large-Scale Distribution Network Considering Reliability

Zhiwei Shi

¹,

Guangzeng You

^2,*,

Linfu Miu

³,

Ning Sun

¹,

Lei Duan

³,

Qianqian Yu

¹,

Chuanliang Xiao

⁴ and

Ke Zhao

⁴

Chuxiong Power Supply Bureau, Yunnan Power Grid Corporation, Chuxiong 675000, China

Planning and Construction Research·Center, Yunnan Power Grid Corporation, Kunming 650217, China

Chuxiong Shuangbai Power Supply Bureau, Yunnan Power Grid Corporation, Chuxiong 675000, China

⁴

School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255000, China

Author to whom correspondence should be addressed.

Processes 2025, 13(2), 354; https://doi.org/10.3390/pr13020354

Submission received: 13 November 2024 / Revised: 27 December 2024 / Accepted: 17 January 2025 / Published: 27 January 2025

(This article belongs to the Section Energy Systems)

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Browse Figures

Figure 1
Flow chart of cluster partition algorithm. "> Figure 2
Simulated runtime of energy storage. "> Figure 3
Topology of the distribution network. "> Figure 4
Results of cluster partition: (a) Cluster partition results based on the comprehensive cluster partition index; (b) Cluster partition results based on the cluster modularity function partition index. "> Figure 5
Comparison of PV planning capacity. "> Figure 6
Comparison of ESS planning capacity. "> Figure 7
Comparison chart of reliability indicators: (a) Comparison of average number of power outages; (b) Comparison of average outage time. "> Figure 7 Cont.
Comparison chart of reliability indicators: (a) Comparison of average number of power outages; (b) Comparison of average outage time. "> Figure 8
The planning of PV capacity in different schemes. "> Figure 9
The planning of energy storage system capacity in different schemes. "> Figure 10
Planning results of different schemes: (a) Scenario 1 planning results; (b) Scenario 2 planning results; (c) Scenario 3 planning results. "> Figure 10 Cont.
Planning results of different schemes: (a) Scenario 1 planning results; (b) Scenario 2 planning results; (c) Scenario 3 planning results. "> Figure 11
Comparison chart of reliability index under different schemes: (a) Average number of power outages; (b) Average outage time. "> Figure 12
Comparison of network planning results under different schemes: (a) Centralized planning results; (b) Planning results based on cluster partition. "> Figure 12 Cont.
Comparison of network planning results under different schemes: (a) Centralized planning results; (b) Planning results based on cluster partition. "> Figure 13
Comparison of PV planning capacity under different schemes. "> Figure 14
Comparison of energy storage planning capacity under different schemes. "> Figure 15
Comparison of reliability indexes under different schemes: (a) Average number of power outages; (b) Average outage time. "> Figure 15 Cont.
Comparison of reliability indexes under different schemes: (a) Average number of power outages; (b) Average outage time. ">

Versions Notes

Abstract

To achieve the optimal planning of grid resource storage for a large-scale distribution network (DN), a cluster partition-based zonal planning method for the DN, considering reliability, is proposed. Firstly, a comprehensive clustering partition index is proposed, which includes the modularity index, power balance index, and node affiliation index. A hybrid genetic–simulated annealing algorithm is employed to perform the cluster partition. Secondly, a three-layer joint expansion planning model based on cluster partitioning is proposed. At the upper level, a route planning model is established to optimize the routing of the cluster. At the intermediate level, a location and capacity planning model for distributed photovoltaics and energy storage is formulated, taking uncertainties into account. By introducing uncertainty parameters, the range of uncertainty for sources and loads is characterized. At the lower level, reliability indices within the clusters are calculated to ensure operational reliability while reducing the conservatism of the optimization outcomes. Finally, the proposed method is applied to a real distribution network in China, demonstrating its effectiveness in improving the economic efficiency of DN planning.

Keywords:

zonal planning; energy storage; distributed photovoltaics; robust optimization; reliability index

1. Introduction

In recent years, photovoltaic (PV) generation systems have rapidly expanded across China. The volatility and intermittency of PV systems have significantly affected load balancing, voltage regulation, and frequency stability within distribution networks [1]. Optimizing DN planning has become crucial for achieving efficient and cost-effective operations, especially with the growing scale of distributed PV systems [2]. Cluster planning can decompose centralized optimization problems into simpler subproblems for individual clusters [3].

Traditionally, DN planning with distributed PV relied on centralized planning models [4], which involve reinforcing or extending DN infrastructure [5] while considering investments in PV facilities [6] and operational costs [7]. Centralized models perform effectively when the share of distributed PV in the network remains relatively low. However, as the penetration of large-scale PV increases, the number of variables grows exponentially [8], resulting in excessive complexity that hampers efficient problem solving [9]. To address the limitations of centralized coordination, cluster partitioning has emerged as an alternative planning method, significantly improving PV utilization [10].

Cluster partition-based DN planning consists of two primary components: cluster partitioning and cluster planning. Existing studies on cluster partitioning generally focus on developing partitioning indices [11] and maintaining power balance within each cluster [12]. The authors of [13] proposed a clustering method based on comprehensive performance indicators. Conventional cluster partitioning algorithms include clustering algorithms [14] and community detection methods [15]. The authors of [16] proposed a comprehensive partitioning index considering both network topology and power balance, employing genetic algorithms (GA) for cluster partitioning. However, most existing indices emphasize active power balance while neglecting reactive power, and current methods often lack computational precision. For complex indices, optimization processes frequently converge to local optima rather than global solutions, limiting the overall effectiveness of DN planning [17].

In the field of cluster planning, most existing studies are based on deterministic scenarios, ignoring the impact of source-load uncertainties on planning outcomes [18]. As the penetration of distributed PV increases, source-load uncertainty mainly impacts planning outcomes by complicating uncertainty modeling in distribution networks and increasing variable dimensions in planning models [19]. Further research is needed to develop cluster planning models under uncertainty to simplify traditional models and maximize local PV utilization [20].

Existing DN planning strategies typically involve reinforcing or expanding the network infrastructure [21], including distributed generation (DG) systems actively managed to optimize outputs [22]. For example, the authors of [23] proposed an extended planning method for electric vehicle parking facilities, while [24] introduced an integrated planning model for power and gas systems structured in two stages: “demand accessibility assessment” and “comprehensive planning”. The authors of [25] propose a distribution network planning method based on power reliability using an ant colony algorithm. The process is divided into two layers: the upper layer focuses on network planning, while the lower layer calculates reliability indices. The two layers iteratively solve the problem to ensure the accuracy of the planning results.

Additionally, the authors of [26] presented a three-phase planning approach for distribution networks. The first phase identifies DG locations, the second optimizes emergency and normal operations, and the third focuses on system resilience [27]. Another expansion model incorporates flexibility constraints to reduce coupling risks and enhance resilience. The authors of [28] propose a cooperative demand response management framework for integrated energy systems (IESs). It employs Nash bargaining theory and the alternating direction method of multipliers to optimize benefit allocation and protect privacy. The authors of [29] propose a two-stage optimization model for allocating air conditioner response capacity. The model is designed to provide peak-valley regulation services. The authors of [30] propose a collaborative planning approach for electric vehicle charging systems and distribution networks, balancing both economic efficiency and reliability. This approach is formulated as a two-stage distributed robust optimization model considering power load uncertainties.

Based on prior research, this paper proposes a three-layer joint expansion planning strategy for grid resource storage systems in distribution networks using cluster partitioning. The main contributions are as follows:

(1): This paper addresses the problems of imbalanced power distribution and uneven cluster scale in existing methods by introducing a comprehensive cluster partitioning index. The proposed index combines modularity, power balance, and node affiliation metrics. To implement the partitioning process, a hybrid genetic–simulated annealing algorithm is employed;
(2): A three-layer joint expansion planning model is proposed. The upper layer involves establishing a line planning model based on cluster partition to optimize the routing of cluster lines. The middle layer determines the location and capacity of resource-storage systems through robust planning. The lower layer focuses on calculating reliability indices within the cluster to ensure operational reliability while reducing the conservatism of the optimization results;
(3): This paper employs a “box uncertainty set” to represent the uncertainties in load and photovoltaic generation, which describes the possible fluctuation range of load and PV output. A parameter for regulating uncertainty is also introduced to oversee the conservativeness of the decision-making process.

