CN107846023A - A kind of improved network loss duty value direct current optimal power flow computational methods - Google Patents

Abstract

The invention discloses an improved DC optimal power flow calculation method for network loss equivalent load, which includes the following steps: (1) introducing equivalent line loss resistance to ground at both ends of the line of the optimal power flow DC model to form network loss, etc. (2) Starting from the AC optimal power flow model, deduce the active power loss formula of the line; (3) Carry out polynomial fitting on the trigonometric function term in the active power loss formula, and use System characteristics Eliminate the voltage amplitude term in the formula; (4) Equivalently distribute the active power loss of the line to the equivalent resistance to ground at both ends of the line to form an improved DC optimal power flow model for equivalent loads of network losses. The accuracy and high efficiency of the model of the present invention are verified in the test. The method provided by the present invention effectively improves the calculation accuracy of the network loss equivalent load model, also ensures the calculation efficiency of the model, and improves the practicality of the network loss equivalent load model. value.

Description

An Improved Calculation Method for DC Optimal Power Flow of Network Loss Equivalent Load

technical field

The invention relates to a linearized optimal power flow calculation method of a power system, in particular to an improved DC optimal power flow calculation method for net loss equivalent load.

Background technique

Optimal power flow calculation (optimal power flow, OPF) was first proposed by the French scholar Carpentier in the 1960s, and it is an important means to ensure the safe and economical operation of the power system. The AC optimal power flow model (alternating current optimal power flow, ACOPF) has strong nonlinear characteristics, and the coupling between its variables is very tight, which leads to low calculation efficiency of the model and cannot meet the online real-time calculation of large-scale systems. need. Therefore, it is particularly important to find a suitable linearized optimal power flow calculation model.

Direct current optimal power flow (DCOPF) is currently the fastest linearized optimal power flow calculation model. However, because the model ignores the influence of network loss factors, it cannot effectively characterize the power flow characteristics of the system, so that the balance node will bear all the difference between the system's power generation and load, which will lead to unreasonable power distribution of the system, resulting in calculation errors of the model too large to be put into practical application.

The study of the DC model considering the network loss is beneficial to improve the calculation accuracy while ensuring the efficiency of the model solution. At present, the relatively complete DC model is the network loss equivalent load DC model, which simulates the influence of network loss on the system by introducing equivalent resistance. However, the existing models mostly use the relationship between the system active power and apparent power to describe the line loss approximately. This method involves the calculation of the proportional factor of the line apparent power amplitude and active power amplitude in the equivalent process. Since this parameter is a constantly changing quantity in the iterative process, the existing models often set it as a fixed value, which has certain shortcomings, which will affect the calculation accuracy of the network loss equivalent load model.

Contents of the invention

Purpose of the invention: Aiming at the above problems, the present invention proposes an improved DC optimal power flow calculation method for network loss equivalent load.

Technical solution: In order to achieve the purpose of the present invention, the technical solution adopted in the present invention is: an improved DC optimal power flow calculation method for network loss equivalent load, including steps:

(1) Introduce the equivalent line loss resistance to ground at both ends of the line of the optimal power flow DC model to form a network loss equivalent load model, and deduce the resistance value of the ground resistance;

(2) Starting from the AC optimal power flow model, deduce the active power loss formula of the line;

(3) Carry out polynomial fitting to the trigonometric function term in the active loss formula, and use the system characteristic to eliminate the voltage amplitude term in the formula;

(4) The active power loss of the line is equivalently distributed to the equivalent resistance to ground at both ends of the line to form an improved DC optimal power flow model for equivalent loads of network losses.

In the step (1), the active power loss of the equivalent ground resistance in the DC optimal power flow model is:

In the formula, P _equ,i and P _equ,j are the active power losses of the equivalent ground resistance on nodes i and j, U _i and U _j are the voltage amplitudes of nodes i and j, r _equ,ij are the two The equivalent resistance to ground at the terminal; P _loss,ij is the active power loss of the branch;

Take U _i ＝U _j ＝1, get:

r _equ,ij =2/P _loss,ij (2).

