A New Method for PMU Deployment Based on the Preprocessed Integer Programming Algorithm

Abstract

To further enhance the deployment efficiency of synchronous phasor measurement units and rationally select deployment configuration schemes, an improved configuration method based on integer programming algorithms is proposed. Based on the existing deployment method of integer programming algorithm, on the one hand, the special conditions of end nodes and zero injection nodes are taken into consideration. By analyzing the corresponding node model matrix, special nodes are given priority processing, achieving the condition simplification of the algorithm model. On the other hand, the evaluation indicators for the construction of schemes with the same minimum deployment quantity in the solution set obtained from the iterative solution of the algorithm are further analyzed and compared, so as to screen out a more reasonable deployment method. After conducting simulation tests on the IEEE-14, IEEE-30, and NE-39 power node systems using the MATLAB platform, the depth-first search algorithm and the improved simulated annealing algorithm were compared with the improved method. Eventually, this method had fewer deployments in a similar number of deployments and less deployment time in a similar number of deployments. The results verified the superiority of this method in terms of time and deployment quantity for PMU deployment problems.

1. Introduction

In recent years, the development of smart grid construction has been constantly advancing. Therefore, achieving effective and efficient full-network status monitoring to ensure the safe and stable operation of the entire system is particularly crucial. The stability of grid operation is related to detection equipment such as PMUs and transformers [1,2,3]. Nowadays, there are many different detection schemes for equipment in power systems from the past, such as the use of image recognition mentioned in [4,5] to detect the status of transformers. In terms of the measurement accuracy of the equipment, references [6,7] further ensure the accuracy of sensor measurement by establishing a more precise model.

In power systems, the deployment methods of measuring equipment have gradually become a hot topic of research. This article mainly focuses on the deployment research of PMUs [8,9,10]. The essence of this research is actually a kind of objective optimization problem, which achieves the optimization of device deployment by combining optimization algorithms [11,12,13]. The descriptions in references [14,15,16] reveal the impact of PMU deployment on the state estimation of power systems.

PMU is a high-precision measuring device in power grids, which operates relying on satellite timing signals [17,18,19]. As a component of the WAMS system, it plays a crucial role in the overall operation, monitoring and control of the distribution network, and is a key link in the current smart grid construction [20,21]. The PMU can detect in real time the voltage phasor, current phasor, frequency, active power, reactive power, and other data in the power network, and promptly transmit the data to the dispatching center [22,23,24]. The existing research on the optimal deployment of PMU mainly focuses on two aspects: the efficiency of the algorithm and its related constraints [25,26,27]. Due to the real-time nature of PMU and the synchronization of all network devices, it is bound to gradually replace the role of SCADA systems in state cognition and become an important support for the development of smart grids. Therefore, how to further achieve the efficient and reasonable deployment of PMUs to realize the full visibility of the power grid is of great significance.

Typically, the deployment configuration schemes of synchronous vector measurement units may vary due to different considerations, such as climatic factors and power consumption factors [28,29]. If all bus nodes in the entire network are equipped with PMUs, the entire system will be directly and completely observable, and the power flow equation of the entire system does not require any iterative calculation and solution. However, due to the high cost of PMU equipment, it is necessary to implement how to rationally deploy PMUs so as to not only reduce the number of PMUs deployed but also ensure the observability of the entire system. Taking into account the changes in the global economy and the demand situation in power production, current research in this area is a point worthy of attention, and it is also a research topic with certain production significance [30,31,32].

Research on the optimal configuration of synchronous phasor measurement units has become a hot topic in the field of power system equipment measurement, and certain research results have been achieved. Reference [33] employs an improved adaptive genetic algorithm, which can calculate crossover probability and mutation probability. However, when the maximum fitness value of the population is equal to the average fitness value, it may fall into local optimum. Reference [34] employed the simulated annealing algorithm. The actual effect of this method is related to the iterative conditions of the algorithm, and the result may not be the optimal situation. Reference [35] adopted an algorithm based on binary particle swarm to achieve deployment optimization, but the algorithm is generally prone to getting stuck in local optimal solutions. Heuristic algorithms have a certain degree of randomness when solving problems, and they may not have a good effect on the final optimization result. Moreover, since their solution effect is related to the initial algorithm setting, the obtained solution may not be the global optimal solution. The integer programming algorithm is a number theory method. It has better adaptability in solving problems under complex conditions and can also achieve good results for complex constraints.

