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Article

The Evaluation Method of the Power Supply Capability of an Active Distribution Network Considering Demand Response

1
Luohe Power Supply Company of State Grid Henan Electric Power Company, Luohe 462000, China
2
School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255000, China
*
Author to whom correspondence should be addressed.
Processes 2024, 12(12), 2719; https://doi.org/10.3390/pr12122719
Submission received: 9 October 2024 / Revised: 11 November 2024 / Accepted: 21 November 2024 / Published: 2 December 2024
(This article belongs to the Section Energy Systems)

Abstract

The accurate quantification of power supply capability (PSC) is crucial for the planning and operation of active distribution networks. To address the issue of PSC quantification, this paper presents a PSC evaluation method of active distribution networks while considering demand response (DR). Firstly, an incentive-based DR model is introduced, followed by the development of a PSC evaluation model that incorporates DR. This model completely describes the PSC of active distribution networks while considering N-1 security constraints. Secondly, a PSC evaluation algorithm based on uniform state-space sampling is presented, enabling the quantification of the complete PSC of active distribution networks with DR. Then, the influence of the load reduction coefficient in DR on the PSC is studied. It is found that the maximum PSC increases with the load reduction coefficient initially and then stabilizes. Furthermore, measures such as appropriately increasing the load reduction coefficient and expanding the bottleneck component capacity are proposed to enhance the PSC. Finally, the effectiveness of the proposed PSC evaluation method is verified by two active distribution networks, CASE1 and CASE2. The proposed method visualizes the complete PSC of active distribution networks considering DR as a curve and quantifies it as an interval value. For CASE1, the complete PSC is quantified within the range of [4.0, 8.4] MVA, and for CASE2, it is quantified within the range of [17.0, 33.0] MVA. The proposed measures effectively enhance the PSC, facilitating the efficient, safe, and low-carbon operation of smart distribution networks.

1. Introduction

With the continuous integration of flexible resources such as distributed generation (DG), traditional passive distribution networks have gradually evolved into active distribution networks [1,2]. The optimized operation of active distribution networks incorporating demand response (DR) [3], electric vehicles [4], and energy storage [5] significantly enhances their reliability, sustainability, and economic efficiency, contributing to the health of both the natural environment and the economy [6,7]. Meanwhile, active distribution networks possess the ability to supply power to loads and accommodate DG [8,9,10]. Power supply capability (PSC) is widely used to evaluate the load-serving capability of distribution networks and serves as an important planning indicator. The accurate evaluation of PSC is essential for the planning, development, and safe operation of active distribution networks [11,12].
Many studies have been conducted on PSC evaluation for distribution networks. Tyagi et al. [13] proposed a look-ahead approach to evaluate the maximum PSC, in which the maximum PSC is the total supply capability (TSC) [11]. In [14], a model and algorithm were presented for evaluating the TSC under normal operational (N-0 security) constraints. Sun et al. [15] introduced a probabilistic method for evaluating the TSC considering electric vehicles. In [16], a critical flow-based method was proposed to calculate the TSC, improving computational speed while maintaining accuracy. Hao et al. [17] developed an evaluation method for the TSC considering energy storage and analyzed the effects of energy storage and DG on enhancing the TSC. Kong et al. [18] proposed a TSC evaluation method considering both cost and DR. In [19], an evaluation method was introduced for TSC under DG economic dispatch, incorporating load static voltage characteristics to enable accurate assessment of the results. Xiao et al. [20] presented an approach for evaluating the TSC considering N-1 security constraints and analyzed the influence of the substation transformer on the TSC. Sarantakos et al. [21] developed a probabilistic method to quantify the TSC of traditional passive distribution networks, also considering N-1 security constraints. In [22], an evaluation approach for TSC taking into account both reliability and N-1 security constraints was proposed. Xu et al. [23] introduced a reliability-based TSC evaluation method considering both N-1 security constraints and DR. In [24], an evaluation model of TSC considering N-k security constraints, network structure, and regional differences was proposed, and the evaluation results are more reasonable. In conclusion, papers [13,14,15,16,17,18,19,20,21,22,23,24] have studied the TSC evaluation of distribution networks considering flexible resources such as DR and DG. However, the TSC can only be obtained under specific load distribution and does not effectively describe the complete PSC of distribution networks under various load and DG distributions.
To evaluate the complete PSC of distribution networks under various load and DG distributions, Xiao et al. [25] introduced the concept, model, and method of the TSC curve for active distribution networks considering N-0 security constraints. The proposed method in [25] can effectively evaluate the complete PSC of active distribution networks under N-0 security but does not consider N-1 security constraints and DR. In [26], a TSC curve model for traditional passive distribution networks considering N-1 security constraints was proposed, enabling the evaluation of the complete PSC under N-1 security but without considering DG and DR. In conclusion, existing TSC curve models for active distribution networks have not yet considered both N-1 security constraints and DR. As a result, existing studies have not yet provided an accurate evaluation of the complete PSC of active distribution networks under these conditions.
To effectively address the aforementioned problems, a PSC evaluation method of active distribution networks considering DR is proposed in this paper. The main contributions are as follows: (1) it proposes a PSC evaluation model considering DR, which depicts the complete PSC of active distribution networks under N-1 security constraints; (2) it develops a PSC evaluation algorithm based on uniform state-space sampling, which visualizes the complete PSC of active distribution networks considering DR as a curve and quantifies it as an interval value; (3) it reveals the influence mechanism of DR on PSC and presents PSC enhancement measures such as appropriately increasing the load reduction coefficient and expanding the bottleneck component capacity.
The rest of this paper is organized as follows. Section 2 presents the PSC evaluation model of active distribution networks considering incentive-based DR and develops the PSC evaluation algorithm based on uniform state-space sampling. The case study and the influences of DR on PSC are introduced and discussed in Section 3, as well as the enhancement measures of PSC. Finally, the conclusion is drawn in Section 4.