2. Cluster Partition Index and Algorithms for Distribution Networks

2.1. Comprehensive Cluster Partition Index

2.1.1. Modularity Index

The modularity index is used to evaluate the coupling degree between nodes, expressed as follows:

ρ_{m} = \frac{1}{2 Ω} \sum_{i \in S} \sum_{j \in S} (v_{i j} - \frac{κ_{i} κ_{j}}{2 Ω}) ϕ (i, j)

(1)

Ω = \frac{\sum_{i \in S} \sum_{j \in S} v_{i j}}{2}

(2)

κ_{i} = \sum_{j \in S} v_{i j}

(3)

where ρ_m is the modularity index,

v_{i j}

is the link weight between node i and node j, S is the DN node set,

Ω

is the sum link weight of the entire network, and

κ_{i}

is the sum of the edge weights connected to node i. The function

ϕ (i, j)

is a cluster judgment function; if node i and node j belong to the same cluster,

ϕ (i, j)

is 1, otherwise,

ϕ (i, j)

is 0. The variable

v_{i j}

represents the electrical coupling intensity between two nodes. The electrical distance is formulated as follows:

L_{Q V}^{i j} = \sqrt{(S_{Q V}^{i 1} - S_{Q V}^{j 1})^{2} + \dots + (S_{Q V}^{i n} - S_{Q V}^{j n})^{2}}

(4)

where

S_{Q V}^{L_{i j}}

represents the relationship between the unit reactive power injected at node j and the corresponding voltage change at node I, and

L_{Q V}^{L_{i j}}

is the electrical distance between the two nodes based on the reactive power–voltage sensitivity matrix. Similarly, the electrical distance

L_{P V}^{L_{i j}}

based on the active power–voltage sensitivity matrix can be derived. Changes in active and reactive power influence node voltages, and the electrical distance can be expressed through the sensitivity matrix as follows:

L^{i j} = \frac{L_{Q V}^{i j} + L_{P V}^{i j}}{2}

(5)

v_{i j} = 1 - \frac{L_{i j}}{m a x (L_{i j})}

(6)

where max(L_ij) is the maximum value of the elements in the electrical distance matrix L_ij.

2.1.2. Cluster Power Balance Index

To assess the capability of a cluster to integrate distributed PV, this paper introduces a cluster power balance index. The active power balance index φ^P is derived from the net power of the cluster and is defined as follows:

φ_{P} = \frac{1}{N_{k}} \sum_{k = 1}^{N_{k}} (1 - \frac{1}{T} \sum_{t = 1}^{T} \frac{P_{k, t}}{\max P_{k, t}})

(7)

where φ_P is the cluster active power balance degree index, N_k is the number of clusters, T is the system simulation time period, and

P_{k, t}

is the net power of cluster k at time t. The purpose of the index φ_P is to evaluate the complementary levels of net power among nodes within the network, ensuring efficient power distribution and enhancing the autonomy of clusters. Similarly, the same approach can be applied to derive the reactive power balance index.

φ_{Q} = \frac{1}{N_{k}} \sum_{k = 1}^{N_{k}} (1 - \frac{1}{T} \sum_{t = 1}^{T} \frac{Q_{need}}{Q_{\sup}})

(8)

where φ_Q is the cluster reactive power balance index, Q_sup is the reactive power supply within the cluster, and Q_need is the reactive power demand within the cluster.

2.1.3. Node Affiliation Index

To maintain a balanced size for each cluster, a node affiliation index is introduced as detailed below:

φ_{M} = \frac{μ (i, V [i])}{μ (i, V - V [i])}

(9)

μ (i, V [i]) = \frac{1}{|V [i]|} \sum_{j \in V [i]} v_{i j}

(10)

μ (i, V - V [i]) = \frac{1}{|V - V [i]|} \sum_{j \in V - V [i]} v_{i j}

(11)

where

V [i]

is the cluster to which node i is assigned,

μ (i, V [i])

represents the affiliation degree of node i to cluster

V [i]

, V represents the set of all clusters in the distribution network,

|V [i]|

represents the sum of the edge number in the cluster; j is a node that is both connected to node i and located within the same cluster,

V - V [i]

represents the set of clusters excluding node I, and

|V - V [i]|

represents the total number of edges in the clusters, excluding

V [i]

. In this case, j represents a node connected to node i but belonging to a different cluster.

Equations (1)–(11) are integrated into the following index:

ϕ = ϖ_{1} ρ_{m} + ϖ_{2} φ_{p} + ϖ_{3} φ_{Q} + ϖ_{4} φ_{M}

(12)

The comprehensive cluster partition index for the distribution network is represented by

ϕ

, where

ϖ_{1} {, ϖ}_{2}, ϖ_{3}

, and

ϖ_{4}

denote the respective weights assigned to each individual index. The desired cluster partition results can be achieved during the partitioning process by adjusting the weights of each index according to the decision maker’s preferences or specific requirements.

2.2. Cluster Partition Method for Distribution Networks

Distribution network planning with distributed photovoltaic systems is a complex mixed-integer optimization problem. This paper uses a hybrid genetic–simulated annealing algorithm to solve this model. The clustering partitioning algorithm process is shown in Figure 1. The implementation steps are as follows:

(1)

Set the initial control parameters;

(2)

Set the initial temperature update counter l = 0. The original network is encoded using its adjacency matrix to generate the initial population

P_{1} (k)

, where k = 0;

(3)

Perform the following 4 steps on the current population until a new population is generated:

Step 1.: The reactive power demand and supply must be balanced, with the constraint $\frac{Q_{n e e d}}{Q_{s u p}} = 1$ . Any individual that fails to meet this constraint will be removed from the population;
Step 2.: The fitness of individuals is calculated using Equation (5) as the indicator. Since the fitness value intuitively represents the quality of a chromosome in the cluster partition scheme, individuals with higher fitness are selected for replication into the next generation, forming the population $P_{l s} (k + 1)$ ;
Step 3.: Following the traditional Genetic Algorithm, the population $P_{l s} (k + 1)$ undergoes crossover and mutation to produce a population $P_{l v} (k + 1)$ . For the mutated population, suboptimal solutions are accepted within limits according to the simulated annealing (SA) acceptance criterion, forming the new population $P_{l z} (k + 1)$ ;
Step 4.: Check whether the genetic generation N has reached the predefined value. If so, proceed to step 4); otherwise, return to Step 3).

(4)

Update the temperature as T_l = rT₀. If the convergence condition T_l < T_end is satisfied, the algorithm stops. Alternatively, the cooling process is applied by updating T_l₊₁ = rT_l, and the procedure returns to the first step;

(5)

Output the optimal solution.

Figure 1. Flow chart of cluster partition algorithm.

