1. Introduction
As the scale and complexity of power systems continue to expand, the safety and stability of their operation have become important topics in electrical engineering research [
1,
2,
3]. Throughout the operation of power systems, factor such as load fluctuations [
4,
5], equipment failures, the uncertainty of renewable energy [
6], or power market transactions could lead to branch flow exceedances, thereby threatening the safety and stability of the entire power system [
7,
8]. Therefore, effectively predicting and controlling branch flow exceedances is key to enhancing the safety and economic efficiency of the grid [
9,
10].
Sensitivity analysis and distribution factor calculations are significant technical methods in power systems. Sensitivity analysis is a crucial tool for studying how changes in input parameters affect system performance and stability [
11], while the distribution factor can predict the impacts of specific network operations on the flow distribution [
12].
Many studies currently utilize sensitivity analysis. In the context of integrating renewable energies, the literature [
13] indicates that the correlation between renewable sources such as wind and solar significantly affects the voltage stability of power systems. The higher the correlation, the greater the sensitivity of renewable energy generation to voltage stability. Reference [
14] proposes a method for wind capacity sensitivity analysis in large-scale grid-connected wind power for multi-energy, strongly coupled integrated energy systems based on unified optimal power flow. Using a sensitivity matrix, the study analyzes the impact of the capacity of the energy-coupling units on wind power absorption capacity. This method provides auxiliary information for integrated energy systems’ safe and stable operation. Reference [
15] introduces a method for calculating the delay margin sensitivity to system parameters. By incorporating the normalization condition of eigenvectors into the differential characteristic equation, a set of sparse linear equations is solved to obtain the delay margin sensitivity to any given parameter, suitable for various power system models and control issues. Reference [
16] conducts a sensitivity analysis on hydropower units under comprehensive operational conditions, showing that the mechanical parameters of generators are most sensitive under power control modes, while electrical parameters are more sensitive under frequency control modes. This analysis aids in understanding the stability and safety of hydropower operations.
Currently, distribution factor is mainly used in power trading research. Literature [
17] provides an algorithm that calculates the cost of electrical energy in transmission networks using alternating current power transfer distribution factors. Reference [
18] introduces a new application of PTDFs in deregulated electricity markets for transmission pricing assessment, establishing a “hybrid analysis and decision framework” for transmission costs and pricing. Reference [
19] proposes an enhanced STF-LODF method to address transmission line congestion. Pricing and sensitivity methods are used for price management while managing the optimal location and voltage instability issues associated with blocking transmission.
Concerning branch power flow exceedance elimination, the literature [
3] employs a branch switching method to identify candidate branches for switching, thereby eliminating branch flow exceedances. Reference [
20] establishes a risk assessment system based on probabilistic flow calculations, identifying key risk branches with flow exceedances. This is crucial for enhancing grid stability and preventing major blackout incidents. Literature [
21] proposes a method that utilizes reinforcement learning and sensitivity analysis to overcome the local optima issues common in artificial intelligence methods, thereby correcting flow exceedance behaviors in grid branches. Reference [
22] develops an optimization-based program to determine the maximum power flow that each branch in the network can withstand, helping to identify branches prone to exceedances.
The methods for branch power flow exceedance elimination discussed in the above literature focus on different aspects and achieve corresponding results. However, they face challenges such as complex modeling, large computational loads, and difficulty in implementation. This paper applies a sensitivity analysis and distribution factors to the issue of branch power flow exceedance and proposes a branch exceedance elimination method based on sensitivity weighting, building upon the foundation of the particle swarm optimization (PSO) algorithm. This method features high operational efficiency, effective optimization results, and suitability for practical engineering applications, enabling rapid and accurate prediction and management of branch power flow exceedances in power systems.
Firstly, the paper introduces the basic theory and computational methods of sensitivity analysis, focusing on power flow sensitivity and voltage sensitivity calculation principles. Applying these theories not only helps us predict changes in power flow within the power network but also provides a scientific basis for the planning and operational optimization of power systems. Subsequently, the paper details the derivation of the calculation principles for the line outage distribution factor (LODF), which is crucial for assessing and predicting the distribution of the power flow in a power grid under specific circumstances. Finally, it utilizes the proposed method for branch power flow exceedance elimination and validates its efficiency and practicality through case studies. This method can handle branch power flow exceedance in power systems more efficiently, ensuring the stable operation of the grid and the safety of power transactions.
