Small-Signal Stability Constrained Optimal Power Flow Model Based on BP Neural Network Algorithm

Round 1

Reviewer 1 Report

1.      The research gap, motivation, and objective are not properly addressed in the literature survey.

2.      The contribution of the work is poor. No contributory mathematical formulations have been found. Most of the mathematical formulations are available in the literature.

The literature review section a be improved with this journal works

https://doi.org/10.1007/s10614-017-9716-2

https://doi.org/10.1016/j.rser.2021.111295

https://doi.org/10.1016/j.est.2019.101057

https://doi.org/10.1007/s12530-019-09271-y

https://doi.org/10.1002/er.6891

https://doi.org/10.1007/s12652-017-0600-7

https://doi.org/10.1016/j.renene.2019.05.008

3.      Although the author has presented the detailed steps of objective functions, the outcome of the work is not appreciable in comparison with the quality of the works published on this topic.

4.      The authors have not justified the reason for choosing the algorithm.

 

5.      The serious drawback with the paper is that there is no comparison of findings with already existing works.

Author Response

Response to Reviewer 1 Comments

 

Point 1: The research gap, motivation, and objective are not properly addressed in the literature survey.

 

Response 1: We thank you for reminding us of this important point. We have added research gap, motivation, and goals to the introduction based on reviewers' suggestions and updated the manuscript with yellow highlighting. The modifications were as follows:

Nowadays, the research of SC-OPF has attained some developments. The literature [9] considered the small-signal stability constraint that the real part of eigenvalue was less than 0, and used the original dual interior point algorithm. The disadvantage of this method is that the calculation is too complicated on the eigenvalue sensitivity. In [10], the real part sensitivity of the eigenvalues was calculated by the numerical method. In [11], a nonlinear semi-definite programming (NLSDP) model was presented to deal with the small-signal stability, which was converted into nonlinear programming and obtained by the inner point method. Li et al. [12] presented a Sequential Quadratic Programming (SQP) approach for SC-OPF to solve the difficulty of convergence. It improved the convergence of the solution by sampling the vicinity of the iteration point each iteration. The literature [13] proposed a sequential approach using sub-problems to handle the original optimization problem. Deepak et al. [14] developed two different convex relaxation approaches, ad-dressing the non-convexity of optimization issues owing to the presence of the nonlinear constraint.

According to the above literature, the gap in the existing research on SC-OPF is major in how to get the small-signal stability index quickly and easily through the system oper-ating parameters in the optimization. The conventional SC-OPF method has to derive the eigenvalue sensitivity calculation formula. It is very difficult for complex systems to derive directly and use the formula during the iterative process, which takes up massive memory and time throughout the optimization. With the development of Artificial Intelligence (AI), AI algorithms have been widely used in power system optimization techniques to effi-ciently process data and implement complex calculations. The literature [15] combines the cuckoo search (CSA) with sunflower optimization (SFO) to improve the performance of OPF solutions. The literature [16] adopts deep reinforcement learning (DRL) to derive rapid and effective OPF decisions. In [17], a Convolutional Neural Network (CNN) is es-tablished to solve OPF problem in the distribution networks. The literature [18] proposes a Deep Neural Network (DNN) method to ensure the feasibility of the OPF-generated solu-tion. Through the successful application of AI techniques in OPF, we believe that the com-bination of AI and SC-OPF has broad prospects, so we try to add AI in SC-OPF to fill the gap of existing study.

 

Point 2: The contribution of the work is poor. No contributory mathematical formulations have been found. Most of the mathematical formulations are available in the literature.

 

Response 2: Thank you very much for the references, which we have carefully studied. According to your suggestion, we have made some modifications in the introduction and and updated the manuscript with yellow highlighting. The main novelty and contributions of this work is listed as follows:

• To overcome the shortcomings of the traditional algorithm that the derivation of small-signal stability index is complicated and computationally intensive, the AI al-gorithm is introduced to solve the SC-OPF issue in this study.
• To determine the minimum damping ratio and the first-order eigenvalue sensitivity from the optimal system, BP neural network is successfully employed.
• The simulation cases on the WSCC-9 bus and IEEE-39 bus test system validate that the BP-SCOPF can achieve optimal solutions.
• The results are compared to previous power flow calculation and economic schedul-ing linear programming method, which shows the superior performance of BP-SCOPF in dealing with small-signal stability constraint.

 

Point 3: Although the author has presented the detailed steps of objective functions, the outcome of the work is not appreciable in comparison with the quality of the works published on this topic.