3. Cluster Partition-Based Three-Layer Planning Strategy for Distribution Networks

3.1. The Line Planning Layer

3.1.1. Objective Function

The cluster line planning layer, serving as the upper layer of the three-layer model, focuses on planning the expansion of distribution network lines based on the forecasted PV generation and load values. The objective function for this layer is defined as follows:

\min f = F_{1} + F_{2}

(13)

F_{1} = \sum_{i, j \in Ω_{s}} [C_{l} \sum_{j \in φ_{i}} D_{i j} x_{i j} \frac{χ {(1 + χ)}^{β}}{{(1 + χ)}^{β} - 1}]

(14)

F_{2} = C_{e} \sum_{t = 1}^{T} \sum_{k = 1}^{N_{k}} \sum_{i = 1}^{Ν_{n, k}} \sum_{j \in φ_{i}} \frac{1}{2} x_{i j} ε_{i j} i_{i j, t}

(15)

where F₁ represents the annual investment expense of the network, F₂ represents the cost of power transmission losses in the network,

Ω_{s}

is the subset of nodes in the subnetwork,

φ_{i}

represents the subset of nodes branching from node i, C_l represents the investment expense for each kilometer of the line, D_ij is the length of the line laid between nodes i and j, x_ij is the 0-1 variable for the line between nodes i and j, with 1 reflecting the installation of the line and 0 reflecting its absence,

χ

is the bank interest rate,

β

is the capital recovery time, C_e is the cost of electricity, N_n,k is the number of nodes contained in cluster k,

ε_{i j}

is the resistance value of the line between nodes i and j, and i_ij_,t is the squared value of the current of the line between nodes i and j at moment t.

3.1.2. Constraints

(1): Power flow constraints containing line 0–1 variables.

Since the joint planning of the grid and distributed resources must incorporate the operational constraints of the distribution network, second-order conic relaxation (SOCR) is employed to formulate constraints on power flow equations involving line variables. The power flow constraints are expressed as follows:

\sum_{i \in Ψ_{j}} (P_{i j, t} - ε_{i j} i_{i j, t}) - x_{i j} P_{L, j, t} + η P_{CH, j, t} - \frac{P_{DS, j, t}}{η} - P_{PV, j, t} + P_{PVf, j, t} = \sum_{c \in φ_{j}} P_{j c, t}

(16)

\sum_{i \in Ψ_{j}} (Q_{i j, t} - δ_{i j} i_{i j, t}) - x_{i j} Q_{L, j, t} + Q_{PV, j, t} + Q_{PVf, j, t} = \sum_{c \in φ_{j}} Q_{j c, t}

(17)

\sum_{j \in φ_{i}} x_{i j} = 1

(18)

v_{j, t} = v_{i, t}^{'} - 2 (ε_{i j} P_{i j, t} + δ_{i j} Q_{i j, t}) + (ε_{i j}^{2} + δ_{i j}^{2}) i_{i j, t}

(19)

0 \leq ν_{i, t}^{'} \leq W x_{i j}

(20)

- W (1 - x_{i j}) + ν_{i, t} \leq ν_{i, t}^{'} \leq ν_{i, t}

(21)

{‖2 P_{i j, t} 2 Q_{i j, t} i_{i j, t} - ν_{i, t}‖}_{2} \leq i_{i j, t} + ν_{i, t}

(22)

x_{i j} v_{\min} \leq ν_{i, t} \leq x_{i j} v_{\max}

(23)

0 \leq i_{i j, t} \leq x_{i j} I_{\max}^{2}

(24)

where ψ_j represents the subset of nodes upstream of node j,

δ_{i j}

represents the line reactance between nodes i and j, P_ij_,t and Q_ij_,t represent the active and reactive power flows in the line between nodes i and j at time t, respectively,

η

represents the storage efficiency during charge and discharge cycles, P_CH,j,t and P_DS,j,t represent the charging and discharging power of the storage system at node j at time t, P_L,j,t and Q_L,j,t represent the active and reactive power demands at node j at time t, respectively, P_PV,j,t represents the actual active power generation of the PV system at node j at time t, and P_PVf,j,t represents the additional active power required to meet demand at node j. Similarly, Q_PV,j,t represents the actual reactive power generation of the PV system at node j, Q_PVf,j,t represents the additional reactive power needed to meet demand, φ_j represents the subset of nodes downstream of node j, v_i_,t represents the square of the voltage magnitude at node i at time t,

v_{i, t}^{'}

represents the square of the voltage magnitude at node i after applying the constraint through the line variable x_ij, W represents a sufficiently large constant, v_max and v_min represent the voltage range limits (upper and lower), and I_max represents the maximum allowable current.

(2): Cluster penetration rate constraint

The definition is as follows:

\sum_{j = 1}^{Ν_{n, k}} S_{PV, j} \leq \sum_{j = 1}^{Ν_{n, k}} S_{L, j}

(25)

where S_PV,j represents the PV capacity available for access at node j, and S_L,j represents the apparent power of the load at node j.

3.2. Location and Capacity Determination Layer for Resource Storage Systems

3.2.1. Objective Function

The resource storage location and capacity determination layer are designed to identify “worst-case” scenarios and optimize resource allocation to minimize total economic costs under these extreme conditions. The objective function governing the intra-cluster siting and capacity allocation is formulated as follows:

\min F = F_{3} - F_{4} + F_{5}

(26)

F_{3} = \sum_{j = 1, j \in Π_{k}}^{N_{n, k}} [C_{PV} S_{PVf, j} \frac{χ {(1 + χ)}^{β}}{{(1 + χ)}^{β} - 1} + C_{OMPV} (S_{PV, j} + S_{PVf, j})]

(27)

F_{4} = (C_{sell} + C_{sub}) \sum_{t = 1}^{T} \sum_{j = 1, j \in Π_{k}}^{N_{n, k}} [ω_{PV, t} (S_{PV, j} + S_{PVf, j}) - P_{L, j, t}]

(28)

F_{5} = \sum_{j = 1, j \in Π_{k}}^{Ν_{n, k}} [\frac{ρ {(1 + ρ)}^{r}}{{(1 + ρ)}^{r} - 1} c_{bat} P_{batt, j}]

(29)

where F₃ is the annual investment and operational costs of PV in cluster k, F₄ is the annual revenue of PV in cluster k, F₅ is the annual investment cost of energy storage systems in cluster k, F₆ is the customer outage loss cost in cluster k, C_PV is the investment cost per megawatt of PV capacity, C_OMPV is the annual fixed maintenance cost per megawatt of PV, Π_k is the set of nodes within cluster k, S_PVf,j is the increased PV capacity of node j in cluster k, C_sell and C_sub are the feed-in tariff and subsidy tariff for distributed generation, respectively,

ω_{PV, t}

is the output limit of PV per megawatt under given sunlight conditions at time t,

ρ

is the discount rate, r is the number of discounted years, c_bat is the investment cost of energy storage unit capacity, and P_batt,j is the storage capacity allocated at node j.

3.2.2. Constraints

The primary purpose of energy storage planning within the cluster is to maximize the utilization of PV energy, which is the energy storage system discharges. This improves the complementarity between generation and load within the cluster.

T_{k, E s s} = {\bar{t}}_{1}

(30)

where T_k_,Ess is the simulated operating hours of the energy storage installed in cluster k, and

\bar{t_{1}}

is the annual PV power generation time in China.

After determining the simulated operating hours of the energy storage system, the storage capacity to be allocated to each cluster is calculated based on these operating hours, as shown in Figure 2.

The constraints are shown below:

\sum_{j \in Π_{k}} η P_{CH, j, t} - \sum_{j \in Π_{k}} \frac{P_{DS, j, t}}{η} - \sum_{j \in Π_{k}} P_{PV, j, t} - \sum_{j \in Π_{k}} P_{PVf, j, t} + \sum_{j \in Π_{k}} P_{L, j, t} = 0, t \in T_{k, E S S}

(31)

s o c_{\min} P_{batt, j} \leq s o c_{j, t} \leq s o c_{\max} P_{batt, j}, j \in Π_{k}, t \in T_{k, E S S}

(32)

s o c_{j, t} = s o c_{j, t - 1} + \frac{η P_{CH, j, t} - \frac{P_{DS, j, t}}{η}}{P_{batt, j}}, j \in Π_{k}, t \in T_{k, E S S}

(33)

0 \leq η P_{CH, j, t} \leq P_{CH, \max}, j \in Π_{k}, t \in T_{k, E S S}

(34)

0 \leq \frac{P_{DS, j, t}}{η} \leq P_{DS, \max}, j \in Π_{k}, t \in T_{k, E S S}

(35)

P_{PVf, j} = \max (P_{PVf, j, t}), t \in T_{k, E S S}

(36)

S_{PVf, j} = P_{PVf, j}, j \in Π_{k}

(37)

where soc_j_,t is the charge state of the energy storage at node j at time t within cluster k, soc_min and soc_max are the minimum and maximum charging states of the energy storage, respectively, and _P_CH,max and _P_DS,max are the maximum power limits of the energy storage charging and discharging, respectively.