2. Sensitivity Calculation
Sensitivity refers to the degree to which a change in one quantity causes a change in a response quantity and is an important indicator of the numerical relationship between these two quantities. Sensitivity analysis is widely applied in power system networks. Through sensitivity calculations, it is possible to predict the power flow and voltage in weak links of the network before equipment is added or removed, thus preventing potential network failures due to improper actions. Additionally, if a fault has already occurred in the system, sensitivity analysis can be utilized to quickly and effectively identify the best strategies for restoring network operations.
2.1. Basic Principles of Sensitivity Calculation
The process of a sensitivity calculation is based on power flow calculations. The calculation process is derived by utilizing the constraint equations and relationships from the power flow analysis, combined with the definitions of various sensitivities.
For any power system network, the characteristic operational equation can be expressed as follows:
In this equation, f(x, u) = 0 represents the node power constraint equation, reflecting the conservation of node power; y = y(x, u) relates the bus voltages to line power flows; u is the control variables, such as generator output power, bus voltage, and the set values of active and reactive power at each node, classified as independent variables; x is the state variables, such as the magnitude and phase angle of the node voltages, classified as dependent variables; and y is the dependent variables that depend on changes in state variables and control variables, such as line power.
When the dependent variables are directly related only to the state variables, Equation (1) can be rewritten as follows:
Assuming the system operates in a steady state, where the state variables and control variables are
, then, in this state, we have the following:
To derive the relationships between the control variables and state variables, Equation (2) is Taylor expanded [
23] around
, and higher-order terms are ignored, resulting in the following:
By substituting Equation (3) into Equation (4), the following can be obtained:
Upon organizing Equation (5), we can derive the following:
In this equation,
Lm represents the sensitivity matrix for changes in
x due to the changes in
u, and
Ln represents the sensitivity matrix for changes in
y due to changes in
u. The specific relationships are as follows:
2.2. Power Flow Sensitivity
Power flow sensitivity reflects the change in line power flow caused by changes in the injected power of the node. Depending on the type of injected power, power flow sensitivity can be divided into active power sensitivity and reactive power sensitivity. Additionally, the observed line power flow can be categorized into active power, reactive power, and apparent power. Therefore, power flow sensitivity is defined as the change in the active power, reactive power, and apparent power of the line caused by the change in active power or reactive power injected into the node.
According to the basic principles of sensitivity, in the calculation of power flow sensitivity,
u acts as the control variable, which is the power injected at the nodes by the load and generators;
x acts as the state variable, which is the bus voltage under steady state conditions;
y acts as the dependent variable, which represents the line power flow. Thus,
f(
x,
u) = 0 is equivalent to the unbalanced quantity in power flow calculation, reflecting the constraints between node injected power and node voltage;
y =
y(
x) represents the relationship between node voltage and line power flow, and using node voltage, the corresponding line power flow can be determined. Therefore,
is the Jacobian matrix,
J, in the power flow calculation [
24].
In
,
f represents the unbalance quantity, noted as follows:
and
u represents the node-injected power, noted as follows:
Derived from the power flow equation of the power system, the following can be obtained:
where
represents the partial derivative of the line power flow with respect to the node voltage, obtained from the power flow equations in rectangular coordinates, as follows:
Taking the partial derivatives of each term in the rectangular coordinate phasor
of the node voltage with respect to
, we can obtain the following:
Substituting Equation (11) into Equation (12), we obtain the following:
Substituting Equations (10)–(12) into Equation (7), we can obtain the following:
In this equation, represents the active power sensitivity for line i-j at node x, and represents the reactive power sensitivity for line i-j at node x, where the real part represents the observed active power, and the imaginary part represents the observed reactive power.
The apparent power
, whose value is
. The formula derivation for the power flow sensitivity of the apparent power as the observed object is as follows:
In this equation, A represents the injected power at the node, which can be either active power or reactive power.
Taking the IEEE 5-bus case as an example, the IEEE 5-bus case consists of 5 nodes and 5 branches, with the branches in the forward direction being 1–5, 2–4, 2–5, 3–4, and 4–5, respectively. Node 1 serves as the slack bus, and its structural topology and network data are shown in
Figure 1.
According to the analytical principles of sensitivity calculation, it is necessary to compute three partial derivatives: , , and .
For , this represents the partial derivative of the node imbalance quantity with respect to the node voltage, which corresponds to the Jacobian matrix in power flow calculations.
For , this represents the partial derivative of the node imbalance quantity with respect to the node injected power. In the IEEE 5-bus case, node 3 is a PV node, and its voltage imbalance quantity’s derivative with respect to the initial injected power is 0, while the rest are 1.
For
, this represents the partial derivative of the line flow with respect to the node voltage, which can be derived and calculated accordingly.