 

Response 3: We greatly appreciate the reviewer’s comments. The innovation of this paper is mainly in improving the optimization process of traditional method, reducing the complexity of sensitivity expression formula in the SC-OPF. Our future study will aim to larger power systems, consider renewable energy sources on the BP-SCOPF model, and strive to optimize BP-SCOPF to give full play to its maximum value.

In addition, according to your suggestion, we have done more work about BP-SCOPF algorithm in Section 4. We have added more test cases with different loads in the WSCC-9 bus system to compare the results of BP-SCOPF and conventional method in the Section 4.1, and updated the manuscript with yellow highlighting. The modifications were as follows:

In order to further demonstrate the good performance of BP-SCOPF model, we compare five cases of different loads in the WSCC-9 bus system. Table 3 specifies the different loads, and Table 4 shows the comparison of the BP-SCOPF and conventional method. From the results, the proposed model can ensure the small-signal stability of the system after changes of loads, and has good economics without the need for a large increase in generator output.

Point 4: The authors have not justified the reason for choosing the algorithm.

 

Response 4: We would like to thank the reviewer for this insightful comment. Based on your suggestion, the reason for choosing the algorithm have been highlighted in the introduction. The modifications are as follows：

Back Propagation (BP) neural network is one of the most extensively used AI algo-rithms [19,20]. It has received considerable attention, and various efforts have focused on it since the 1980s [21,22]. Now the BP algorithm has good generalization, strong robustness and rich theoretical basis, and does well in nonlinear problems exceeding other algorithms [23,24], so it can accurately obtain the non-linear fitted relationship between the system variables and the key eigenvalues in assessing small-signal stabil-ity[25,26]. The traditional eigenvalue analysis methods require a large number of derivations and tedious calculations, in contrast, the advantage of the BP method is reducing the complexity, because there is no derivation of the eigenvalue formula, and the corresponding sensitivity is obtained directly after training. Therefore, we creatively propose to incorporate a BP neural network into SC-OPF, where the BP algorithm is used to calculate the first-order eigenvalue sensitivity during the iteration.

 

Point 5: The serious drawback with the paper is that there is no comparison of findings with already existing works.

 

Response 5: We thank you for your careful reading of our paper and providing us with some keen scientific insight. According to your suggestion, we have compared the proposed method in this paper with existing works in the introduction and case study, and updated the manuscript with yellow highlighting. The modifications are as follows：

Nowadays, the research of SC-OPF has attained some developments. The literature [9] considered the small-signal stability constraint that the real part of eigenvalue was less than 0, and used the original dual interior point algorithm. The disadvantage of this method is that the calculation is too complicated on the eigenvalue sensitivity. In [10], the real part sensitivity of the eigenvalues was calculated by the numerical method. In [11], a nonlinear semi-definite programming (NLSDP) model was presented to deal with the small-signal stability, which was converted into nonlinear programming and obtained by the inner point method. Li et al. [12] presented a Sequential Quadratic Programming (SQP) approach for SC-OPF to solve the difficulty of convergence. It improved the convergence of the solution by sampling the vicinity of the iteration point each iteration. The literature [13] proposed a sequential approach using sub-problems to handle the original optimization problem. Deepak et al. [14] developed two different convex relaxation approaches, addressing the non-convexity of optimization issues owing to the presence of the nonlinear constraint.

According to the above literature, the gap in the existing research on SC-OPF is major in how to get the small-signal stability index quickly and easily through the system oper-ating parameters in the optimization. The conventional SC-OPF method has to derive the eigenvalue sensitivity calculation formula. It is very difficult for complex systems to derive directly and use the formula during the iterative process, which takes up massive memory and time throughout the optimization. With the development of Artificial Intelligence (AI), AI algorithms have been widely used in power system optimization techniques to effi-ciently process data and implement complex calculations. The literature [15] combines the cuckoo search (CSA) with sunflower optimization (SFO) to improve the performance of OPF solutions. The literature [16] adopts deep reinforcement learning (DRL) to derive rapid and effective OPF decisions. In [17], a Convolutional Neural Network (CNN) is es-tablished to solve OPF problem in the distribution networks. The literature [18] proposes a Deep Neural Network (DNN) method to ensure the feasibility of the OPF-generated solu-tion. Through the successful application of AI techniques in OPF, we believe that the com-bination of AI and SC-OPF has broad prospects, so we try to add AI in SC-OPF to fill the gap of existing study.