(1): PV capacity constraints for each node within a cluster:

$0 \leq P_{PVf, j} \leq P_{L, j}^{\max}$

(38)

where $P_{L, j}^{\max}$ is the maximum value of load demand at node j in the cluster k.
(2): Balance constraints for power distribution and voltage-current constraints are given in (16), (17), and (19).

This paper employs a box-type uncertainty set. A randomness parameter is introduced, which regulates the conservative nature of the plan by adjusting its value. A higher value of this parameter leads to a more conservative solution, whereas a lower value results in a more aggressive scheme:

\{\begin{cases} U = {u = {[P_{L, j, t} Q_{L, j, t} P_{PV, j, t}]}^{T}| \\ P_{L, j, t} \in [{\bar{P}}_{L, j, t} - Δ P_{L, j, t}, {\bar{P}}_{L, j, t} + Δ P_{L, j, t}], \sum_{j = 1}^{N_{a}} \frac{|P_{L, j, t} - {\bar{P}}_{L, j, t}|}{Δ P_{L, j, t}} \leq Γ_{PL}; \\ Q_{L, j, t} \in [{\bar{Q}}_{L, j, t} - Δ Q_{L, j, t}, {\bar{Q}}_{L, j, t} + Δ Q_{L, j, t}], \sum_{j = 1}^{N_{b}} \frac{|Q_{L, j, t} - {\bar{Q}}_{L, j, t}|}{Δ Q_{L, j, t}} \leq Γ_{QL}; \\ P_{PV, j, t} \in [{\bar{P}}_{PV, j, t} - Δ P_{PV, j, t}, {\bar{P}}_{PV, j, t} + Δ P_{PV, j, t}], \sum_{e = 1}^{N_{e}} \frac{|P_{PV, j, t} - {\bar{P}}_{PV, j, t}|}{Δ P_{PV, j, t}} \leq Γ_{PV};} \end{cases}

(39)

where U is the box uncertainty set, P_L,j,t, Q_L,j,t and P_PV,j,t are the uncertain variables of active load power, reactive load power and photovoltaic output at node j within cluster k, respectively,

{\bar{P}}_{L, j, t}

{\bar{Q}}_{L, j, t}

, and

{\bar{P}}_{PV, j, t}

are the predicted values of active load power, reactive load power and PV output of node j within cluster k, respectively,

∆ P_{L, j, t}

∆ Q_{L, j, t}

, and

{∆ P}_{PV, j, t}

are the fluctuation deviations of active load power, reactive load power and photovoltaic output of node j within cluster k, N_a, N_b, and N_e are the total number of active load nodes, reactive load nodes and PV nodes, respectively, and

Γ_{PL}

Γ_{QL}

, and

Γ_{PV}

are uncertainty adjustment parameters introduced for active load power, reactive load power, and photovoltaic output, respectively. These parameters take integer values ranging from 0 to N_a, 0 to N_b, and 0 to N_e. They are the uncertain regulation parameters introduced for active load power, reactive load power, and photovoltaic output.

3.3. Reliability Calculation Layer

3.3.1. Objective Function

E C O S T = \sum_{j = 1, j \in Π_{k}}^{N_{n, k}} \bar{P_{j}} λ_{j} r_{j} + H_{j}

(40)

\begin{array}{l} \bar{P_{j}} = \sum_{t = 1}^{T} \sum_{j = 1, j \in Π_{k}}^{N_{n, k}} \frac{P_{L, j, t} + η P_{CH, j, t} - \frac{P_{DS, j, t}}{η} + P_{PV, j, t} + P_{PVf, j, t}}{N_{n, l}} \end{array}

(41)

H_{j} = \sum_{j = 1, j \in Π_{k}}^{N_{n, k}} P_{j} (1 - \partial_{k}) t_{j}^{o f f}

(42)

where ECOST is the customer outage loss cost within cluster k,

\bar{P_{j}}

represents the average load value in node j, N_n_,l represents the total number of connected branches within cluster k, H_j is the loss cost caused by an outage of average node load j,

t_{j}^{o f f}

is the maximum interruption duration for the interruptible average load in node j, and P_j represents the maximum interruptible load in node j.

3.3.2. Constraints

Considering the unique challenges in calculating reliability indices for a clustered distribution network model under islanded operation and network structure uncertainty, a method based on the minimum path algorithm to assess cluster planning reliability indices is proposed. First, the equivalence method is applied to model transformers and branch lines in the network as equivalent components. In the case of cluster islanding, if a cluster’s subnetwork is connected to the main network through only one branch node, the system will send a command to disconnect the tie switch between the cluster and the main network when an external component failure occurs. This action isolates the cluster, enabling it to enter islanded operation. The tie switch could be either an isolator or a circuit breaker.

Case 1: The cluster tie switch is a circuit breaker

Minimum Road:

λ_{j} = (1 - \partial_{k}) e_{k} + \sum_{j = 1}^{N_{n, k}} e_{j} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p}

(43)

r_{j} = \frac{(1 - \partial_{k}) e_{k} r_{k}^{g} + \sum_{j = 1}^{N_{n, k}} e_{j} r_{j}^{g} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p} r_{p}^{g}}{(1 - \partial_{k}) e_{k} + \sum_{j = 1}^{N_{n, k}} e_{j} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p}}

(44)

Case 2: The cluster tie switch is an isolating switch

Minimum Road:

λ_{j} = e_{k^{1}} + \sum_{j = 1}^{N_{n, k}} e_{j} + \sum_{p = 1}^{N_{p, k}} e_{p}

(45)

r_{j} = \frac{r_{b r e a k} + (1 - \partial_{k}) e_{k^{1}} r_{k}^{g} + \sum_{j = 1}^{N_{n, k}} e_{j} r_{j}^{g} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p} r_{p}^{g}}{(1 - \partial_{k}) e_{k^{1}} + \sum_{j = 1}^{N_{n, k}} e_{j} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p}}

(46)

where λ_j is the failure rate (average number of outages) of the node j, r_j is the average outage time of the failure of the node j, α_k is the probability of cluster islanding, which physically represents the probability that the PV and energy storage within cluster k can meet load demand when the cluster enters islanded operation, e_k is the failure rate of the circuit breaker, e_j is the malfunction rate of element j, e_p is the malfunction rate of the PV, N_n,k is the count of nodes in cluster k, N_p,k is the total of PV nodes within cluster k,

r_{i}^{g}

is the failure recovery time of element I,

r_{p}^{g}

is the failure recovery time of PV nodes, r_break is the operating time of the isolator, e_k1 is the failure rate of the isolator, and μ_j represents the average outage length of the failure of the node j.