2. Methodology

This section first introduces an incentive-based DR model and presents the PSC evaluation model of active distribution networks considering DR. Subsequently, a PSC evaluation algorithm based on uniform state-space sampling is proposed.

2.1. Evaluation Model of PSC Considering DR

2.1.1. Modeling of Incentive-Based DR

The incentive-based DR model is derived by minimizing the total loss of users as the objective function, with the response cost, electricity charges, and benefits serving as constraints.
The objective function of the incentive-based DR model is formulated in Equation (1).
S = min ( C + C B )
where S represents the total loss of users, C denotes the response cost of users, C refers to the electricity charges of users, and B is the response benefits of users.
The security constraints satisfied by the incentive-based DR model are shown in Equations (2)–(5).
Equation (2) represents the constraint for the response benefits of users. It indicates that users receive compensation for electricity interruptions when satisfying response requirements and compensation for self-reduced electricity consumption when failing to satisfy the response requirements.
B = Q a A i ( Q Q a ) Q A i ( Q < Q a )
where Ai represents the interruption compensation per unit of electricity consumption after the user reduces the load, Qa denotes the amount of electricity consumption the supplier requires to be interrupted, and Q refers to the actual amount of electricity consumption interrupted by the user.
Equation (3) represents the constraint for the response cost of users. A quadratic function, as described in [27], is utilized for quantification, offering both computational simplicity and sufficient accuracy.
C = K 1 Q 2 + K 2 Q K 2 Q θ
where K1 and K2 are constant coefficients. θ represents the willingness of users to endure a power outage, taking a random value within the range of (0,1). K 2 Q θ differentiates the power outage costs among users because θ reflects user subjectivity. A lower θ value indicates a higher marginal cost.
Equation (4) represents the constraint for electricity charges following user response, indicating the costs that users must pay for their electricity consumption.
C = ρ i ( Q i Q )
where ρ i represents the electricity price during the i-th time period and Qi is the original electricity consumption of users per time unit.
Equation (5) represents the constraint on electricity consumption interruption. It indicates that the amount of electricity consumption interrupted by the user does not exceed the maximum reduction specified in the agreement of users.
0 Q n Q i
where n represents the maximum load reduction ratio specified in the agreement of users.
It should be pointed out that when an N-1 contingency occurs in the distribution network, users are permitted to voluntarily reduce or interrupt a certain proportion of their load to ensure compliance with the N-1 security constraint. Consequently, the DR model represented by Equations (1)–(5) is equivalently simplified as shown in Equation (6).
S L , i = ( 1 λ i ) S L , i
where S L , i denotes the apparent power amplitude of the i-th load node following an N-1 contingency in the distribution network and S L , i represents the apparent power amplitude of the i-th load node during normal operation. λ i is the coefficient representing the proportion of load that the user of the i-th load node is willing to reduce or interrupt, referred to as the load reduction coefficient in this paper, where 0 < λ i < 1 . A value of λ i = 1 indicates that the user of the i-th load node allows for a complete power outage after the N-1 contingency, while λ i = 0 represents that the user does not permit any load reduction.