In these results, the odd rows represent the sensitivity when there is a unit change in the active power injection at a node, known as the active power sensitivity; the even rows represent the sensitivity when there is a unit change in the reactive power injection at a node, known as the reactive power sensitivity. It should be noted that in the calculation results, each term is in a complex form, where the real part represents the active power and the imaginary part represents the reactive power.
2.3. Voltage Sensitivity
Voltage sensitivity reflects the changes in line power flows caused by changes in the magnitude of the generator node voltages. Similarly observed objects include active power, reactive power, and apparent power. Therefore, voltage sensitivity is defined as follows: when there is a change in the voltage magnitude at a generator node, it causes changes in the active, reactive, and apparent power flows in the line.
Voltage sensitivity analysis is similar to power flow sensitivity analysis, with the only difference being in the control variables u. For voltage sensitivity, u represents the voltage at the generator node.
For , thi is also the Jacobian matrix, J, used in power flow calculations.
For
,
f represents the unbalanced quantity, which is still noted as Equation (8).
u is the node voltage of the generator, and for convenience in calculating the value of
, let
u represent the voltage magnitude at all nodes, noted as follows:
To find the relationship between the unbalanced quantity and voltage magnitude, represent the node voltage in polar coordinates and derive the expression for the unbalanced quantity in polar coordinates as follows:
In this equation, if node i represents a PV node, then should be replaced with 0.
Deriving the partial derivative of the unbalanced quantity with respect to the voltage magnitude at the nodes yields a new matrix, denoted as follows:
In this matrix, 0 represents the partial derivative of the reactive unbalance at equilibrium nodes with respect to the voltage magnitude. The expressions for other elements in the matrix are as follows:
- 1.
When i = j:
- 2.
When i ≠ j:
When
i is a PV node, and
j is any node, the following applies:
where
represents the partial derivative of the line power flow with respect to node voltage, consistent with the calculation of the power flow sensitivity, aligning with the reasoning process in Equation (12).
By incorporating the Jacobian matrix from the Newton–Raphson power flow calculation method with Equations (12) and (18) into Equation (7), we can obtain the following:
Following the above calculation process, the voltage sensitivity of each node to the power flow in the lines can be determined. However, it should be noted that typically only the slack and PV nodes have generators connected, so the voltage sensitivity calculated at the PQ nodes may be considered irrelevant and can be disregarded. Additionally, each element in Equation (22) is divided into real and imaginary parts, where the real part represents the observed active power, and the imaginary part represents the observed reactive power. Similar to the power flow sensitivity, when the observed object is apparent power, the derivations can also be based on Equation (15), in which represents the sensitivity of the line’s active power to node voltage, and represents the sensitivity of the line’s reactive power to node voltage.
Taking the IEEE 5-bus case as an example, again, similar to power flow sensitivity, it is necessary to calculate the three partial derivatives , , and . Considering that the only difference between voltage sensitivity and power flow sensitivity lies in the control variables, the calculations for and are the same as those for power flow sensitivity; only needs to be computed.
For , this is the partial derivative of the node imbalance quantity with respect to the voltage amplitude at the generator nodes. From the principle analysis, to understand the relationship between the imbalance quantity and the voltage amplitude, the node voltage needs to be represented in polar coordinates. Referring to Equations (18)–(21), the derivative of the imbalance quantity with respect to the voltage amplitude is derived and, by substituting the data, the derivative for all nodes can be calculated. For the slack node, its power imbalance with respect to the voltage amplitude is generally not considered and should be omitted. Also, it is important to note that typically only the slack and PV nodes have generators connected, so the voltage sensitivity calculated at PQ nodes is irrelevant and can be disregarded. Therefore, for this case, the result of this derivative ultimately has only two columns of data.
After calculating these three partial derivatives, the data are substituted into Equation (22) to obtain the following:
where the two data points represent the voltage sensitivities of line 1–5 with respect to nodes 1 (slack node) and 3 (PV node), respectively. Here, the real part represents the observed object as active power, and the imaginary part as reactive power.
3. Distribution Factor Calculation
The study of distribution factor is based on power flow equations operating at steady-state points, and from this basis, it explores the relationships among variables of interest. These variables often reflect the operating conditions of the power network and may arise from changes in bus active power injections, adjustments in inter-regional power transactions, or the opening and closing of branches. Through distribution factor, one can clearly and intuitively observe the impact of various power transactions on the system [
25,
26].