In the step (2), the active power expression in the AC optimal power flow model is:

In the formula, i and j are the node numbers at both ends of the line, θ _ij = θ _i - θ _j is the voltage phase angle difference between node i and node j, θ _i is the voltage phase angle of node i, θ _j is the voltage of node j Phase angle, g _ij and b _ij are the conductance and susceptance of the line respectively, and P _ij is the active power flowing from node i to j;

The line active power loss of the AC optimal power flow model is:

In described step (3), polynomial fitting is carried out to the trigonometric function item in active loss formula:

Substitute formula (5) into formula (4):

In the formula, U _ij =U _i -U _j is the voltage amplitude difference at both ends of the line;

Because U _ij << θ _ij , ignoring the U _ij item in formula (6):

In the step (4), the power balance equation in the DC optimal power flow model is:

In the formula, ΔP _i is the active power imbalance of node i, P _Gi is the active output of generator i, P _Di is the active load of node i, is the node admittance matrix element established by the reciprocal of branch reactance, θ _j is the voltage phase angle of node j;

Combining formula (1), formula (2) and formula (7), the network loss equivalent load P _equ,i of node i can be obtained as:

Subtracting the network loss equivalent load of each node, the power balance equation in the improved network loss equivalent load model is:

In the formula, f(x) is the objective function of the optimal power flow, n _b is the number of nodes, n _g is the number of generators, a _2i , a _1i and a _0i are the consumption characteristic parameters of the i-th generator; θ _i is the voltage phase angle of node i; θ _i , is the lower limit and upper limit of the voltage phase angle of node i; P _Gi is the active output of the i-th generator; P _Gi , It is the lower limit and upper limit of the active output of the i generator.

Beneficial effects: The improved network loss equivalent load DC optimal power flow model of the present invention re-deduces the active power loss formula from the ACOPF model, and uses the system operating characteristics to eliminate the voltage amplitude item in the active power loss formula, making it suitable for DCOPF The model eliminates the dependence of the existing network loss equivalent load model on the accuracy of the scale factor of apparent power and active power, effectively improves the calculation accuracy of the model, ensures the calculation efficiency of the model, and effectively improves the network loss, etc. The range of applicability of the value loading model.

Description of drawings

Figure 1 is a schematic diagram of a network loss equivalent load model.

Detailed ways

The technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

As shown in Figure 1, it is the network loss equivalent load model, which improves the DC optimal power flow to improve the calculation accuracy of the DC optimal power flow. The main idea of the network loss equivalent load model is to consider the line loss factor on the basis of the DC optimal power flow model, and evenly distribute the line loss as the loss of the equivalent resistance to ground at both ends of the line. In Figure 1, r _equ,ij is the equivalent resistance to ground at both ends of the access branch; P _loss,ij is the active loss of the branch.

In the DC optimal power flow model of network loss equivalent load, U _i ≈ U _j ≈ 1, where U _i and U _j are the voltage amplitudes of each node respectively. Therefore, when r _equ,ij =2/P _loss,ij , the active power consumed by each equivalent ground resistance is (1/2)P _loss,ij , and at this time, satisfy P _ij =P _ji +P _{loss ,ij} , P _ij is the active power flowing from node i to j, which is in line with the actual power flow of the line. Therefore, as long as a reasonable line loss expression is found, the calculation accuracy of the DC optimal power flow model can be effectively improved.

The existing methods mostly use the relationship between the active power and apparent power of the line to describe the line loss approximately. This method involves the calculation of the proportional factor between the apparent power amplitude and the active power amplitude of the line in the equivalent process. Since this parameter is a constantly changing quantity in the iterative process, and the existing methods mostly set it as a fixed value, there are certain deficiencies, which will affect the calculation accuracy of the network loss equivalent load model.

The present invention re-deduces the active power loss formula of the line starting from the AC optimal power flow model, adopts polynomial fitting for the trigonometric function items in the AC optimal power flow model to effectively equivalence, and uses the operating characteristics of the system to eliminate the voltage in the active power loss formula The magnitude term makes it suitable for the DC optimal power flow model. Apply the calculation formula of active power loss derived above to the DC optimal power flow model of network loss equivalent load, thereby eliminating the dependence of the existing network loss equivalent load model on the accuracy of the scale factor of apparent power and active power, effectively improving The calculation accuracy of the model is improved.

The improved DC optimal power flow calculation method for network loss equivalent load proposed by the present invention specifically includes steps:

(1) Deduce the DC optimal power flow model, and introduce the equivalent line loss resistance to the ground at both ends of the line of the DC model to form the equivalent load model of the network loss, and deduce the value of the ground resistance r _equ,ij ;

As shown in Figure 1, the active power loss of the equivalent resistance to ground can be written as:

In the formula, P _equ,i and P _equ,j are the active power losses of the equivalent ground resistance on nodes i and j respectively, U _i and U _j are the voltage amplitudes of each node respectively, r _equ,ij is the access branch The equivalent resistance to ground at both ends of the circuit; P _loss,ij is the active power loss of the branch circuit.