Set G as the variable to determine whether there is an isolation switch or not; if the branch node is an isolation, the switch is 1. Otherwise, it is 0. Then, the above equation can be written as follows:

Minimum Road:

λ_{j} = (1 - G) [(1 - \partial_{k}) e_{k} + \sum_{j = 1}^{N_{n, k}} e_{j} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p}] + G (e_{k^{1}} + \sum_{j = 1}^{N_{n, k}} e_{j} + \sum_{p = 1}^{N_{p, k}} e_{p})

(47)

\begin{array}{l} r_{j} = (1 - G) [\frac{(1 - \partial_{k}) e_{k} r_{k}^{g} + \sum_{j = 1}^{N_{n, k}} e_{j} r_{j}^{g} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p} r_{p}^{g}}{(1 - \partial_{k}) e_{k} + \sum_{j = 1}^{N_{n, k}} e_{j} + \partial_{k} \sum_{p = 1}^{N_{p, k}} e_{p}}] + \\ G [\frac{r_{b r e a k} + \partial_{k} (\sum_{p = 1}^{N_{p, k}} \sum_{i = 1, i \notin j}^{N_{n, k}} r_{j}^{g} λ_{i} r_{p}^{g} λ_{p} (e_{i} + e_{p}) (r_{j}^{g} + r_{p}^{g}))}{\partial_{k} (\sum_{p = 1}^{N_{p, k}} \sum_{i = 1, i \notin j}^{N_{n, k}} λ_{i} λ_{p} (e_{i} + e_{p}))}] \end{array}

(48)

In the event of a fault within the cluster, if the conditions for islanded operation are not met, it becomes necessary to account for the maximum allowable interruption duration and the maximum interruptible load capacity for interruptible loads.

(1): Maximum interruption time for interruptible average load at node j.

$t_{j}^{o f f} \leq t_{j}^{\max}$

(49)

where $t_{j}^{m a x}$ is the maximum interruption time limit for the interruptible average load at node j.
(2): Maximum interruptions for interruptible average load at node j.

$0 \leq P_{j} \leq P_{j}^{\max}$

(50)

where $P_{j}^{m a x}$ is the maximum interruption limit for the interruptible average load at node j.
(3): Power balance constraints and voltage-current constraints are given in (16), (17), and (19), and PV and load uncertainties are given in (39).

3.4. Cluster Partition-Based Three-Layer Planning Model for Distribution Networks

The three-layer planning model constructed in the previous section belongs to a non-deterministic polynomial-hard (NP-hard) problem, and the model is formulated in its compact form as follows:

The upper-layer objective function is as follows:

\min_{x} c^{T} x

(51)

The mid-layer objective function is as follows:

\max_{u \in U} \min_{y \in Ω (x, u)} d^{T} y

(52)

The lower-layer objective function is as follows:

\max_{u \in U} \min_{y_{2} \in Ω (x, u, y)} d^{T} y_{2}

(53)

The constraints are as follows:

P x = W

(54)

Ω (x, u) = \{\begin{cases} B x + C y \leq D \\ R_{x} u = O y \\ S_{u} y \leq V \\ ‖G y‖ \leq H^{T} y \end{cases}

(55)

Ω (x, u, y) = \{\begin{cases} B x + C y + C_{2} y_{2} \leq D \\ R_{x} u + O y = O_{2} y_{2} \end{cases}

(56)

\{\begin{cases} x = {[x_{i j}]}^{T} \\ y = {[v, v^{'}, P_{s}, Q_{s}, I, P_{PV}, P_{PVf}, P_{CH}, P_{DS}, P_{batt}]}^{T} \\ y_{2} = {[\bar{P_{j}}, H_{j}]}^{T} \end{cases}

(57)

where x, y, and u are the optimization vectors. The optimization vector of the upper layer is x. The optimization vectors of the lower layer are y and u, which describe the real and reactive power outputs of renewable energy under various scenarios, as well as the power flow variables and the uncertain variables related to PV output, active and reactive load. c and d are the matrices of coefficients pertaining to the objective function, respectively, P, B, C, R_x, S_u, G, H, and C₂ are the coefficient structures linked to the respective constraints, D, O, V, W, and O₂ are steady vertical vectors,

Ω (x, u)

is the feasible region of the lower-layer variables y given a set of x and u, and

Ω (x, u, y)

) is the permissible solution space given a set of x, u, and y.

4. Example Analysis

4.1. Actual Distribution Network

To evaluate the practicality of the proposed theory, a real-world distribution network is utilized as a case study for simulation-based validation. The network has a load capacity of 22.8 MVA and a PV power access capacity of 7.5 MW. The geographical layout of the network is depicted in Figure 3. Each PV installation has a capacity of 500 kW, located at the following nodes: 11, 15, 18, 24, 32, 35, 38, 44, 48, 51, 54, 64, 67, 71, 75, 77, 80, 81, 82, and 83. Detailed information regarding the distance relationships between the network lines is provided in Table 1. The total planned line length for the network is 329.82 km.

4.2. Cluster Partition and Analysis of Planning Results

In order to illustrate the superiority of the proposed comprehensive cluster partition index, the proposed comprehensive cluster partition index in this paper is compared with the cluster modularity function partition index. Both cluster partition algorithms use the genetic–simulated annealing algorithm. To achieve a balance among multiple objectives and reduce the impact of a single indicator on the overall partitioning performance. The weight of the proposed comprehensive index is taken as

ϖ_{1}

ϖ_{2}

ϖ_{3}

ϖ_{4}

= 0.25. The clustering results based on the proposed comprehensive index are shown in Figure 4a, and the clustering results using the modularity function classification index are displayed in Figure 4b.

Figure 4a,b indicate that the cluster partitioning based on the modular function partition index results in isolated nodes forming separate clusters. In contrast, the cluster partition results based on the comprehensive partition index proposed in this paper show a more balanced distribution of nodes across clusters. This indicates that the cluster sizes and node distribution under the comprehensive partition index are more balanced and better suited for subsequent cluster control and operation.

Figure 5 and Figure 6 illustrate the PV planning capacity and energy storage capacity under dynamic and static cluster partitions, respectively. Figure 5 demonstrates that the PV planning capacity under the dynamic cluster partition method proposed in this chapter is higher. In contrast, static cluster partition under complex network conditions fails to meet the requirements for reasonable distribution and management of distributed generation. As shown in Figure 6, the energy storage capacity under the proposed method is lower, demonstrating that this approach better aligns with the needs of the distribution network by optimally allocating photovoltaic capacity and energy storage to achieve a balanced supply–demand state. As a result, the required allocation of energy storage systems is significantly reduced. Based on Figure 5 and Figure 6, it can be concluded that the method proposed in this chapter allows for greater PV integration with less energy storage, ensuring the optimal utilization of distributed energy resources.

Based on the average number of outages and average outage time under the dynamic partition of clusters and under the static partition of clusters, as shown in Figure 7, which is derived from Figure 7a,b, among the 75 load points, they are all load points under the method of this chapter; the average outage time and the average number of outages have all declined. Specifically, nodes 78, 79, 81, and 84 were more effectively partitioned into clusters due to changes in access points, resulting in longer power supply durations and a substantial rise in reliability indices, with the average outage time decreasing by 78%, 62%, 72%, and 52%, and the average number of outages decreasing by 70%, 59%, 65%, and 55%, respectively. The above analysis demonstrates that the planning methodology proposed in this chapter has resulted in an effective improvement in power supply reliability.

Table 2 provides a detailed comparison of the performance of two cluster partition methods—the dynamic and static cluster partition methods—across several key indicators. Specifically, the comparison covers PV investment and operation and maintenance (O and M) costs (A, unit: CNY 10,000), photovoltaic annual income (B, unit: CNY 10,000), the network annual investment costs (C, unit: CNY 10,000), network losses (D, unit: CNY 10,000), the annual investment costs of energy storage (E, unit: CNY 10,000), and the user outage loss costs (F, unit: CNY 10,000) and the total cost (G, unit: CNY 10,000).

In the planning process, the upper-level planning first completes the line planning and passes the results as initial conditions to the middle-level planning. After receiving the information from the upper-level, the middle-level planning makes decisions on the location and capacity of distributed PV and energy storage systems. At the same time, the middle-level planning feeds back the parameters of PV, energy storage, and load to the upper-level planning to satisfy power flow constraints and cluster penetration rate constraints. On the other hand, the parameters of PV, energy storage, and load are passed to the lower-level planning. After determining the distributed PV and energy storage, and line routing, the lower-level calculates the reliability indicators based on the given parameters and returns the results to the middle-level. The upper, middle, and lower levels iterate repeatedly until the optimal result is obtained.