Back Propagation (BP) neural network is one of the most extensively used AI algo-rithms [19,20]. It has received considerable attention, and various efforts have focused on it since the 1980s [21,22]. Now the BP algorithm has good generalization, strong robustness and rich theoretical basis, and does well in nonlinear problems exceeding other algorithms [23,24], so it can accurately obtain the non-linear fitted relationship between the system variables and the key eigenvalues in assessing small-signal stability[25,26]. The traditional eigenvalue analysis methods require a large number of derivations and tedious calculations, in contrast, the advantage of the BP method is reducing the complexity, because there is no derivation of the eigenvalue formula, and the corresponding sensitivity is obtained directly after training. Therefore, we creatively propose to incorporate a BP neural network into SC-OPF, where the BP algorithm is used to calculate the first-order eigenvalue sensitivity during the iteration.

Reviewer 2 Report

1. The concept of the paper is very good. 

2. The literature part is not well written. Modify this part with providing recent papers.

3. The contribution of the paper needs to be highlighted clearly in the introduction part of the paper.

4. Result parts need to be improved by providing brief explanations.

5. Compare the results with other established methods.

6. The conclusion section is not supported by the results. Modify these parts.

7. The standard of the languages is also average. Modify this.

Author Response

Response to Reviewer 2 Comments

 

Point 1: The concept of the paper is very good. 

 

Response 1: We gratefully thank you for your approval of our paper! In our future study, we will further apply the proposed methods to solve SC-OPF problem in larger power systems, consider renewable energy sources in the BP-SCOPF model, and strive to optimize BP-SCOPF so as to give full play to its maximum value.

 

Point 2: The literature part is not well written. Modify this part with providing recent papers.

 

Response 2: We thank you for your careful reading of our paper and providing us with some keen scientific insight. According to your suggestion, we have added three recent pieces of literature about AI combined OPF for reference and corrected inappropriate words in the manuscript. The modifications were as follows:

Nowadays, the research of SC-OPF has attained some developments. The literature [9] considered the small-signal stability constraint that the real part of eigenvalue was less than 0, and used the original dual interior point algorithm. The disadvantage of this method is that the calculation is too complicated on the eigenvalue sensitivity. In [10], the real part sensitivity of the eigenvalues was calculated by the numerical method. In [11], a nonlinear semi-definite programming (NLSDP) model was presented to deal with the small-signal stability, which was converted into nonlinear programming and obtained by the inner point method. Li et al. [12] presented a Sequential Quadratic Programming (SQP) approach for SC-OPF to solve the difficulty of convergence. It improved the convergence of the solution by sampling the vicinity of the iteration point each iteration. The literature [13] proposed a sequential approach using sub-problems to handle the original optimization problem. Deepak et al. [14] developed two different convex relaxation approaches, ad-dressing the non-convexity of optimization issues owing to the presence of the nonlinear constraint.

According to the above literature, the gap in the existing research of SC-OPF is major in how to get the small-signal stability index quickly and easily through the system opera-tion parameters in the optimization calculation. Conventional SC-OPF method has to de-rive the eigenvalue sensitivity calculation formula. It is very difficult for complex systems to derive directly and use the formula repeatedly during the iterative process, which takes up massive memory and time throughout the optimization. With the development of Arti-ficial Intelligence (AI), AI algorithms have been widely used in power system optimization techniques to efficiently process data and implement complex calculations. The literature [15] combines the cuckoo search (CSA) with sunflower optimization (SFO) to improve the performance of OPF solutions. The literature [16] adopts deep reinforcement learning (DRL) to derive rapid and effective OPF decisions. In [17], a Convolutional Neural Net-work (CNN) is established to solve OPF problem in the distribution networks. The litera-ture [18] proposes a Deep Neural Network (DNN) method to ensure the feasibility of the OPF generated solution. Through the successful application of AI in OPF, we believe that the combination of AI and SC-OPF has broad prospects, so we try to add AI in SC-OPF to fill the gap of existing study.

The Back Propagation (BP) neural network is one of the most extensively used AI al-gorithms [19,20]. It has received considerable attentions, and various works have focused on it since the 1980s [21,22]. Now the BP algorithm has good generalization, strong robustness and rich theoretical basis and does well in nonlinear problems exceeding other algorithms [23,24], so it can obtain the non-linear fitted relationship between the system variables and the key eigenvalues accurately in assessing small-signal stability[25,26]. The traditional eigenvalue analysis methods require a large number of derivation and tedious calculations, in contrast, the advantage of the BP method is reducing the complexity, because there is no derivation of the eigenvalue formula, and the corresponding sensitivity is obtained directly after training. Therefore, we creatively propose to incorporate a BP neural network into SC-OPF, where the BP algorithm is used to calculate the first-order eigenvalue sensitivity during the iteration.