2.1.2. Modeling of PSC Considering DR

In existing studies, the TSC curve has been utilized to evaluate the complete PSC of active distribution networks without taking into account DR and N-1 security constraints. This paper proposes a TSC curve model for active distribution networks that considers both DR and N-1 security constraints to evaluate the complete PSC.
The TSC curve is defined as a curve formed by the total load of all secure boundary points of active distribution networks satisfying security constraints in ascending order [25]. In this curve, the horizontal axis represents the serial number of the sorted points, and the vertical axis corresponds to the respective total load values. A secure boundary point exhibits criticality. Criticality indicates that any infinitesimal increase in load of a secure operating point results in overload [25]. As the secure boundary points form the TSC curve, these points are referred to as TSC curve points in this paper.
The TSC curve model of active distribution networks considering DR is formulated in Equation (7).
T SC = i , Val ( S L , i ) Val ( S L , i ) Val ( S L , i + 1 ) Val ( S L , i ) = k = 1 n S L , k i 1 , 2 , 3
where TSC represents the TSC curve, the PSC Val(SL,i) of an operating point is the total load, SL,i is the power amplitude vector of the load node, SL,k represents the power amplitude of the k-th load node, and i denotes the serial number of the point.
The security constraints satisfied by the TSC curve model considering DR are shown in Equations (8)–(12).
Equation (8) represents the state-space constraint, indicating that the power of load nodes and DG nodes during actual operation remains within a certain range. The state space is defined as the bounded set consisting of all operating points that satisfy Equation (8).
W = S L S DG = S L , 1 , S L , 2 , , S L , n T S DG , 1 , S DG , 2 , , S DG , m T 0 S L , i S L , i max , i L S DG , i max S DG , i 0 , i G
where W is the operation point composed of SL and SDG, SL is the power amplitude vector of the load node, SDG is the power amplitude vector of the DG node, SL,n is the power amplitude of the n-th load node, SDG,m is the power amplitude of the m-th DG node, S L , i max denotes the upper limit of SL,i, S DG , i max represents the upper limit of SDG,i, L refers to the set of all load nodes, and G denotes the set of all DG nodes.
Equation (9) represents the normal operational (N-0 security) constraint. It indicates that the absolute value of the power of lines and substation transformers during a normal operational state does not exceed their capacity.
| S B , i | = | j Λ B , i ( S L , j + S DG , j ) | c B , i , i B | S T , i | = | j Λ T , i ( S L , j + S DG , j ) | c T , i , i T
where SB,i is the power amplitude of the i-th line, Λ B , i is the set of all the downstream nodes of line i, cB,i is the capacity of line i, B denotes the set of all lines, ST,i is the power amplitude of the i-th substation transformer, Λ T , i is the set of all the downstream nodes of substation transformer i, cB,i is the capacity of substation transformer i, and T represents the set of all substation transformers.
Equation (10) represents the N-1 security constraint, which means that when an N-1 contingency occurs in the distribution network the capacity of lines and substation transformers in the reconfigured network must be no less than the absolute value of their power. Following an N-1 contingency, the distribution network is reconfigured into a new topology to restore the power supply to non-fault areas, with the power balance equations adjusted accordingly.
| S B , i ( k ) | = | j Λ B , i ( k ) [ ( 1 λ i ) S L , j + S DG , j ] | c B , i , i B | S T , i ( k ) | = | j Λ T , i ( k ) [ ( 1 λ i ) S L , j + S DG , j ] | c T , i , i T ψ k Ψ , ψ k F , ψ k T
where Ψ = { ψ 1 , , ψ i , , ψ k } represents the set of N-1 contingencies, ψk denotes the occurrence of an N-1 contingency of component k (line or substation transformer). SB,i(k) and ST,i(k) represent the power of line i and substation transformer i, respectively, following an N-1 contingency of component k. Λ B , i ( k ) refers to the set of all the downstream nodes of line i after an N-1 contingency, while Λ T , i ( k ) refers to the set of all the downstream nodes of substation transformer i under the same conditions. λ i denotes the load reduction coefficient, reflecting the effectiveness of user participation in DR.
The secure boundary points are subject to the criticality constraint of Equation (11). This constraint indicates that all load variables of the secure boundary points are restricted by equality constraints, meaning no further increase in any load is possible. The criticality is ensured by the absence of purely zero columns in the equality constraint matrix K L , e q . Equation (11) is derived by converting certain inequalities from Equations (8)–(10) into equalities and removing redundant constraints. It incorporates both forward and reverse power flow constraints. The equality constraint indicates that once the forward power flow of a component reaches its capacity limit, no further load increases are allowed. In this paper, components that reach the forward power flow capacity limit under the equality constraint are defined as bottleneck components. The identification of bottleneck components is of significant importance and value for enhancing PSC.
W i Ω Ω = { β j | j = 1 , 2 , 3 } β j = K L , eq ( 1 λ i ) S L + K DG , eq S DG = c eq | K L , i eq ( 1 λ i ) S L + K DG , ieq S DG | c ieq
where Ω denotes the set of all security boundaries, and β j represents the j-th security boundary. K L , eq and K DG , eq are the load coefficient matrix and DG coefficient matrix of the equality constraints, respectively, with elements of either 0 or 1, determined by the network structure. c eq represents the vector of constants in the equality constraints, with elements corresponding to the capacity of the respective substation transformer or feeder. K L , ieq and K DG , ieq are the load coefficient matrix and DG coefficient matrix of the inequality constraints, respectively, while c ieq represents the vector of constants in the inequality constraints.
Equation (12) represents the voltage constraint, which means that the voltage offset of all secure boundary points remains within a secure range.
U Δ U Δ U Δ +
where U Δ represents the voltage offset vector at the nodes, U Δ + and U Δ denote the maximum and minimum voltage offset, respectively.
It is important to note that the proposed TSC curve model differs significantly from existing models [25,26] in the following distinct features:
  • The proposed TSC curve model accounts for DR, enabling accurate evaluation of the complete PSC of distribution networks considering DR. In contrast, existing TSC curve models [25,26] do not consider DR and therefore cannot accurately evaluate the complete PSC of such networks.
  • The proposed TSC curve model considers both N-0 and N-1 security constraints, which is suitable for evaluating the complete PSC of active distribution networks accounting for N-1 contingencies. In contrast, the TSC curve model presented in [25] only considers N-0 security constraints and is not applicable to N-1 contingency scenarios. The TSC curve model proposed in [26] accounts for N-1 security constraints but does not consider DG and is therefore applicable only to traditional passive distribution networks, not to active distribution networks.
In summary, compared with the proposed approach, the limitations of previous works are as follows: the previous models [25,26] are not applicable for evaluating the complete PSC of active distribution networks that concurrently consider DR and N-1 contingencies, as these models have not yet accounted for both DR and N-1 security constraints simultaneously.

2.2. Evaluation Algorithm of PSC Considering DR

The TSC curve, which delineates the complete PSC of the distribution network, is constituted by an infinite number of TSC curve points. Nevertheless, the TSC curve points are infinite in number. State-space sampling can obtain a finite number of sampling points, which serves to represent the complete TSC curve points. In accordance with this, this section presents an evaluation algorithm based on uniform state-space sampling to precisely quantify the complete PSC of active distribution networks considering DR through the calculation of the TSC curve. The proposed algorithm is illustrated in Figure 1 and includes the following four steps.
Step 1: obtain the operating points.
X operating points are obtained by performing uniform sampling with a step size of σ within the state space of load and DG, which effectively represent the complete operating points.
X = i = 1 n ( 1 + S L , i max σ ) i = 1 m ( 1 + S DG , i max σ )
where S L , i max and S DG , i max are the power upper limits of SL,i, and SDG,i, respectively.
Step 2: calculate the TSC curve points.
TSC curve points are first identified from the sampled operating points in Step 1 that satisfy the criticality constraint in Equation (11). For each TSC curve point, AC power flow and voltage offsets are calculated using the OpenDSS power flow solver, and the effective TSC curve points are then identified based on the voltage constraint in Equation (12).
Step 3: visualize the TSC curve.
The total load Val(WL,i) for effective TSC curve points is calculated according to Equation (7), and the TSC curve is visualized by sorting Val(WL,i) in ascending order.
Step 4: quantify the TSC curve indices.
The maximum PSC of the active distribution network considering DR is the TSC. The average PSC and the minimum PSC of the active distribution network accounting for DR are TSC ¯ and TSCmin, respectively.
Note that the time complexity of the proposed algorithm based on uniform state-space sampling is O ( ( 1 + S L , i max σ ) n ( 1 + S DG , i max σ ) m ) , where n is the number of SL,i, and m is the number of SDG,i.