In the case of distribution networks with many planned lines, load nodes, and substations, if the source-grid-load-storage planning results are output directly, it may lead to issues such as cluster node modularity, power imbalance between nodes, and reduced node membership. Therefore, the static clustering method cannot meet the clustering requirements in such situations. Hence, this paper will perform dynamic clustering on the model after the initial planning is completed. If the result is consistent with the first clustering result, it will be output; otherwise, the model will be re-solved with the initial clustering results.

In addition, in order to analyze more intuitively, this chapter defines the cluster static partition method as Scheme 1, while the cluster dynamic partition method proposed in this chapter is defined as Scheme 2.

As shown in Table 2, the scheme based on the dynamic cluster partition algorithm proposed in this chapter results in a 3.5% increase in annual PV investment and operational costs, an 11.8% increase in annual PV revenue, a 4.7% reduction in line planning costs, and a 30.1% reduction in network loss expenses. Additionally, annual investment costs for energy storage systems are reduced by 3.1%, while customer outage loss expenses decrease by 7.4%, resulting in a 10.1% overall reduction in total planning costs. The dynamic cluster partition method proposed in this chapter provides flexibility in adjusting cluster boundaries, making it more advantageous for distribution network planning. In contrast, the static cluster partition method struggles to adapt to the complexities of contemporary distribution networks. This demonstrates that the cluster partition algorithm proposed in this chapter effectively enhances the distribution network’s capacity for distributed PV integration, improves network reliability, reduces planning costs, and better adapts to the complexities.

4.3. Analysis of the Optimisation Results of the Planning Scheme

Three sets of uncertainty control parameters were selected for comparison based on the network connection scheme: energy storage access capacity, PV access capacity, and total planning cost. The uncertainty regulation parameters for active and reactive load power in the three models,

Γ^{PL}

Γ^{QL}

, and the PV output power uncertainty regulation parameter

Γ^{PV}

are selected as follows:

Option 1:

Γ^{PL} = Γ^{QL} = 0

Γ^{PV} = 0

, where all uncertain parameters are taken as predicted values;

Option 2:

Γ^{PL} = Γ^{QL} = 30

Γ^{PV} = 10

, where 30 nodes of active and reactive load power are set to the lower limit of the forecasted range, and the output of 10 PV nodes is set to the maximum value within the forecast range;

Option 3:

Γ^{PL} = Γ^{QL} = 75

Γ^{PV} = 20

, representing the worst-case scenario under all robust planning schemes, where 75 nodes of active and reactive load power are set to the minimum of the prediction interval and 20 nodes of PV output are set to the maximum.

Figure 8 and Figure 9 show the PV planning capacity and energy storage access capacity under the comparison of 3 groups of scenarios.

According to Figure 8, the PV planning capacity gradually decreases as the parameter for adjusting uncertainty increases. This is mainly due to increasingly severe conditions. To ensure power supply reliability and normal operation of the cluster, it is necessary to reduce PV access and keep the PV penetration of the cluster at a low level. Further observation of Figure 9 reveals that as the planning scenarios become more conservative, the energy storage capacity gradually increases. This is due to the fact that compared to Scenario 1, Scenarios 2 and 3 have smaller active power loads and reactive power loads, as well as larger PV outputs. Thus, more storage systems are needed to balance the uncertainty of the PV outputs.

Figure 10a–c shows the planning results. By comparing Figure 10a–c, It is evident that the connection results of nodes 78, 80, 81, 83, and 84 under different scenarios change as the uncertainty regulation parameter increases. The results of the planning typically expand the power supply range to ensure the grid operates reliably and stability.

The results of the average number of outages under the 3 groups of scenarios are shown in Figure 11a,b. From Figure 11a,b, it is evident that as the uncertainty adjustment parameter increases, the average number of outages and the average outage duration of the nodes under different scenarios rise. As uncertainty increases, the obtained scenarios become increasingly conservative, leading to a decrease in the reliability indexes of the nodes while the average outage duration and average number of outages also rise. When the degree of uncertainty increases, it is necessary to increase the degree of conservatism of the scheme to guarantee safe and economical operation.

The specific costs are presented in Table 3, where A represents the annual PV investment and O&M costs (CNY 10,000), B represents the annual PV revenue (CNY 10,000), C represents the line planning costs (CNY 10,000), D represents the annual network loss costs of the clusters (CNY 10,000), E represents the annual energy storage investment costs (10,000 ¥), F represents the user outage loss costs (CNY 10,000), and G represents the total cost (10,000 ¥).

As Table 3 shows, as uncertainty regulation increases, annual PV revenue decreases despite lower PV investment and O and M costs. Meanwhile, network investment, network losses, energy storage costs, and customer outage losses rise. This occurs because accounting for uncertainties in planning leads to more conservative scenarios. While this reduces risks, it increases outage losses, driving up total costs.

4.4. Computational Performance Analysis of the Planning Scheme

The cluster partition-based distribution network planning method proposed in this chapter is compared with the centralized planning method that does not partition the clusters and directly performs the overall calculations and

Γ^{PL} = Γ^{QL} = 30

Γ^{PV} = 10

. Table 4 and Table 5 show the comparison of the computation of the overall cost and time of planning under the two methods, and the network connection scheme, PV access capacity, storage access capacity, and user reliability indexes are shown in Figure 12, Figure 13, Figure 14 and Figure 15.

Figure 12 shows the connection results of nodes 79, 84, and 85 under the proposed planning scheme differ from those under the centralized planning scheme. The power supply line lengths for nodes 79 and 84 are reduced by 2.94 km and 4.00 km, respectively, under the proposed planning scheme compared to the centralized planning scheme, which shortens the power supply range, enables more efficient power transmission over shorter distances, enhances power supply reliability, and effectively reduces the economic costs.

Figure 13 and Figure 14 show that the cluster-based planning scheme allows for higher PV capacity and lower storage access compared to the centralized scheme, promoting greater PV integration and distributed power consumption.

Figure 15a,b shows that, compared to the centralized planning method, the reliability indexes in the proposed planning method have improved, primarily because the method first divides the clusters, which ensures reasonable power transmission between nodes and increases the probability of islanding operation. This allows the nodes to maintain power supply for a longer duration and reduces the likelihood of outages, thereby enhancing the reliability of the planning scheme. Additionally, the energy supplier and the small-micro industrial park profits have increased, with the most notable increase observed in social surplus profits.

From Table 4, it can be concluded that the cluster partition-based distribution network planning scheme is 23.21 s faster than the centralized planning scheme, thus indicating that the cluster partition-based planning scheme is faster and superior in terms of computational speed.

Scenario A represents the centralized planning scheme, and Scenario B represents the planning scheme based on cluster partition. According to the data presented in Table 5, it can be found that compared with the centralized planning scheme, the cluster-based planning scheme improves the annual PV investment and O and M costs and annual PV revenue by 14.1% and 16.3%, respectively. At the same time, the scheme also shows significant advantages in terms of annual network investment cost, network loss, annual investment cost of energy storage, and user outage loss cost, which are reduced by 3.3%, 35.9%, 17.7%, and 16.8%, respectively. This result shows that the planning scheme based on cluster partition not only optimizes the economics of distribution network planning and reduces the planning cost but also effectively reduces the losses caused by power outages and improves the reliability of distribution network operation.