Reference [36] uses the integer programming algorithm to solve the PMU deployment problem, but the setting of its constraint conditions is rather complex. Reference [37] employed an improved integer programming method to address the configuration issue of PMUs, optimizing the algorithm by simplifying the constraint conditions. However, it failed to take into account the special nodes of the power system, leaving room for further improvement in practical application fields. Reference [38] took zero-injection nodes into consideration, but its method has deficiencies in the setting of constraint conditions for zero-injection nodes in terms of condition setting. In the problem of PMU optimization configuration, most of the literature does not take into account the situation of minimizing the number of analog channels.

In this paper, aiming at the complete observability of the power system state and the minimization of the number of configured PMUs, an improved PMU configuration method is proposed based on the existing integer programming algorithm. The optimal deployment of PMUs is achieved by setting preprocessing conditions and evaluation indicators. On the one hand, the special conditions of the end nodes and the zero injection nodes are taken into consideration. By analyzing the corresponding node model matrix and preprocessing the special nodes, the conditions of the algorithm model are simplified and the time complexity of the algorithm is reduced. On the other hand, a multi-scheme solution is achieved by improving the solution method, and evaluation indicators are constructed for the solution set. Through further analysis and comparison, a more reasonable deployment method is selected. This study conducted simulation calculations on the IEEE-14, IEEE-30, and NE-39 node systems, compared the algorithm proposed in this paper with different algorithms, and verified the effectiveness and rationality of the method proposed in this paper.

2. Materials and Methods

2.1. Power System and Its Observability

2.1.1. Observability Requirements

The prerequisite for the deployment of PMU in the power grid is to meet the observability of the system. For each node, it has state quantities such as amplitude and phase Angle. If the state of each node in a power system can be obtained through direct or indirect observation, then the power system is overall observable [39]. The observability conditions of power systems are usually described from two aspects: “algebraic observability” and “topological observability” [40].

(1)

The algebra is considerable

From an algebraic perspective, the observability of a power system with n nodes and m measurements can be expressed by Equation (1).

$$\mathit{z}=\mathit{H}\mathit{x}+\mathit{v},$$

(1)

Among them, z is the m-dimensional measurement vector; H is the Jacobian matrix of m × (2n − 1) dimensional measurements; x is a 2n − 1 dimensional voltage state vector; v is the m-dimensional measurement error vector, and the elements in the vector conform to the normal distribution of N(0, σ_i²).

When the Jacobian matrix H is measured to be of full Rank and in a good state, that is, it satisfies the condition Rank(H) = 2n − 1, then it is determined that the system satisfies algebraic observability.

(2)

The topology is considerable

From the perspective of graph theory, a power system can be regarded as a graph composed of n vertices and m edges, G = (V, E), where V represents the set of vertices corresponding to the busbars within the system and E represents the set of edges corresponding to the branches within the system. The deployment of the measurement network constitutes a measurement subgraph G′ = (E′, V′), which corresponds to the conditions of V′∈ V and E′∈ E. When the subgraph G′ contains all the points of graph G, it is determined that the system satisfies topological observability.

2.1.2. Configuration Rules Under Observable Conditions

In the power system, there exist zero-injection nodes, which refer to nodes without generators or load connection. The net injection power of these nodes is zero, and they are usually used as transmission nodes, thus having certain particularities.

For a zero-injection node, it and its adjacent nodes can jointly form an N-node set. If N − 1 nodes are observable, then all nodes can be observable through calculation, as shown in Figure 1.

Figure 1. Zero injection node sizable situation conditions.

According to the observability requirements of nodes, the following rules can be obtained:

(1)

When node i is configured with a PMU, the node itself can directly measure and achieve observability.