3. Results and Discussion

This section first demonstrates the proposed method by a modified IEEE active distribution network (CASE1) and then further verifies it by an IEEE RBTS BUS4 active distribution network (CASE2). The complete PSC is first evaluated, followed by an analysis of the influence of DR on PSC and potential enhancement measures.

3.1. Case Overview

The proposed method is verified using a modified IEEE active distribution network (CASE1) illustrated in Figure 2. CASE1 comprises two IEEE 33-node distribution networks, derived from [25], interconnected through a tie switch (TS), and includes two 33/11 kV substation transformers (T1 and T2). Each substation transformer has a capacity of 6.0 MVA, and the feeder capacity is 4.0 MVA. Each branch length is 0.25 km, with an impedance per unit length of 0.09 + j0.08 Ω/km. The longest path, between nodes 0 and 17, is 4.25 km. Following the method in [25], adjacent nodes are merged to form load nodes (L1, L2, and L3) and DG nodes (DG1 and DG2). The power range for load nodes is [0, 4.0] MVA, and that for DG nodes is [−2.0, 0] MVA. Load node L2 participates in DR, with a load reduction coefficient of λ = 0.1 . The simplified IEEE active distribution network after equivalent merging is depicted in Figure 3. In China’s low voltage distribution network, national standards mandate voltage offsets with maximum and minimum values of +7% and −7%, respectively. The experimental simulation environment is setup as follows: the computer is equipped with an Intel Core i7-10510U CPU @ 4.10 GHz processor, 16 GB of memory, and operates on Microsoft Windows 10, and the simulation platform is Matlab R2012a.

3.2. PSC Evaluation Process and Results

3.2.1. Implementation of Proposed Model

The TSC curve for CASE1 considering DR is calculated based on the proposed model described in Section 2, and the results are as follows.
T SC = i , Val ( S L , i ) Val ( S L , i ) Val ( S L , i + 1 ) Val ( S L , i ) = k = 1 3 S L , k i 1 , 2 , 3
The security constraints for the TSC curve of Equation (14) are shown in Table 1.
Note that the N-1 security constraints in Table 1 correspond to the new reconfigured topology following an N-1 contingency. For instance, when an N-1 contingency occurs in substation transformer T1 or branch B1, SL,1, SDG,1, and SL,2, are transferred to substation transformer T2 and branch B5 through N-1 network reconfiguration (where the B1 branch switch is open and the tie switch (TS) is closed). Accordingly, the first N-1 security constraint | S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 | min ( 4.0 , 6.0 ) indicates that the capacities of substation transformer T2 and branch B5 must be no less than the absolute value of their power.
On the other hand, the equality constraint S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 = 4.0 in Table 1 covers all load variables, which ensures that the calculated TSC curve satisfies the criticality constraint. Furthermore, the right-hand side of the equality constraint represents the 4.0 MVA capacity of branches B1 and B5, where the forward power flow reaches the capacity limit first. As a result, branches B1 and B5 are identified as the bottleneck components.
After eliminating the redundant constraints [28] in Table 1, the expression for the TSC curve is obtained, as shown in Equation (15).
T SC = i , Val ( S L , i ) Val ( S L , i ) Val ( S L , i + 1 ) Val ( S L , i ) = k = 1 3 S L , k i 1 , 2 , 3 W i = S L , i S DG , i = S L , 1 , S L , 2 , , S L , n T S DG , 1 , S DG , 2 , , S DG , m T Ω Ω = { β 1 } β 1 = S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 = 4.0 | ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 | 4.0 | ( 1 λ ) S L , 2 + S L , 3 + S DG , 2 | 4.0 | S L , 1 + ( 1 λ ) S L , 2 + S DG , 1 + S DG , 2 | 4.0 | S L , 1 + S L , 2 + S DG , 1 | 4.0 0 S L , i 4.0   i = 1 , 2 , 3 2.0 S DG , j 0   j = 1 , 2   7 % U Δ % 7 %

3.2.2. Simulation Calculation Process and Results

The PSC of CASE1 is evaluated using the algorithm described in Section 2.2, and the simulation process and results are as follows.
Step 1: X = i = 1 3 ( 1 + 4 0.1 ) i = 1 2 ( 1 + 2 0.1 ) = 30394161 operating points are obtained by uniformly sampling the state space of load and DG with a step size of σ = 0.1.
Step 2: For each sampled operating point, the effective TSC curve points that satisfy both the criticality and voltage constraints in Equation (15) are identified. A total of 20173 effective TSC curve points are finally obtained and are presented in Table 2.
Step 3: the TSC curve is visualized by sorting Val(SL,i) in ascending order based on Table 2 and is shown in Figure 4.
Figure 4 illustrates that the complete PSC of CASE1 considering DR is evaluated by the proposed method as a TSC curve rather than a single TSC value (8.4 MVA). The TSC curve of CASE1 ranges from 4.0 MVA to 8.4 MVA. It indicates that, under various load and DG distributions, the complete PSC of all secure boundary points is within the range of [4.0, 8.4] MVA; specifically, it cannot be less than the minimum PSC of 4.0 MVA nor exceed the maximum PSC of 8.4 MVA. Note that the TSC curve displays numerous distinct horizontal segments, resulting from multiple TSC curve points corresponding to the same PSC value.
Step 4: the TSC curve indices are quantified based on Table 2 and are presented in Table 3.
Table 3 provides a further quantification of the complete PSC for CASE1, presenting it as a range between 4.0 MVA and 8.4 MVA instead of a singular value. Consequently, the minimum and maximum PSC of CASE1 considering DR are 4.0 MVA and 8.4 MVA, respectively, with an average PSC of 5.97 MVA. Note that the computational time for CASE1 is 57.264 s based on the proposed algorithm.