The separate analyses of economic efficiency and reliability reveal a significant correlation between the two in dynamic partitioning and cluster planning schemes. Dynamic partitioning significantly reduces the frequency and duration of node outages. For instance, the average outage time and frequency of nodes 78, 79, 81, and 84 decreased by approximately 70% and 65%, respectively. At the same time, by optimizing resource allocation and reducing network losses, the dynamic partitioning scheme lowers total planning costs, reducing them by about 10.1% compared to static partitioning. Additionally, user outage loss costs decrease by 7.4%. This indicates that improved power supply reliability not only optimizes power supply capability but also indirectly reduces economic losses caused by outages. From the data in Table 3 and Table 5, it can be observed that as the conservatism of the scheme increases, the costs of the scheme also rise. This suggests that when higher reliability is required, the system needs to allocate additional energy storage and other resources to ensure power supply stability, and this increase in costs impacts the system’s economic efficiency. Nonetheless, the dynamic partitioning method achieves a balance between the proportion of photovoltaic integration and system economic efficiency by optimizing the matching relationship between photovoltaic generation and load, thereby reducing line investment costs and network losses. The comparison between cluster planning and centralized planning further demonstrates that cluster planning improves power supply reliability by optimizing power resource allocation and enhancing synergy between nodes. It also effectively reduces economic losses caused by outages while lowering system investment and operating costs. This dual optimization highlights the high coupling and mutual reinforcement between economic efficiency and reliability in cluster planning schemes.

5. Conclusions

In addressing the planning issues of large-scale distributed renewable energy integration, it is vital to reasonably plan the capacity and location of the distributed renewable energy sources that connect to the distribution network, as well as the routing of the lines within the distribution network. This paper studies this issue from the perspectives of joint planning that considers source-load uncertainty, dynamic clustering, and reliability indicators, with the main research findings as follows:

To address the current shortcomings in cluster partitioning regarding the lack of reactive power indicators and cluster size metrics, this paper proposes a comprehensive clustering index that fully considers electrical distance-based modularity indicators, active power balance relationships, reactive power supply-demand relationships, and cluster size. To prevent the clustering results from falling into local optima, an improved genetic algorithm is used for cluster partitioning. The clusters partitioned based on these four indicators and the improved algorithm are more reasonable and can effectively increase the proportion of integrated distributed renewable energy;
A three-tier planning model that considers reliability indicators is constructed. This model couples the distribution network planning process with the calculation of reliability indicators. The lower-level operational model that considers reliability indicators ensures that the planning of the distribution network maintains its economic viability while guaranteeing reliability. Given the situation where there are many planned lines and numerous nodes in the distribution network, a concept of dynamic clustering is proposed to ensure compact clustering and maximize power transmission within the clusters.

Author Contributions

Z.S.: Responsible for resources, formal analysis, program compilation and writing original draft. G.Y.: Responsible for writing review and editing. L.M.: Responsible for methodology and project administration. N.S.: Responsible for obtaining the experimental data. L.D.: Responsible for investigation and resources. Q.Y.: Responsible for funding acquisition and resources. C.X.: Responsible for resources and formal analysis. K.Z.: Responsible for resources and formal analysis. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Innovation Program of China Southern Power Grid Co., Ltd., “Collaborative Optimization and Control Technology of New Distribution System with High Proportion of Distributed Wind Power and Distributed Photovoltaic” Topic 4: Development and Demonstration of a New Intelligent Dispatching Platform for Distribution Networks with High-proportion Distributed Photovoltaic Clusters (YNKJXM20222435).

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

Authors Zhiwei Shi, Ning Sun, and Qianqian Yu are employed by the Chuxiong Power Supply Bureau of Yunnan Power Grid Corporation. Author Guangzeng You is employed by the Planning and Construction Research·Center of Yunnan Power Grid Corporation. Authors Linfu Miu and Lei Duan are employed by the Chuxiong Shuangbai Power Supply Bureau of Yunnan Power Grid Corporation. Authors Chuanliang Xiao and Ke Zhao are employed by the Shandong University of Technology. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Abbreviations

DN	Distribution network	GA	Genetic Algorithms
PVs	Distributed photovoltaics	DG	Distributed Generation
PV	Photovoltaic	IES	Integrated energy systems
SA	Simulated Annealing	SOCR	Second-order conic relaxation
NP-hard	Non-deterministic polynomial-hard
Formulas
ρ_m	Modularity index	$v_{i j}$	Link weight between node i and node j
S	DN node set.	$Ω$	Sum link weight of the entire network
$κ_{i}$	Sum of the edge weights connected to node i	$S_{Q V}^{L i j}$	Relationship between the unit reactive power injected at node j and the corresponding voltage change at node i
$L_{Q V}^{L i j}$	Electrical distance between based on the reactive power–voltage sensitivity matrix	φ_P	Cluster active power balance degree index
N_k	Number of clusters	T	System simulation time period
$P_{k, t}$	Net power of cluster k at time t	φ_Q	Cluster reactive power balance index
Q_sup	Reactive power supply within the cluster	Q_need	Reactive power demand within the cluster
$V [i]$	Cluster to which node i is assigned	$μ (i, V [i])$	Affiliation degree of node i to cluster $V [i]$
V	Set of all clusters in the distribution network	$\|V [i]\|$	Sum of the edge number in the cluster
$V - V [i]$	Set of clusters excluding node i	$\|V - V [i]\|$	Total number of edges in the clusters excluding $V [i]$
$φ_{M}$	Node affiliation index	$ϖ_{1}$	Weights assigned to modularity index
$ϖ_{2}$	Weights assigned to cluster active power balance degree index	$ϖ_{3}$	Weights assigned to cluster reactive power balance index
$ϖ_{4}$	Weights assigned to node affiliation index	$ϕ$	Comprehensive cluster partition index
F₁	Annual investment expense	F₂	Cost of power transmission losses
$Ω_{s}$	Subset of nodes in the subnetwork	$φ_{i}$	Subset of nodes branching from node i
C_l	Investment expense for each kilometer of the line	D_ij	Length of the line laid between nodes i and j
x_ij	0-1 variable	$χ$	Bank interest rate
$β$	Capital recovery time	C_e	Cost of electricity
N_n,k	Number of nodes contained in cluster k	$ε_{i j}$	Resistance value of the line
i_ij_,t	Squared value of the current	ψ_j	Subset of nodes upstream of node j
$δ_{i j}$	Line reactance between nodes i and j	P_ij_,t	Active power flows between nodes i and j at time t
Q_ij_,t	Reactive power flows between nodes i and j at time t	$η$	Storage efficiency
P_CH,j,t	Charging power of the storage system at node j at time t	P_DS,j,t	Discharging power of the storage system at node j at time t
P_L,j,t	Active power demands at node j at time t	Q_L,j,t	Reactive power demands at node j at time t
P_PV,j,t	Actual active power generation of the PV system at node j at time t	P_PVf,j,t	Additional active power required to meet demand at node j
Q_PV,j,t	Actual reactive power generation of the PV system at node j at time t	Q_PVf,j,t	Additional reactive power required to meet demand at node j
φ_j	Subset of nodes downstream of node j	v_i_,t	Square of the voltage magnitude at node i at time t
$v_{i, t}^{'}$	Square of the voltage magnitude at node i after applying the constraint through the line variable x_ij	W	Sufficiently large constant
v_max	Maximum voltage.	v_min	Minimum voltage
I_max	Maximum allowable current	S_PV,j	PV capacity available for access at node j
S_L,j	Apparent power of the load at node j	F₃	Annual investment and operational costs of PV in cluster k
F₄	Annual revenue of PV in cluster k	F₅	Annual investment cost of energy storage systems in cluster k
F₆	Customer outage loss cost in cluster k	C_PV	Investment cost per megawatt of PV capacity
C_OMPV	Annual fixed maintenance cost per megawatt of PV	Π_k	Set of nodes within cluster k
S_PVf,j	Increased PV capacity of node j in cluster k	C_sell	Feed-in tariff for distributed generation
C_sub	Subsidy tariff for distributed generation	$ω_{PV, t}$	Output limit of PV per megawatt under given sunlight conditions at time t
$ρ$	Discount rate	r	Number of discounted years
c_bat	Investment cost of energy storage unit capacity	P_batt,j	Storage capacity allocated at node j
T_k_,Ess	Simulated operating hours of the energy storage installed in cluster k	$\bar{t_{1}}$	Annual PV power generation time in China
soc_j_,t	Charge state of the energy storage at node j at time t	soc_min	Minimum charging states of the energy storage
soc_max	Maximum charging states of the energy storage	_P_CH,max	Maximum power limits of the energy storage charging
_P_DS,max	Maximum power limits of the energy storage discharging	$P_{L, j}^{\max}$	Maximum value of load demand at node j
U	Box uncertainty set	P_L,j,t	Uncertain variables of active load power at node j
Q_L,j,t	Uncertain variables of reactive load power at node j	P_PV,j,t	Uncertain variables of photovoltaic output at node j
${\bar{P}}_{L, j, t}$	Predicted values of active load power at node j	${\bar{Q}}_{L, j, t}$	Predicted values of reactive load power at node j
${\bar{P}}_{PV, j, t}$	Predicted values of PV output at node j	$∆ P_{L, j, t}$	Fluctuation deviations of active load power of node j
$∆ Q_{L, j, t}$	Fluctuation deviations of reactive load power of node j	${∆ P}_{PV, j, t}$	Fluctuation deviations of photovoltaic output of node j
N_a	Total number of active load nodes	N_b	Total number of reactive load nodes
N_e	Total number of PV nodes	$Γ_{PL}$	Uncertainty adjustment parameters introduced for active load power
$Γ_{QL}$	Uncertainty adjustment parameters introduced for reactive load power	$Γ_{PV}$	Uncertainty adjustment parameters introduced for photovoltaic output
ECOST	Customer outage loss cost within cluster k	$\bar{P_{j}}$	Average load value in node j
N_n_,l	Total number of connected branches within cluster k	H_j	Loss cost caused by outage of average node load j
$t_{j}^{o f f}$	Maximum interruption duration for the interruptible average load in node j	P_j	Maximum interruptible load in node j
λ_j	Average number of outages	r_j	Average outage time of the failure of the node j
α_k	Probability of cluster islanding	e_k	Failure rate of the circuit breaker
e_j	Malfunction rate of element j	e_p	Malfunction rate of the PV
N_p,k	Total of PV nodes within cluster k	$r_{i}^{g}$	Failure recovery time of element i
$r_{p}^{g}$	Failure recovery time of PV nodes	r_break	Operating time of the isolator
e_k1	Failure rate of the isolator	μ_j	Average outage length of the failure of the node j
$t_{j}^{m a x}$	Maximum interruption time limit for the interruptible average load at node j.	$P_{j}^{p m a x}$	Maximum interruption limit for the interruptible average load at node j.