In DCOPF, U _i ≈U _j ≈1, so there is 1/r _equ,ij =0.5P _loss,ij , so:

r _equ,ij =2/P _loss,ij (2)

(2) Starting from the AC optimal power flow model, deduce the active power loss formula of the line;

In the ACOPF model, ignoring the line-to-ground impedance, its active power can be expressed as:

In the formula, i and j are the node numbers at both ends of the line, θ _ij = θ _i - θ _j is the voltage phase angle difference between node i and node j, θ _i is the voltage phase angle of node i, θ _j is the voltage of node j Phase angle, g _ij and b _ij are the conductance and susceptance of the line respectively, and P _ij is the active power flowing from node i to j.

According to the definition of line loss, the active loss of line ij in the AC model is:

(3) Carry out polynomial fitting to the trigonometric function term in the loss formula, and use the system characteristic to eliminate the voltage amplitude term in the formula;

Equation (4) contains some trigonometric function terms, which makes the coupling of voltage amplitude and phase angle very tight, which is not convenient for simplification of the equation. Therefore, according to the characteristic that the phase angle difference at both ends of the system line is usually between -π/6 and π/6, the present invention uses the MATLAB fitting toolbox to fit it.

The specific fitting process is to sample 100 points at equal intervals between -π/6 and π/6, store their values in matrix X, and find the cosine of each point value, store them in matrix Y, and The matrices X and Y are imported into the MATLAB fitting toolbox, so that the following equivalent relationship can be obtained:

For the convenience of subsequent expression, let 0.49=C. Substituting formula (5) into formula (4) can get:

In the formula, U _ij =U _i -U _j is the voltage amplitude difference at both ends of the line.

Since the node voltage of the power system is generally maintained at about 1.0pu during operation, the value of the voltage amplitude difference between the beginning and the end of the line is relatively small. After a large number of calculation examples, it can be obtained that U _ij << θ _ij , so the U _ij term in formula (6) can be ignored. At this time, formula (6) can be written as:

(4) Equivalently allocate the line loss to the equivalent ground resistance at both ends of the line to form an improved network loss equivalent load DC optimal power flow model;

In the DC optimal power flow model, the power balance constraint can be written as:

In the formula, ΔP _i is the active power imbalance of node i, P _Gi is the active output of generator i, P _Di is the active load of node i, is the node admittance matrix element established by the reciprocal of branch reactance, θ _j is the voltage phase angle of node j.

It can be seen from Figure 1 that the network loss equivalent load model is equivalent to introducing the network loss equivalent load on each node on the basis of the DC model. Therefore, the power balance equation in the network loss equivalent load model needs to subtract the The network loss equivalent load.

Combining formula (1), formula (2) and formula (7), the network loss equivalent load P _equ,i of node i can be obtained as:

After the above derivation, the improved network loss equivalent load DC optimal power flow model can be expressed in the following form:

In the formula, f(x) is the objective function of the optimal power flow, n _b is the number of nodes, n _g is the number of generators, a _2i , a _1i and a _0i are the consumption characteristic parameters of the i-th generator; θ _i is the voltage phase angle of node i; θ _i , is the lower limit and upper limit of the voltage phase angle of node i; P _Gi is the active output of the i-th generator; P _Gi , It is the lower limit and upper limit of the active output of the i generator.

(5) Verify the accuracy and efficiency of the model in the test set.

In order to verify the advantages of the present invention compared with the prior art, the calculation accuracy of the model of the present invention is verified, and at the same time, the improved DC optimal load model based on the network loss equivalent load model is compared with the comparative document 1 (He Tianyu, Wei Zhinong, Sun Guoqiang, etc. Power flow algorithm [J]. Power System Automation, 2016, 40(6): 58-64) and the DCOPF model for comparison. For the convenience of description, the ACOPF model is defined as AC, the DCOPF model is defined as DC, the model described in Comparative Document 1 is defined as M_1, and the model described in the present invention is defined as M_2.

The present invention uses the primal dual interior point method to solve each model, and implements algorithm programming on the MATLAB 2014a platform. Carry out numerical example tests on IEEE 300-node system, Polish 2383-node system, Polish 2736-node system and a large 8304-node system.