3.3. Analysis of the Influence of DR on PSC

DR affects the distribution of load magnitudes, thereby influencing the PSC of the distribution network. To investigate the influence of DR on PSC, the proposed method is employed to evaluate the PSC of the active distribution network under different load reduction coefficients λ .

3.3.1. TSC Curve Indices Under Different Load Reduction Coefficients

The TSC curve indices of CASE1 under different λ are presented in Table 4.
(1) TSC under different load reduction coefficients.
The relationship between the maximum PSC (TSC) and the load reduction coefficient λ is illustrated in Figure 5 based on Table 4.
Table 4 and Figure 5 illustrate that the maximum PSC (TSC) is 8.0 MVA when DR is not considered ( λ = 0 ). Upon accounting for DR, the TSC increases linearly from 8.0 MVA to 10.0 MVA as the load reduction coefficient λ increases within the range of (0, 0.5]. When λ further increases within the range of (0.5, 1.0], the TSC remains constant at 10.0 MVA without further increase. The reasons are as follows:
According to the distribution network structure and Equation (15), CASE1 satisfies the constraints S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 = 4.0 , S L , 1 + S L , 2 + S DG , 1 = 4.0 , S DG , 1 = 2.0 , and 2.0 S L , 3 4.0 when reaching the TSC. From this, it can be derived that S L , 3 = ( λ S L , 2 S DG , 2 ) and S L , 1 + S L , 2 + S L , 3 = 6.0 + ( λ S L , 2 S DG , 2 ) . Meanwhile, the total load at TSC is expressed as TSC = Val ( S L , i ) = S L , 1 + S L , 2 + S L , 3 = 6.0 + ( λ S L , 2 S DG , 2 ) = 6.0 + S L , 3 .
As λ increases linearly within the range of [0, 0.5], ( 1 λ ) S L , 2 decreases linearly from 4.0 MVA to 2.0 MVA. To satisfy the criticality constraint S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 = 4.0 , S L , 3 = ( λ S L , 2 S DG , 2 ) increases linearly from 2.0 MVA to 4.0 MVA, resulting in TSC = 6.0 + S L , 3 to rise linearly from 8.0 MVA to 10.0 MVA.
As λ continues to increase within the range of (0.5, 1.0], S L , 3 = ( λ S L , 2 S DG , 2 ) theoretically could continue to rise greater than 4.0 MVA. However, in practice, S L , 3 is constrained by the capacity limit of the bottleneck component B5 of 0 S L , 3 4.0 (see Equation (15)), so S L , 3 consistently remains at 4.0 MVA. Consequently, TSC = 6.0 + S L , 3 remains constant at 10.0 MVA.
(2) TSCmin under different load reduction coefficients.
Table 4 illustrates that the minimum PSC (TSCmin) remains constant at 4.0 MVA as the load reduction coefficient λ increases. This is because S L , 2 is always equal to 0 when TSCmin is achieved, which makes it independent of λ at load node L2.

3.3.2. TSC Curve Under Different Load Reduction Coefficients

The TSC curves of CASE1 under different λ are visualized, as shown in Figure 6.
Figure 6 shows that as the load reduction coefficient λ increases within the range [0, 0.5], the peak of the TSC curve rises progressively from 8.0 MVA to 10.0 MVA. As λ continues to increase, the peak of the cyan TSC curve at λ = 0.6 no longer increases compared to the magenta curve at λ = 0.5 . This observation aligns with the results presented in Table 4 and Figure 5.
In summary, the influence law of DR on the TSC curve is as follows: as the load reduction coefficient λ increases, the TSC first increases and subsequently remains constant. The influence mechanism is as follows: As λ increases, the power at load nodes involved in DR after the N-1 contingency decreases to ( 1 λ ) S L , i , which leads to a decline in the forward power flow through the bottleneck component. To ensure the full loading of the bottleneck component, the power at certain load nodes must increase, which causes a rise in the peak of the TSC curve and an increase in the TSC. However, as λ continues to increase, the power at load nodes becomes constrained by the capacity of the bottleneck component, which inhibits any further rise in the peak of the TSC curve, resulting in the TSC remaining constant.