 

Point 3: The contribution of the paper needs to be highlighted clearly in the introduction part of the paper.

Response 3: Thank you very much for your constructive suggestion. We have made some modifications in the introduction according to the reviewer’s suggestion and updated the manuscript with yellow highlighting. The novelty and contributions of this study have been highlighted in the introduction. The modifications are as follows：

This method is abbreviated as BP-SCOPF. The main contributions of this work is listed as follows:

• To overcome the shortcomings of the traditional algorithm that the derivation of small-signal stability index is complicated and computationally intensive, the AI al-gorithm is introduced for the first time to solve the SC-OPF issue in this study.
• To determine the minimum damping ratio and the first-order eigenvalue sensitivity from the optimal system, BP neural network is successfully employed.
• The simulation cases on the WSCC-9 bus and IEEE-39 bus test system validate that the BP-SCOPF can achieve optimal solutions.
• The results are compared to previous power flow calculation and economic schedul-ing linear programming method, which shows the superior performance of BP-SCOPF in dealing with small-signal stability constraint.

 

Point 4: Result parts need to be improved by providing brief explanations.

 

Response 4: We thank you for reminding us of this important point. According to your suggestion, we discuss how direction and magnitude of the change in operating parameters modified in the Section 3.2, and updated the manuscript with yellow highlighting. The modifications were as follows:

The constraint of the damping ratio is added in the process of solving the correction equation, and the sensitivity of the minimum damping ratio to generator output is used to predict the direction and magnitude of generator output variation so that the constraint on the degree of variation of operating parameters can be obtained without crossing the boundary and converging after several iterations. Besides, we have made the following modifications in Section 4.1 to give brief explanations:

Table 2 shows the comparison results in the standard load. PF is the power flow cal-culation that only considers the power balance. The minimum damping ratio of PF is 0.0057, which fails to satisfy the safety requirement of 0.03. OPF is the conventional opti-mal power flow, and the total active power decreases 2MW, but the minimum damping ratio is also reduced to 0.0034, which is detrimental to the long-term stable operation. In the solution of BP-SCOPF, the total active power is between the results of the above meth-ods and the critical minimum damping ratio is improved to 0.0302. The approximate sensitivity to generator over 30 iterations is shown in Figure 6. The sensitivity fluctuates widely and there is no definite pattern. The direction and magnitude of the generator changes are determined by the value of sensitivity. The sensitivity to PG2 is counted as negative 21 times, so the PG2 drops to the lower bound in the result. The sensitivity to PG3 is all larger than 0 and the sensitivity to PG1 fluctuates above and below 0, thus PG3 in-creases more notably than PG1. The optimal solution verifies the feasibility of BP-SCOPF method.

 

Point 5: Compare the results with other established methods

 

Response 5: We would like to thank the reviewer for this insightful comment. Based on your suggestion, we have added more test cases in the WSCC-9 bus system to compare the results of BP-SCOPF with the conventional method in the Section 4.1, and updated the manuscript with yellow highlighting. The modifications were as follows:

To further demonstrate the good performance of BP-SCOPF model, we compare five cases of different loads in the WSCC-9 bus system. Table 3 specifies the different loads, and Table 4 shows the comparison of the BP-SCOPF and conventional methods. From the results, the critical damping ratio of BP-SCOPF model all exceeds 0.03, with a total power increase of less than 2 MW over the OPF. This explains that BP-SCOPF can ensure the small-signal stability of the system after load changes, and has good economics without the need for a large increase in generator output.

Point 6: The conclusion section is not supported by the results. Modify these parts.