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Figure 2. Simulated runtime of energy storage.

Figure 3. Topology of the distribution network.

Figure 4. Results of cluster partition: (a) Cluster partition results based on the comprehensive cluster partition index; (b) Cluster partition results based on the cluster modularity function partition index.

Figure 5. Comparison of PV planning capacity.

Figure 6. Comparison of ESS planning capacity.

Figure 7. Comparison chart of reliability indicators: (a) Comparison of average number of power outages; (b) Comparison of average outage time.

Figure 8. The planning of PV capacity in different schemes.

Figure 9. The planning of energy storage system capacity in different schemes.

Figure 10. Planning results of different schemes: (a) Scenario 1 planning results; (b) Scenario 2 planning results; (c) Scenario 3 planning results.

Figure 11. Comparison chart of reliability index under different schemes: (a) Average number of power outages; (b) Average outage time.

Figure 12. Comparison of network planning results under different schemes: (a) Centralized planning results; (b) Planning results based on cluster partition.

Figure 13. Comparison of PV planning capacity under different schemes.

Figure 14. Comparison of energy storage planning capacity under different schemes.

Figure 15. Comparison of reliability indexes under different schemes: (a) Average number of power outages; (b) Average outage time.

Table 1. Inter-line distances.

Inter-Node Routes	Distance/km	Inter-Node Routes	Distance/km	Total Length of Line to be Planned
29–77	22.37	49–81	21.62
5–77	19.31	1–82	22.50
6–78	15.31	56–82	24.71
7–78	13.89	8–83	23.83	329.82
1–79	26.41	75–83	29.31
7–79	29.35	1–84	11.31
3–80	9.35	22–84	15.31
21–80	7.12	55–85	7.57
1–81	24.32	60–85	6.23

Table 2. Planning costs in different clustering partitioning schemes.

Programmatic	A	B	C	D	E	F	G
1	1148.13	414.91	436.51	1065.10	913.20	714.41	2662.44
2	1177.74	428.44	430.10	835.41	894.39	685.23	2394.43

Table 3. Planning costs in different uncertainty schemes.

Programmatic	A	B	C	D	E	F	G
1	1224.17	461.17	430.03	687.53	881.00	614.94	2176.50
2	1177.74	428.44	430.10	835.41	894.39	685.23	2394.43
3	1049.95	406.73	431.21	1146.54	1057.53	772.38	2637.42

Table 4. The computing time in different algorithms.

Planning Methodology	Calculation Time (s)
Planning results based on cluster segmentation	32.41
Centralized planning results	55.62

Table 5. Planning costs in different planning schemes.

Programmatic	A	B	C	D	E	F	G
A	1069.50	410.47	434.54	1135.41	1022.57	762.87	2814.42
B	1177.74	428.44	430.10	835.41	894.39	685.23	2394.43

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Shi, Z.; You, G.; Miu, L.; Sun, N.; Duan, L.; Yu, Q.; Xiao, C.; Zhao, K. Zonal Planning for a Large-Scale Distribution Network Considering Reliability. Processes 2025, 13, 354. https://doi.org/10.3390/pr13020354

AMA Style

Shi Z, You G, Miu L, Sun N, Duan L, Yu Q, Xiao C, Zhao K. Zonal Planning for a Large-Scale Distribution Network Considering Reliability. Processes. 2025; 13(2):354. https://doi.org/10.3390/pr13020354

Chicago/Turabian Style

Shi, Zhiwei, Guangzeng You, Linfu Miu, Ning Sun, Lei Duan, Qianqian Yu, Chuanliang Xiao, and Ke Zhao. 2025. "Zonal Planning for a Large-Scale Distribution Network Considering Reliability" Processes 13, no. 2: 354. https://doi.org/10.3390/pr13020354

APA Style

Shi, Z., You, G., Miu, L., Sun, N., Duan, L., Yu, Q., Xiao, C., & Zhao, K. (2025). Zonal Planning for a Large-Scale Distribution Network Considering Reliability. Processes, 13(2), 354. https://doi.org/10.3390/pr13020354

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Zonal Planning for a Large-Scale Distribution Network Considering Reliability

Abstract

1. Introduction

2. Cluster Partition Index and Algorithms for Distribution Networks

2.1. Comprehensive Cluster Partition Index

2.1.1. Modularity Index

2.1.2. Cluster Power Balance Index

2.1.3. Node Affiliation Index

2.2. Cluster Partition Method for Distribution Networks

3. Cluster Partition-Based Three-Layer Planning Strategy for Distribution Networks

3.1. The Line Planning Layer

3.1.1. Objective Function

3.1.2. Constraints

3.2. Location and Capacity Determination Layer for Resource Storage Systems

3.2.1. Objective Function

3.2.2. Constraints

3.3. Reliability Calculation Layer

3.3.1. Objective Function

3.3.2. Constraints

3.4. Cluster Partition-Based Three-Layer Planning Model for Distribution Networks

4. Example Analysis

4.1. Actual Distribution Network

4.2. Cluster Partition and Analysis of Planning Results

4.3. Analysis of the Optimisation Results of the Planning Scheme

4.4. Computational Performance Analysis of the Planning Scheme

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

References

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