3.4. Enhancement Measures for PSC

Based on the influence law and mechanism revealed in Section 3.3, this section presents two effective measures for enhancing the PSC as follows:
(1)
Measure 1: increase the load reduction coefficient λ within a certain range, as λ affects load distribution and subsequently influences the TSC curve.
(2)
Measure 2: expand the capacity of the bottleneck component, as its capacity represents a critical factor in constraining the TSC curve.
The two measures mentioned above are implemented separately in CASE1 in Section 3.1. Measure 1 involves increasing λ from 0.1 to 0.5. Measure 2 entails expanding the capacity of bottleneck components (B1 and B5) from 4.0 MVA to 5.0 MVA with λ = 0.1 . The corresponding TSC curves under these two measures are shown in Figure 7.
Figure 7 demonstrates that both proposed measures significantly enhance the TSC curve. Following the implementation of Measure 1, the highest point of the magenta TSC curve rises, with the maximum value increasing from 8.0 MVA to 10.0 MVA, resulting in a 2.0 MVA improvement in the TSC and a 0.59 MVA increase in average PSC. Upon the application of Measure 2, the blue TSC curve exhibits an overall improvement, as its maximum value increases from 8.0 MVA to 9.4 MVA, leading to a 1.4 MVA enhancement in the TSC. Furthermore, the minimum value rises from 4.0 MVA to 5.0 MVA, indicating a 1.0 MVA improvement in minimum PSC, while the average PSC increases by 1.02 MVA.
The advantages and corresponding applicable scenarios of the two proposed measures for enhancing the PSC are outlined as follows:
(1)
Measure 1 significantly enhances the maximum PSC (i.e., TSC) when the load reduction coefficient λ increases in DR. This measure is particularly applicable in scenarios where incentive policies promote an increase in the load reduction coefficient at specific nodes through DR.
(2)
Measure 2 effectively enhances the overall TSC curve by addressing specific bottleneck components without necessitating substantial additional investment in capacity expansion. This measure is particularly suited to distribution network upgrade projects involving the replacement of components.

3.5. IEEE RBTS BUS4 Case Verification

As shown in Figure 8, the proposed method is further verified by CASE2, an IEEE RBTS BUS4 active distribution network [29]. CASE2 comprises three 33 kV/11 kV substations, seven 10 kV feeders, 38 load nodes, and 4 DG nodes. Each substation transformer has a capacity of 9.0 MVA, the feeder capacity is 6.0 MVA, and the power range for DG nodes is [−3.0, 0] MVA. Each feeder length is 3.0 km, with an impedance per unit length of 0.09 + j0.08 Ω/km. Following the method in [25], the load nodes on the same feeder are merged to form load nodes. The power range for load nodes is [0, 6.0] MVA. Load node L3 participates in DR, with a load reduction coefficient of λ = 0.2 . The simplified IEEE RBTS BUS4 active distribution network after equivalent merging is depicted in Figure 9.
The TSC curve for CASE2 considering DR is calculated based on the proposed model introduced in Section 2, and the expression for the TSC curve is shown in Equation (16).
T SC = i , Val ( S L , i ) Val ( S L , i ) Val ( S L , i + 1 ) Val ( S L , i ) = k = 1 7 S L , k i 1 , 2 , 3 W i = S L , i S DG , i = S L , 1 , S L , 2 , , S L , n T S DG , 1 , S DG , 2 , , S DG , m T Ω Ω = { β 1 , β 2 } β 1 = S L , 2 + ( 1 λ ) S L , 3 + S L , 4 + S DG , 2 + S DG , 3 = 9.0 S L , 2 + ( 1 λ ) S L , 3 + S L , 6 + S DG , 2 = 9.0 S L , 1 + S L , 7 + S DG , 1 = 6.0 S L , 5 + S L , 7 + S DG , 4 = 6.0 | ( 1 λ ) S L , 3 + S L , 4 + S DG , 2 + S DG , 3 | 6.0 | S L , 2 + ( 1 λ ) S L , 3 | 9.0 | S L , 2 + S L , 6 | 6.0 | S L , 2 + S L , 3 + S DG , 2 | 9.0 0 S L , i 6.0   i = 1 , 2 , 3 , 4 , 5 , 6 , 7 3.0 S DG , j 0   j = 1 , 2 , 3 , 4   β 2 = ( 1 λ ) S L , 3 + S L , 4 + S DG , 2 + S DG , 3 = 6.0 S L , 2 + S L , 6 = 6.0 S L , 1 + S L , 7 + S DG , 1 = 6.0 S L , 5 + S L , 7 + S DG , 4 = 6.0 | S L , 2 + ( 1 λ ) S L , 3 + S L , 4 + S DG , 2 + S DG , 3 | 9.0 | S L , 2 + ( 1 λ ) S L , 3 + S L , 6 + S DG , 2 | 9.0 | S L , 2 + ( 1 λ ) S L , 3 | 9.0 | S L , 2 + S L , 3 + S DG , 2 | 9.0 0 S L , i 6.0   i = 1 , 2 , 3 , 4 , 5 , 6 , 7 3.0 S DG , j 0   j = 1 , 2 , 3 , 4   7 % U Δ % 7 %
Based on the expression of the TSC curve in Equation (16), the PSC of CASE2 is evaluated by the algorithm introduced in Section 2.2, and 1560 effective TSC curve points are finally calculated as presented in Table 5.
Based on Table 5, the visualized TSC curve and the quantitative TSC curve indices are presented in Figure 10 and Table 6, respectively.
Figure 10 shows that the complete PSC of CASE2 considering DR is evaluated by the proposed method as a TSC curve rather than a single TSC value (33.0 MVA), with the TSC curve for CASE2 ranging from 17.0 MVA to 33.0 MVA. This indicates that, under various load and DG distributions, the complete PSC for all secure boundary points is within the range of [17.0, 33.0] MVA.
Table 6 offers a more detailed quantification of the complete PSC for CASE2, presenting it as a range between 17.0 MVA and 33.0 MVA rather than a singular value. As a result, the minimum and maximum PSC for CASE2 considering DR are 17.0 MVA and 33.0 MVA, respectively, with an average PSC of 24.67 MVA. Note that the computational time for CASE2 is 271.112 s based on the proposed algorithm.