(2)

When node i is configured with a PMU, the state quantities of its adjacent nodes can be calculated, and the implementation of its adjacent nodes is considerable.

(3)

When node i is injected into a node with zero, in the set of N nodes formed by itself and adjacent nodes, as long as N − 1 nodes achieve observability, the remaining node can be calculated, and all N nodes achieve observability.

2.1.3. Average Channel Index

The analog channels of PMU can be divided into voltage analog channels and current analog channels. Typically, the voltage analog channel is used to measure the voltage phasor at the installation point, while the current analog channel is used to measure the current phasor at the adjacent branch of the installation point. For each PMU, if the number of its connection channels can be kept as few as possible, the entire system will have better stability. Therefore, for the same node system, even if the same number of PMUs are used in different deployment schemes, the deployment locations of different schemes are also worth exploring. In this paper, the final optimal deployment method is selected by constructing the average channel index of PMU, that is: the average number of connected channels = the total number of adjacent channels connected to PMU/the number of PMU installations.

2.2. Improve the Integer Programming Deployment Method After Preprocessing

Compared with heuristic algorithms, integer programming algorithms are deterministic algorithms, and the results they obtain are deterministic optimal solutions. Under normal circumstances, mathematical models are simple and intuitive. To meet actual needs, only the objective function and constraints need to be modified. The traditional deployment method that combines planning algorithms generally takes the observability premise and associated parameters of all nodes within the system as the objective function and constraint conditions. For the target of the power system with the fewest observable PMU devices installed throughout the network and the fewest average number of connected channels, the basic form of the target equation is shown in Equation (2). Its fundamental main objective is to minimize the funds required for the deployment plan as much as possible.

$$\left\{\begin{array}{l}\min {\displaystyle \sum _{\mathit{j}=1}^{\mathit{n}}{\mathit{x}}_{\mathit{j}}}\\ \\ \min {\displaystyle \frac{{\mathit{P}}_{\mathit{w}}}{{\mathit{P}}_{\mathit{n}}}}\end{array}\right.,$$

(2)

In this formula: n represents the total number of nodes in the system, x_j = 1 indicates that a PMU is installed at node j, x_j = 0 indicates that a PMU is not installed, P_w represents the total number of connection channels of the PMU, and P_n represents the number of PMUs installed.

The constraint equation that satisfies the observable condition of non-zero injection nodes is shown in Equation (3).

$$\mathit{F}(\mathit{i})={\displaystyle \sum _{\mathit{j}\in \mathit{i}}{\mathit{a}}_{\mathit{i}\mathit{j}}{\mathit{x}}_{\mathit{j}}}\ge 1,$$

(3)

In this formula, when F(i) ≥ 1, it indicates that the non-zero injection node i can be observed; a_ij is the corresponding element in the node adjacency matrix A. The node adjacency matrix A represents the connection status among various nodes in the power system, and matrix A is an n × n matrix. When node i is connected to node j, a_ij = 1; when node i is not connected to node j, a_ij = 0. That is shown in Equation (4) below.

$${\mathit{a}}_{\mathit{i}\mathit{j}}=\left\{\begin{array}{l}1\\ 1\\ 0\end{array}\right.\begin{array}{c}\mathit{i}=\mathit{j}\\ \mathit{i} \mathrm{is\; connected\; to} \mathit{j}\\ \mathit{i} \mathrm{is\; not\; connected\; to} \mathit{j}\end{array},$$

(4)

The representation of the adjacency matrix A is shown in Equation (5) below.