Table 1

Table 1 gives the calculation results of AC, DC, M_1 and M_2, where the relative error refers to the relative error between the model and the AC model. It can be seen from Table 1 that M_1 and M_2 can effectively improve the calculation accuracy of the DC model. At the same time, because M_2 expresses the approximate formula of the line loss more accurately, the model effectively improves the calculation accuracy of the network loss equivalent load model, and successfully controls the calculation error within five thousandths. For the IEEE 300 node system and Polish 2736 node system, M_2 even controls the error to about 5/10,000.

Table 2

Table 2 shows the calculation time required for different models to solve the OPF problems of different systems, so as to verify the influence of the line active loss calculation method of the present invention on the calculation model solution efficiency. It can be seen from the results that thanks to the powerful computing power of MATLAB, the AC model can effectively converge within 50 iterations and 7s when solving the OPF problems of most systems. However, for the 8304-node large system tested by the present invention, the AC model needs to be iterated 796 times, and the convergence time is about 165s, which is far beyond the requirements of online applications for computational efficiency. Therefore, the present invention has important practical significance for the exploration of the DCOPF model. Among several models, since the DC model greatly simplifies the system power balance equation, it has the least number of iterations and the highest calculation efficiency, and the efficiency of the model is also significantly improved for small systems with a small number of nodes. The number of iterations and calculation time of M_1 and M_2 are basically the same, which shows that the method of the present invention will not affect the solution efficiency of the network loss equivalent load model. Since there are residual nonlinear parts in both models, the number of iterations and calculation time are slightly longer than those of the DC model. But even so, its improvement of OPF calculation efficiency is still very significant. Even for a large system with 8304 nodes, the calculation time is still controlled within 2s, which is 99% lower than that of the AC model.

In summary, M_2 effectively improves the calculation accuracy of the network loss equivalent load model, and does not affect its solution speed, so it has higher practical application value than M_1.

Claims (5)