4. Conclusions

To accurately quantify the complete PSC, this paper proposes a PSC evaluation method of active distribution networks considering DR. The main conclusions are summarized as follows:
(1)
The proposed PSC evaluation model considering incentive-based DR effectively describes the complete PSC of active distribution networks considering both N-1 security and DR.
(2)
The proposed PSC evaluation algorithm based on uniform state-space sampling visualizes the complete PSC of active distribution networks considering DR as a TSC curve and quantifies it as an interval value rather than a single TSC value. For instance, the complete PSC for CASE1 is quantified within the range of [4.0, 8.4] MVA, whereas for CASE2, it is quantified within the range of [17.0, 33.0] MVA.
(3)
The influence law and mechanism of the load reduction coefficient in DR on PSC are revealed. As the load reduction coefficient λ increases, the TSC first increases and then remains constant. For instance, as λ increases from 0 to 1.0, the TSC for CASE1 first increases linearly from 8.0 MVA to 10.0 MVA, then remains constant at 10.0 MVA.
(4)
The two proposed measures of increasing the load reduction coefficient and expanding the capacity of bottleneck components can effectively enhance the PSC of active distribution networks. For instance, as λ increases from 0.1 to 0.5, the TSC and average PSC of CASE1 increase by 2.0 MVA and 0.59 MVA, respectively. Similarly, as the capacity of bottleneck components expands from 4.0 MVA to 5.0 MVA, the TSC, average PSC, and minimum PSC of CASE1 increase by 1.4 MVA, 1.02 MVA, and 1.0 MVA, respectively.
The accurate evaluation of the PSC of smart distribution networks considering both DR and DG holds significant importance and value for system planning and operation. The enhancement of PSC is integral to achieving the safe, efficient, and low-carbon operation of smart distribution networks. In the future, the quantitative evaluation of the PSC of active distribution networks, considering various flexible resources such as energy storage and electric vehicles, as well as the high-efficiency evaluation algorithm, will be investigated.

Author Contributions

Conceptualization, N.L. and S.S.; methodology, N.L. and L.Z.; software, N.L., L.Z. and Y.Q.; validation, N.L., S.S. and Y.Q.; investigation, N.L. and J.W.; writing—original draft preparation, N.L.; writing—review and editing, N.L., S.S., L.Z. and Y.Q.; visualization, Y.Q.; supervision, S.S. and Y.Q.; project administration, L.Z. and J.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research is supported by the Science and Technology Project of State Grid Henan Electric Power Company (Data-Driven Power Supply Capability Enhancement and Risk Prevention Technology for Active Distribution Networks, No. 5217G0240006).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare that this study received funding from Luohe Power Supply Company of State Grid Henan Electric Power Company. The funder had the following involvement with the study: Ning Li, Sipei Sun, Liang Zhang and Jianjun Wang. The authors declare no conflicts of interest.

Acronym

SymbolFull formMeaning
PSCPower supply capabilityThe total load of the secure boundary point.
TSCTotal supply capabilityThe maximum load that a distribution network can serve under a given security criterion, i.e., maximum PSC.
TSC curveTotal supply capability curveA curve formed by the total load of all secure boundary points with criticality.
DRDemand responseA mechanism that enables consumers to adjust their electricity consumption in response to external signals to help balance supply and demand in the energy grid.
DGDistributed generationThe generation of electricity from multiple, small-scale energy sources located close to the point of consumption, which can be connected to the grid or operate independently (off-grid).
TSTie switchAn electrical device used to link different parts of a power network, enabling the transfer of electrical power between them.