$$\mathit{A}=\left[\begin{array}{c}{\mathit{A}}_1\\ {\mathit{A}}_2\\ {\mathit{A}}_3\\ ⋮\\ {\mathit{A}}_{\mathit{n}}\end{array}\right]=\left[\begin{array}{ccccc}{\mathit{a}}_{11}& {\mathit{a}}_{12}& {\mathit{a}}_{13}& \dots & {\mathit{a}}_{1\mathit{n}}\\ {\mathit{a}}_{21}& {\mathit{a}}_{22}& {\mathit{a}}_{23}& \dots & {\mathit{a}}_{2\mathit{n}}\\ {\mathit{a}}_{11}& {\mathit{a}}_{11}& {\mathit{a}}_{11}& \dots & {\mathit{a}}_{3\mathit{n}}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathit{a}}_{\mathit{n}1}& {\mathit{a}}_{\mathit{n}2}& {\mathit{a}}_{\mathit{n}3}& \dots & {\mathit{a}}_{\mathit{n}\mathit{n}}\end{array}\right]\begin{array}{c}\\ \\ \\ \\ ,\end{array}$$

(5)

Considering the characteristics of the end node and the zero injection node, certain preprocessing is carried out on the solution model in this paper. From the perspective of the cost of PMU installation, the end node is usually not regarded as an installation node in the power system. When the adjacent nodes meet certain observability conditions, the state parameters of the end node can be obtained through calculation. Therefore, in this paper, it is considered to first identify the end nodes, and then, respectively, identify the situations where the end nodes are connected to regular nodes or zero-injection nodes. For the case of connecting to regular nodes, pre-deployment is set. For the case of connecting to zero-injection nodes, deployment conditions are set in combination with the characteristics of zero-injection nodes, thereby reducing the overall number of constraint conditions.

In terms of algorithmic solution, for the solution set of the deployed schemes, calculate the average number of connection channels for each scheme, and ultimately select the scheme with the smallest average number of channel connections. The overall method process is shown in Figure 2.

Figure 2. Deployment flowchart.

3. Simulation and Results

The simulation in the text runs in the MATLAB R2023b environment. The IEEE-14 node, IEEE-30 node and NE-39 node were, respectively, compared and simulated for verification.

3.1. IEEE-14

IEEE-14 node system as shown in Figure 3.

Figure 3. IEEE-14 node model.

According to node data, the zero-injection node in the IEEE-14 node system is node 7, and the terminal node within the system is node 8. During the preprocessing stage, according to the deployment rules, when all the adjacent nodes of node 7 except node 8 are observable, nodes 7 and 8 meet the observability condition. Therefore, the node association parameter matrix is set to zero for node 8 at this time, and the association matrix is updated. For other nodes, generate constraint conditions based on the node association matrix:

$$\left\{\begin{array}{l}\mathit{F}(1)={\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_5\ge 1\\ ⋮\\ \mathit{F}(7)={\mathit{x}}_4+{\mathit{x}}_7+{\mathit{x}}_9\ge 1\\ \mathit{F}(9)={\mathit{x}}_4+{\mathit{x}}_7+{\mathit{x}}_9+{\mathit{x}}_{10}+{\mathit{x}}_{14}\ge 1\\ ⋮\\ \mathit{F}(14)={\mathit{x}}_9+{\mathit{x}}_{13}+{\mathit{x}}_{14}\ge 1\end{array}\right.\begin{array}{c}\\ \\ \\ \\ \\ ,\end{array}$$

(6)

After solving, the preferred deployment nodes are 2, 6, and 9 nodes. Table 1 shows the comparison of the deployment results of the algorithm in this paper with those of other algorithms.

Table 1. Comparison of IEEE-14 node deployment algorithms.

Algorithm	PMU Deployment Quantity	PMU Deployment Node Location	Average Algorithm Time
Improved algorithm in the text	3	2\6\9	0.02 s
Depth-first search algorithm	6	2\4\6\8\10\14	0.004 s
Improve the simulated annealing algorithm	4	2\6\7\9	0.03 s

It can be seen from the results in Table 1 that under the IEEE-14 node system conditions, the number of PMUs required for deployment in this paper is the least and the average completion time of the algorithm is relatively short.

3.2. IEEE-30

IEEE-30 node system as shown in Figure 4.

Figure 4. IEEE-30 node model.