 

Response 6: We greatly appreciate the reviewer’s comments. According to your suggestion, we have modified the conclusion section and updated the manuscript with yellow highlighting. The modifications were as follows:

This article proposes the BP-SCOPF model considering the economics and stability of the power system. The BP neural network is firstly applied to the SC-OPF optimization step, which fills the algorithm of solving eigenvalue sensitivity problem in the OPF field. Given the complex sensitivity expression formula in the traditional interior point method, BP neural network is introduced to optimize the calculation of sensitivity, and it makes the small-signal stability index and sensitivity more concise and flexible. The experi-mental results on the 9-bus and 39-bus systems show that the BP-SCOPF model can im-prove the critical minimum damping ratio to 0.03 and calculate the optimal solution to meet the small-signal stability. BP algorithm accurately fits the mapping relationship be-tween the generator power and the minimum damping ratio of the system and rapidly calculates the approximate sensitivity of the critical damping ratio in the correction equa-tion during the BP-SCOPF iteration, which shows BP algorithm has the features of good nonlinear fitting, fast-fitting speed and high precision. Compared with LPC method, the BP-SCOPF method has a superior capability to break out of the partial extreme value and conduct a global search. Our future study will aim to larger power systems, consider re-newable energy sources on the BP-SCOPF model, and strive to optimize BP-SCOPF to give full play to its maximum value.

 

Point 7: The standard of the languages is also average. Modify this.

 

Response 7: We thank you for your careful reading of our paper and providing us with some keen scientific insight. We have refined the whole paper, removed inappropriate words and sentences, tried to be specific and accurate in the description of the model, and added more analysis and comparison of the results in the case study.

Reviewer 3 Report

1.     1.Why activation function not applied for output layer (Equation 9). Also provide the type of activation function used for hidden layer.

2.  Mention the reason for taking different thresholds that is  and  for output and hidden layers. What is the threshold values used for hidden and output layers?

3. Provide the comparison table of   obtained through constructed BP-SCOPF and conventional method for some loading conditions taken for test cases. If possible provide the computational timings also.

4.  Some more explanation is needed on how generation rescheduling takes place with approximate sensitivity method due to violation of small signal stability index i.e. how direction and magnitude of the change in operating parameters modified.

5.   Provide equation numbers in the algorithm steps and flow chart of BP-SCOPF.

6. From equation 14 the inputs are active power ?_?? and reactive power ?_?? of generator, active power ?_??and reactive power ?_??of node load, active power ?_?? and reactive power ?_?? of line. Since there are three generators, three loads and three transmission lines, the number of input neurons should be 18. Why 30 is taken and what are the other inputs.

 

 

 

Author Response

Response to Reviewer 3 Comments

 

Point 1: Why activation function not applied for output layer (Equation 9). Also provide the type of activation function used for hidden layer.

 

Response 1: We gratefully thank you for the precious time the reviewer spent making constructive remarks. In the BP model, Equation 9 is used to calculate the output value of the rth neuron in the output layer. The output variable is the minimum damping ratio in this model, so the number of output layers is 1, and the solution can be obtained without the activation function when using Equation 9. The type of activation function used for hidden layer is the Sigmoid activation function. The Sigmoid is the most widely used class of activation functions and has an exponential shape, which is closest to a neuron in a physical sense. The output of the sigmoid has good properties and can be represented as a probability or used for normalization. According to your suggestion, We have added this clarification in the Section 2.2 and updated the manuscript with yellow highlighting.

 

Point 2: Mention the reason for taking different thresholds that is and for output and hidden layers. What is the threshold values used for hidden and output layers?

 

Response 2: We thank you for your careful reading of our paper and providing us with some keen scientific insight. In a BP neural network, the number of input layers is determined by the number of input variables, and the number of the output layers is determined by the number of output variables. The number of the hidden layers is determined by the simulation model. The basic principle of selecting the number the hidden layers is as follows: the number the hidden layers is as small as possible while satisfying the accuracy. Bwcause if the number of nodes in the hidden layer is too small, the network will not be able to establish complex judgment boundaries, train a suitable network, and have poor error tolerance, but if the number is too large, the training time will be too long, the generalization ability of the network will be reduced, and the error may not be optimal. After many extensive trials, the best performance of BP model is obtained when the number of hidden layers is 9 and the number of output layers is 1.

 

Point 3: Provide the comparison table of obtained through constructed BP-SCOPF and conventional method for some loading conditions taken for test cases. If possible provide the computational timings also.

Response 3: Thank you very much for your constructive suggestion. According to your suggestion, we have added some test cases with different loads in the WSCC-9 bus system to compare the results of BP-SCOPF and conventional method in the Section 4.1, and updated the manuscript with yellow highlighting. The modifications in Section 4.1 were as follows:

In order to further demonstrate the good performance of BP-SCOPF model, we compare five cases of different loads in the WSCC-9 bus system. Table 3 specifies the different loads, and Table 4 shows the comparison of the BP-SCOPF and conventional method. From the results, the proposed model can ensure the small-signal stability of the system after changes of loads, and has good economics without the need for a large increase in generator output.