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Figure 1. Flowchart of the PSC evaluation algorithm based on uniform state-space sampling.
Figure 1. Flowchart of the PSC evaluation algorithm based on uniform state-space sampling.
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Figure 2. CASE1: a modified IEEE active distribution network considering DR.
Figure 2. CASE1: a modified IEEE active distribution network considering DR.
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Figure 3. Simplified IEEE active distribution network considering DR.
Figure 3. Simplified IEEE active distribution network considering DR.
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Figure 4. TSC curve of CASE1.
Figure 4. TSC curve of CASE1.
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Figure 5. The relationship between TSC and load reduction coefficient λ for CASE1.
Figure 5. The relationship between TSC and load reduction coefficient λ for CASE1.
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Figure 6. The TSC curve of CASE1 under different load reduction coefficients λ .
Figure 6. The TSC curve of CASE1 under different load reduction coefficients λ .
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Figure 7. TSC curves of CASE1 under two measures.
Figure 7. TSC curves of CASE1 under two measures.
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Figure 8. CASE2: an IEEE RBTS BUS4 active distribution network considering DR.
Figure 8. CASE2: an IEEE RBTS BUS4 active distribution network considering DR.
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Figure 9. Simplified IEEE RBTS BUS4 active distribution network considering DR.
Figure 9. Simplified IEEE RBTS BUS4 active distribution network considering DR.
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Figure 10. TSC curve of CASE2.
Figure 10. TSC curve of CASE2.
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Table 1. The security constraints of the TSC curve for CASE1.
Table 1. The security constraints of the TSC curve for CASE1.
Security ConstraintExpression
State-space constraint 0 S L , i 4.0   i = 1 , 2 , 3 , 2.0 S DG , j 0   j = 1 , 2
Normal operational constraint | S L , 1 + S L , 2 + S DG , 1 | min ( 4.0 , 6.0 ) , | S L , 2 + S DG , 1 | 4.0 , | S L , 2 | 4.0 ,
| S DG , 2 | 4.0 , | S L , 3 + S DG , 2 | min ( 4.0 , 6.0 )
N-1 security constraint | S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 | min ( 4.0 , 6.0 ) ,
| ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 | min ( 4.0 , 6.0 ) , | S L , 1 | min ( 4.0 , 6.0 ) ,
| ( 1 λ ) S L , 2 + S L , 3 + S DG , 2 | min ( 4.0 , 6.0 ) , | S L , 1 + S DG , 1 | min ( 4.0 , 6.0 ) ,
| S L , 1 + ( 1 λ ) S L , 2 + S DG , 1 + S DG , 2 | min ( 4.0 , 6.0 ) , | S L , 3 | min ( 4.0 , 6.0 )
Criticality constraint W i = S L , i S DG , i = S L , 1 , S L , 2 , , S L , n T S DG , 1 , S DG , 2 , , S DG , m T Ω , Ω = { β 1 } ,
β 1 = S L , 1 + ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 = 4.0 | ( 1 λ ) S L , 2 + S L , 3 + S DG , 1 + S DG , 2 | 4.0 | ( 1 λ ) S L , 2 + S L , 3 + S DG , 2 | 4.0 | S L , 1 + ( 1 λ ) S L , 2 + S DG , 1 + S DG , 2 | 4.0 | S L , 1 + S L , 2 + S DG , 1 | 4.0 0 S L , i 4.0   i = 1 , 2 , 3 2.0 S DG , j 0   j = 1 , 2
Voltage constraint 7 % U Δ % 7 %
Table 2. TSC curve points of CASE1.
Table 2. TSC curve points of CASE1.
TSC Curve PointsSL,1 (MVA)SL,2 (MVA)SL,3 (MVA)SDG,1 (MVA)SDG,2 (MVA)Val(SL,i) (MVA)
W10.00.04.00.00.04.0
W20.10.03.90.00.04.0
W30.20.03.80.00.04.0
W40.30.03.70.00.04.0
W50.40.03.60.00.04.0
W109373.81.01.2−0.9−1.06.0
W109380.04.02.10.0−1.76.1
W109390.12.04.00.0−1.96.1
W201692.73.02.6−2.0−2.08.3
W201702.83.02.5−2.0−2.08.3
W201712.93.02.4−2.0−2.08.3
W201723.03.02.3−2.0−2.08.3
W201732.04.02.4−2.0−2.08.4
Table 3. TSC curve indices of CASE1.
Table 3. TSC curve indices of CASE1.
TSC (MVA) TSC ¯ (MVA)TSCmin (MVA)PSC Range (MVA)
8.45.974.0[4.0, 8.4]
Table 4. The TSC curve indices of CASE1 under different load reduction coefficients λ .
Table 4. The TSC curve indices of CASE1 under different load reduction coefficients λ .
λ TSC (MVA) TSC ¯ (MVA)TSCmin (MVA) λ TSC (MVA) TSC ¯ (MVA)TSCmin (MVA)
0.008.05.894.00.5510.06.314.0
0.058.25.864.00.6010.06.604.0
0.108.45.974.00.6510.06.344.0
0.158.65.984.00.7010.06.534.0
0.208.86.134.00.7510.06.704.0
0.259.06.214.00.8010.06.734.0
0.309.26.224.00.8510.06.384.0
0.359.46.164.00.9010.06.614.0
0.409.66.404.00.9510.06.334.0
0.459.86.224.01.0010.06.824.0
0.5010.06.564.0
Table 5. TSC curve points of CASE2 (MVA).
Table 5. TSC curve points of CASE2 (MVA).
TSC Curve PointsSL,1SL,2SL,3SL,4SL,5SL,6SL,7SDG,1SDG,2SDG,3SDG,4Val(SL,i)
W10.04.05.01.00.01.06.00.00.00.00.017.0
W20.04.05.01.01.01.06.00.00.00.0−1.018.0
W30.04.05.02.00.01.06.00.00.0−1.00.018.0
W40.05.05.01.00.01.06.00.0−1.00.00.018.0
W51.04.05.01.00.01.06.0−1.00.00.00.018.0
W9746.05.05.02.06.01.00.00.0−1.0−1.00.025.0
W9750.03.05.06.03.03.06.00.0−1.0−3.0−3.026.0
W9762.03.05.05.03.02.06.0−2.00.0−3.0−3.026.0
W15565.03.05.06.05.03.04.0−3.0−1.0−3.0−3.031.0
W15575.03.05.06.06.03.03.0−2.0−1.0−3.0−3.031.0
W215586.03.05.06.06.03.02.0−2.0−1.0−3.0−2.031.0
W15596.04.05.05.06.02.03.0−3.0−1.0−3.0−3.031.0
W15606.03.05.06.06.03.03.0−3.0−1.0−3.0−3.032.0
Table 6. TSC curve indices of CASE2.
Table 6. TSC curve indices of CASE2.
TSC (MVA) TSC ¯ (MVA)TSCmin (MVA)PSC Range (MVA)
33.024.6717.0[17.0, 33.0]
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Li, N.; Sun, S.; Zhang, L.; Wang, J.; Qu, Y. The Evaluation Method of the Power Supply Capability of an Active Distribution Network Considering Demand Response. Processes 2024, 12, 2719. https://doi.org/10.3390/pr12122719

AMA Style

Li N, Sun S, Zhang L, Wang J, Qu Y. The Evaluation Method of the Power Supply Capability of an Active Distribution Network Considering Demand Response. Processes. 2024; 12(12):2719. https://doi.org/10.3390/pr12122719

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Li, Ning, Sipei Sun, Liang Zhang, Jianjun Wang, and Yuqing Qu. 2024. "The Evaluation Method of the Power Supply Capability of an Active Distribution Network Considering Demand Response" Processes 12, no. 12: 2719. https://doi.org/10.3390/pr12122719

APA Style

Li, N., Sun, S., Zhang, L., Wang, J., & Qu, Y. (2024). The Evaluation Method of the Power Supply Capability of an Active Distribution Network Considering Demand Response. Processes, 12(12), 2719. https://doi.org/10.3390/pr12122719

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