Any change in the status of a branch, whether it is opened or closed, will cause a change in the power flow distribution of the entire system, which requires an indicator to measure the degree to which each component of the system is affected by changes in the state of the branch. line outage distribution factor (LODF) reflects the change in active power flow in other branches when a specific line in the power network is disconnected. LODF uses the change in the power flow of the selected line as a baseline to calculate the percentage change in the power flow of other components relative to this baseline, typically used to indicate the impact of the disconnection of one branch on the power flow in other branches of the network.
The LODF is generally calculated using the direct current method. Let the active power flow in branch
l under the base state be denoted as
; if branch
l is disconnected, the power flow on branch
k will change accordingly. This change is denoted as
, and the relationship between them is as follows:
In this equation, represents the line outage distribution factor.
In the analysis using the direct current method, the sensitivity equation relating the change in active power injection to the change in the angular displacement of node voltages can be derived [
27], as follows:
In this equation,
B is the admittance matrix obtained under the DC method, and
X is the inverse matrix of
B. In the construction of matrix
B, it is assumed that the reactance in the line (
i,
j) is much greater than the resistance, and all shunt branches to ground are neglected, with the transformer per-unit ratios consistently considered to be 1. The elements in the matrix are as follows:
Note that when the line’s starting or ending point is a slack node, both its self-admittance, , and mutual admittance,, are 0.
Assuming that the disconnection of a branch does not cause a change in the injected power at the nodes, before the disconnection, taking a stable state as the baseline, the change in active power at each node is 0, and the line power flow in the branch,
, is denoted as
, with the direction of the power flow going from
i to
j. When the branch
is disconnected, changes occur in the power at the nodes on both ends of the branch; the change at node
i is denoted as
and at node
j as
. Hence, the change in injected power at the new network nodes is the following:
Which can be expressed as the following:
In this equation, is the node–branch incidence vector, where rows correspond to node numbers and columns to branches. Assuming the direction of line power flow is from the starting node toward the ending node, the value of at the starting node of the branch is set to 1, indicating only power injection there; at the ending node, it is set to −1, indicating only power output; the other values are 0, representing a balance of injection and output.
For example, in the IEEE 5-bus case, it is possible to represent the following:
By excluding the slack node, the result can be expressed as follows:
In the matrix M, each column represents the node–branch association column vector corresponding to the branch in that column.
When the branch
in the network is disconnected, the new network’s admittance matrix will also change, specifically,
and
will be reduced by the admittance
of branch
l;
and
become 0, which is equivalent to adding the admittance
of branch
l. Using
, the modification to the admittance matrix can be represented as follows:
The new admittance matrix then becomes the following:
Substituting Equation (29) into Equation (24) results in the following:
For the line
, before disconnection, the line power flow is the following:
When the line is disconnected, changes in the node voltage angles lead to a change in the line power flow to the following:
Combining Equations (27), (29), (30), and (32), the change in power flow for line
k is as follows:
From Equation (23), the expression for the LODF is the following:
Using the matrix inversion lemma
, Equation (34) can be written as follows:
Letting
, then it is as follows:
where
represents the mutual impedance between nodes
k and
l;
represents the self-impedance at port
l.
5. Conclusions
This study, through the integrated use of sensitivity analysis and distribution factors, has designed a branch exceedance elimination method based on sensitivity-weighted PSO algorithm. This method offers the advantages of high operational efficiency, effective optimization results, and ease of application in practical engineering projects, playing a key role in preventing and addressing branch flow exceedances in power systems.
The article thoroughly analyzes and derives power flow sensitivity, voltage sensitivity, and LODF. Sensitivity analysis enables the prediction of power flow responses at various nodes within the power system due to power or voltage changes, providing a scientific basis and priority determination for generator output scheduling after branch exceedances. Additionally, the calculation of the distribution factor offers a quantitative assessment of the impact on the power flow distribution of the power system under specified circumstances, enhancing the predictive capability for system stability. Through the LODF results, it is possible to more accurately identify key lines that may affect system stability following a specific line fault, thus providing a scientific basis for grid operations. This helps prioritize which lines require special attention in emergency response plans. Additionally, this also emphasizes the need to consider the interdependencies among lines when designing and implementing grid operation strategies, particularly in highly interconnected networks.
The article demonstrates how the sensitivity and distribution factor can predict and control the branch power flow exceedance in the power system through simulation of actual power network cases. By combining theory and simulation, the designed method was verified to effectively and practically address branch power flow exceedance issues in power systems, also highlighting its practical application value in power system optimization and fault recovery processes. Through this research, the paper provides an effective analytical and operational strategy for branch power flow management in a power system, contributing significantly to enhancing the safety and reliability of power systems both theoretically and practically.