1. An improved network loss equivalent load DC optimal power flow calculation method, characterized in that: comprising the steps: (1) Introduce the equivalent line loss resistance to ground at both ends of the line of the optimal power flow DC model to form a network loss equivalent load model, and deduce the resistance value of the ground resistance; (2) Starting from the AC optimal power flow model, deduce the active power loss formula of the line; (3) Carry out polynomial fitting to the trigonometric function term in the active loss formula, and use the system characteristic to eliminate the voltage amplitude term in the formula; (4) The active power loss of the line is equivalently distributed to the equivalent resistance to ground at both ends of the line to form an improved DC optimal power flow model for equivalent loads of network losses. 2. The improved network loss equivalent load DC optimal power flow calculation method according to claim 1, characterized in that: in the step (1), the active loss of the equivalent ground resistance in the DC optimal power flow model is: <mrow><msub><mi>P</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mfrac><msubsup><mi>U</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>r</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub></mfrac><mo>=</mo><msub><mi>P</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mfrac><msubsup><mi>U</mi><mi>j</mi><mn>2</mn></msubsup><msub><mi>r</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>P</mi><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow> In the formula, P _equ,i and P _equ,j are the active power losses of the equivalent ground resistance on nodes i and j, U _i and U _j are the voltage amplitudes of nodes i and j, r _equ,ij are the two The equivalent resistance to ground at the terminal; P _loss,ij is the active power loss of the branch; Take U _i ＝U _j ＝1, get: r _equ,ij =2/P _loss,ij (2). 3. The improved network loss equivalent load DC optimal power flow calculation method according to claim 1, characterized in that: in the step (2), the active power expression in the AC optimal power flow model is: <mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>U</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><msub><mi>g</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>U</mi><mi>i</mi></msub><msub><mi>U</mi><mi>j</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>U</mi><mi>i</mi></msub><msub><mi>U</mi><mi>j</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow> In the formula, i and j are the node numbers at both ends of the line, θ _ij = θ _i - θ _j is the voltage phase angle difference between node i and node j, θ _i is the voltage phase angle of node i, θ _j is the voltage of node j Phase angle, g _ij and b _ij are the conductance and susceptance of the line respectively, and P _ij is the active power flowing from node i to j; The line active power loss of the AC optimal power flow model is: <mrow><msub><mi>P</mi><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><msubsup><mi>U</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>U</mi><mi>j</mi><mn>2</mn></msubsup><mo>)</mo></mrow><mo>-</mo><mn>2</mn><msub><mi>g</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>U</mi><mi>i</mi></msub><msub><mi>U</mi><mi>j</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>.</mo></mrow> 4. the improved network loss equivalent load direct current optimal power flow calculation method according to claim 1, is characterized in that: in described step (3), polynomial fitting is carried out to the trigonometric function term in active loss formula: <mrow><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>-</mo><msubsup><mi>C&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow> Substitute formula (5) into formula (4): <mrow><msub><mi>P</mi><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>U</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>Cg</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>&amp;theta;</mi><mrow><mi>i</mi>mi><mi>j</mi></mrow><mn>2</mn></msubsup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow> In the formula, U _ij =U _i -U _j is the voltage amplitude difference at both ends of the line; Because U _ij << θ _ij , ignoring the U _ij item in formula (6): <mrow><msub><mi>P</mi><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>2</mn><msub><mi>Cg</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mo>.</mo></mrow> 5. The improved network loss equivalent load DC optimal power flow calculation method according to claim 1, characterized in that: in the step (4), the power balance equation in the DC optimal power flow model is: <mrow><msub><mi>&amp;Delta;P</mi><mi>i</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>D</mi><mi>i</mi></mrow></msub><mo>+</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mi>b</mi></msub></mrow>munderover><msubsup><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow><mo>*</mo></msubsup><msub><mi>&amp;theta;</mi><mi>j</mi></msub><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow> In the formula, ΔP _i is the active power imbalance of node i, P _Gi is the active output of generator i, P _Di is the active load of node i, is the node admittance matrix element established by the reciprocal of branch reactance, θ _j is the voltage phase angle of node j; Combining formula (1), formula (2) and formula (7), the network loss equivalent load P _equ,i of node i can be obtained as: <mrow><msub><mi>P</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><munder><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>&amp;Element;</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>&amp;NotEqual;</mo><mi>i</mi></mrow></munder><mrow><mo>(</mo><msub><mi>P</mi><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>,</mo><mi>i</mi><mi>j</mi></mrow></msub><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><munder><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>&amp;Element;</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>&amp;NotEqual;</mo><mi>i</mi></mrow></munder><mrow><mo>(</mo><msub><mi>Cg</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msubsup><mi>&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mrow> Subtracting the network loss equivalent load of each node, the power balance equation in the improved network loss equivalent load model is: <mfenced open = "" close = ""><mtable><mtr><mtd><mrow><mi>o</mi><mi>b</mi><mi>j</mi></mrow></mtd><mtd><mrow><mi>min</mi><mi></mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mi>g</mi></msub></munderover><mrow><mo>(</mo><msub><mi>a</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><msubsup><mi>P</mi><mrow><mi>G</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mi>a</mi><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mi>P</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>a</mi><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mn>mfenced> <mrow><mtable><mtr><mtd><mrow><mi>s</mi><mo>.</mo><mi>t</mi><mo>.</mo></mrow></mtd><mtd><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msub><mi>&amp;Delta;P</mi><mi>i</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>D</mi><mi>i</mi></mrow></msub><mo>+</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mi>b</mi></msub></munderover><msubsup><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow><mo>*</mo></msubsup><msub><mi>&amp;theta;</mi><mi>j</mi></msub><mo>-</mo><msub><mi>P</mi><mrow><mi>e</mi><mi>q</mi><mi>u</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo></mrow></mtd><mtd><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2...</mn><msub><mi>n</mi><mi>b</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><munder><mi>P</mi><mo>&amp;OverBar;</mo></munder><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>&amp;le;</mo><msub><mi>P</mi><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>&amp;le;</mo><msub><mover><mi>P</mi><mo>&amp;OverBar;</mo></mover><mrow><mi>G</mi><mi>i</mi></mrow></msub><mo>,</mo></mrow></mtd><mtd><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2...</mn><msub><mi>n</mi><mi>g</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><munder><mi>&amp;theta;</mi><mo>&amp;OverBar;</mo></munder><mi>i</mi></msub><mo>&amp;le;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub><mo>&amp;le;</mo><msub><mover><mi>&amp;theta;</mi><mo>&amp;OverBar;</mo></mover><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2...</mn><msub><mi>n</mi><mi>b</mi></msub></mrow></mtd></mtr></mtable></mfenced></mtd></mtr></mtable><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>10</mn><mo>)</mo></mrow></mrow> In the formula, f(x) is the objective function of the optimal power flow, n _b is the number of nodes, n _g is the number of generators, a _2i , a _1i and a _0i are the consumption characteristic parameters of the i-th generator; θ _i is the voltage phase angle of node i; θ _i , is the lower limit and upper limit of the voltage phase angle of node i; P _Gi is the active output of the i-th generator; P _Gi , It is the lower limit and upper limit of the active output of the i generator.