According to the node data, the zero injection nodes in the IEEE-30 node system are nodes 6, 9, 22, 25, 27, and 28, and the terminal nodes existing in the system are nodes 11, 13, and 26. In the preprocessing stage, according to the deployment rules, for the classification processing of end nodes, the nodes connected to the zero injection node are node 11 and node 26, and the corresponding connected end nodes are node 9 and node 25. Therefore, node 12 is a node that must be deployed, and thus the constraint conditions for nodes 4, 12, 13, 14, 15, and 16 can all be omitted. At this point, the node association parameter matrix sets nodes 11 and 26 to zero and updates the association matrix.

For the remaining other nodes, generate corresponding constraint conditions based on the node association matrix:

$$\left\{\begin{array}{l}{\mathit{x}}_{12}=1\\ \mathit{F}(1)={\mathit{x}}_1+{\mathit{x}}_2+x_3\ge 1\\ \mathit{F}(2)={\mathit{x}}_1+{\mathit{x}}_2+{\mathit{x}}_4+{\mathit{x}}_5+{\mathit{x}}_6\ge 1\\ \mathit{F}(3)={\mathit{x}}_1+{\mathit{x}}_3+{\mathit{x}}_4\ge 1\\ \mathit{F}(5)={\mathit{x}}_2+{\mathit{x}}_5+{\mathit{x}}_7\ge 1\\ ⋮\\ \mathit{F}(30)={\mathit{x}}_{27}+{\mathit{x}}_{29}+{\mathit{x}}_{30}\ge 1\end{array}\right.\begin{array}{c}\\ \\ \\ \\ \\ \\ ,\end{array}$$

(7)

After solving, it is found that under the condition of the same average number of PMU channels, the preferred deployment nodes are 3, 7, 10, 12, 19, 24, and 29 nodes, and the average number of channels is 2.571.

In Table 2, the partial solution sets obtained from the solution and the corresponding average number of channels are listed. Table 3 shows the comparison of the deployment of the method proposed in this paper with other algorithms.

Table 2. Comparison of different scenarios for IEEE-30 node system.

Plan	PMU Deployment Location (First Line)/ Number of PMU Connection Channels (Second Line)	Average PMU Channel
1	3\7\10\12\19\24\29	2.571
	2\2\5\4\1\2\2
2	1\2\10\12\19\24\29	2.857
	2\4\5\4\1\2\2
3	2\4\10\12\19\24\30	3.000
	4\3\5\4\1\2\2

Table 3. Comparison of IEEE-30 node deployment algorithms.

Algorithm	PMU Deployment Quantity	PMU Deployment Node Location	Average Algorithm Time
Improved algorithm in the text	7	3\7\10\12\ 19\24\29	0.02 s
Depth-first search algorithm	10	1\5\6\10\11\12\ 18\24\26\27	0.004 s
Improve the simulated annealing algorithm	7	1\5\10\12\ 18\24\27	0.1 s

As can be seen from Table 2, even for different schemes with the same number of PMU deployments, the average number of channels varies. From Table 3, it can be observed that the improved method proposed in this paper has different advantages and disadvantages over other algorithms in terms of deployment quantity and time. Under the same number of deployments, the average completion time of the algorithm is shorter.

3.3. NE-39 Nodes

NE-39 node system as shown in Figure 5.

Figure 5. NE-39 node model.

According to the node data, the zero injection nodes in the NE-39 node system are nodes 2, 5, 6, 10, 11, 13, 14, 17, 19, and 22, and there are end nodes 30, 31, 32, 33, 34, 35, 36, 37, and 38 within the system. During the preprocessing stage, according to the deployment rules, the end nodes are classified. The end nodes connected to the zero injection nodes are 30, 31, 32, 33, and 35, and the corresponding zero injection nodes connected are 2, 6, 10, 19, and 22. Therefore, nodes 20, 23, 25, and 29 are the nodes that must be deployed. Therefore, the constraints on itself and the adjacent nodes of the corresponding connection can be omitted. At this time, the node association parameter matrix is set to zero for nodes 30, 31, 32, 33, and 35, and the association matrix is updated. For the remaining other nodes, generate constraint conditions based on the node association matrix:

$$\left\{\begin{array}{l}{\mathit{x}}_{20}=1\\ {\mathit{x}}_{23}=1\\ {\mathit{x}}_{25}=1\\ {\mathit{x}}_{29}=1\\ ⋮\\ \mathit{F}(18)={\mathit{x}}_3+{\mathit{x}}_{17}+{\mathit{x}}_{18}\ge 1\\ \mathit{F}(21)={\mathit{x}}_{16}+{\mathit{x}}_{21}+{\mathit{x}}_{22}\ge 1\\ ⋮\\ \mathit{F}(29)={\mathit{x}}_{26}+{\mathit{x}}_{28}+{\mathit{x}}_{29}+{\mathit{x}}_{38}\ge 1,\end{array}\right.\begin{array}{c}\\ \\ \\ \\ \\ \\ \\ \\ \end{array}$$

(8)

After solving, the obtained deployment nodes are 3, 6, 13, 16, 20, 23, 25, 29, and 39 nodes. Table 4 shows the comparison of the deployment results of the algorithm in this paper with those of other algorithms.

Table 4. Comparison of NE-39 node deployment algorithms.

Algorithm	PMU Deployment Quantity	PMU Deployment Node Location	Average Algorithm Time
Improved algorithm in the text	9	3\6\13\16\20\23\25\29\39	0.03 s
Depth-first search algorithm	16	2\4\6\8\10\12\16\18\20\ 22\26\33\36\37\38\39	0.004 s
Improve the simulated annealing algorithm	12	2\8\10\11\16\22\ 26\34\36\37\38\39	0.2 s

It can be seen from the results in Table 4 that under the NE-39 node system condition, the number of PMUs required for deployment in this paper is the least and the average completion time of the algorithm is shorter.

4. Discussion

This study shows that by preprocessing the deployment conditions, the efficiency of the deployment algorithm can be improved to a certain extent. Meanwhile, adjusting the algorithm’s solution mode can make the deployment plan more rational. This research represents a further alteration to the traditional algorithm deployment problem, making the issues to be addressed more organized. These results are of certain help for the study of the PMU deployment field, providing a new approach to solving the PMU deployment optimization problem, rather than being limited to simply replacing different algorithms to change the optimization effect of the PMU deployment problem.

This study provides some new research ideas for the deployment of PMUs, but it also has certain limitations. The current research mainly focuses on the improvement of deployment algorithms under static distributed network conditions. However, in actual application scenarios, there may be situations where the node attributes change due to the variation in switch states. Therefore, future research in this field can consider further in-depth exploration and analysis of deployment issues under complex conditions.

5. Conclusions

This paper mainly studies the optimization problem of PMU deployment. Traditional deployment algorithms have certain room for improvement in the optimization methods of PMU deployment. This paper makes certain improvements on the traditional deployment algorithm. Starting from the observability of the power system, a deployment method that comprehensively considers the preprocessing conditions and the number of PMU connection channels is proposed. On the one hand, compared with traditional methods, this method reduces the overall constraints through preprocessing conditions, thereby improving the performance of the deployment algorithm. On the other hand, the solution method of the algorithm has been changed. For multiple deployment methods, the average number of connection channels is used as the condition for screening the final deployment plan, making the final plan more economical.

(1)

In the test case of the IEEE-14 node model, it can be seen that compared with other deployment algorithms, the improved algorithm only takes 0.02 s to obtain the solution, and the number of deployed PMUs is only 3 to meet the global observability condition.

(2)

In the test case of the EIEE-30 node model, it can be seen that there are multiple solutions obtained by the improved algorithm. It only takes 0.02 s to obtain the optimal solution, and the number of deployed PMUs is only 7 to meet the global observability condition.

(3)

In the test case of the NE-39 node model, it can be seen that the improved algorithm only takes 0.03 s to obtain the solution, and the number of deployed PMUs only needs 9 to meet the global observability condition.

In conclusion, through the comparison of simulation tests, it can be seen that the deployment algorithm, through certain pre-processing of data and the setting of algorithm strategies, can ultimately make the improved method have certain advantages in the solution time and deployment quantity of the PMU deployment optimization scheme.