	PL1/MW	PL2/MW	PL3/MW
Case0	125	90	100
Case1	110	110	110
Case2	130	100	120
Case3	150	110	130
Case4	160	115	125

Table 3. Five cases of different loads in the WSCC-9 bus system.

Table 4. Comparison of three power flow calculation with different loads.

	PF		OPF		BP-SCOPF
	ζPF	ΣPGi/MW	ζOPF	ΣPGi/MW	ζBP	ΣPGi/MW
Case0	0.0057	319.64	0.0034	317.64	0.0302	319.19
Case1	0.0065	334.49	0.0054	332.92	0.0306	334.23
Case2	0.0071	354. 41	0.0056	353.17	0.0303	354. 31
Case3	0.0083	395.66	0.0067	393.98	0.0305	394.98
Case4	0.0105	406.06	0.0074	404.31	0.0308	405.29

 

The modifications in Section 4.3 were as follows:

Table 10 provides the computational time of the 9-bus and 39-bus systems in the standard load conditions. For a deterministic system, only one training of the BP is required, as the load and generator variations are already considered in the samples, as described in Section 4.1, which reduces the memory launching the bp algorithm. The total BP-SCOPF time of 39-bus system is 19.14s, compared with 157.38s for the conventional method in the literature [30], which proves that the calculation speed of the proposed method is more faster

Table 10. The computational time of the procedure iterations

	BP training /s	Iteration round	Average iteration /s	Total BP-SCOPF/s
9-bus system	1.36	30	0.17	5.12
39-bus system	2.01	58	0. 33	19.14

 

Point 4: Some more explanation is needed on how generation rescheduling takes place with approximate sensitivity method due to violation of small signal stability index i.e. how direction and magnitude of the change in operating parameters modified.

 

Response 4: We thank you for reminding us of this important point. According to your suggestion, we discuss how direction and magnitude of the change in operating parameters modified in the Section 3.2, and updated the manuscript with yellow highlighting. The modifications were as follows: The constraint of the damping ratio is added in the solving process, and the approximate sensitivity of the minimum damping ratio to generator output is used to predict the direction and magnitude of generator output variation, according to generation rescheduling methods[29] so that the constraint on the degree of variation of operating parameters can be obtained without crossing the boundary and converging after several iterations.

[29] Chung, C. Y.; Wang, L.; Howell, F.; Kundur, P. Generation rescheduling methods to improve power transfer capability constrained by small-signal stability. IEEE Transactions on Power Systems, 2004,19(1), 524-530. doi: 10.1109/TPWRS.2003.820700

 

Point 5: Provide equation numbers in the algorithm steps and flow chart of BP-SCOPF.

 

Response 5: We would like to thank the reviewer for this insightful comment. Based on your suggestion, we have updated the algorithm steps and flow chart of BP-SCOPF in Section 3 and revised the manuscript with yellow highlighting. The modifications were as follows:

This article uses the BP neural network to optimize calculation of eigenvalue sensitivity in the SC-OPF. The specific realization steps are as follows:

• Read the system operation data from text files.
• Change the generator output under the principle of system power balance to get the input and output of the BP model, as Equation (14).
• Train and test the BP model using samples, judging the fitting performance according to the curve, further, Equation (12) and (13) are as error analysis evaluation indicators.
• Take the Equation (1) as the objective function of SC-OPF and set the inequality constraints and small-signal stability constraint, as Equations (3) and (6).
• Carry on BP-SCOPF iterative computation, with calculating eigenvalue sensitivity by BP algorithm, as show in Equation (16). Predict direction and magnitude of operating parameters by the approximate sensitivity, and then optimize variables during the iterative process.
• Check the eigenvalues and the minimum damping ratio. If all prespecified constraints are satisfied, stop the run and output the optimal solution; otherwise, return to step 3.

The model framework is shown in Figure 2.

Figure 2. The Flow chart of BP-SCOPF Model.

 

Point 6: From equation 14 the inputs are active power ??? and reactive power ??? of generator, active power ???and reactive power ???of node load, active power ??? and reactive power ??? of line. Since there are three generators, three loads and three transmission lines, the number of input neurons should be 18. Why 30 is taken and what are the other inputs.

 

Response 6: We greatly appreciate the reviewer’s comments. The inputs are active power ??? and reactive power ??? of generator, active power ??? and reactive power ??? of node load, active power ??? and reactive power ??? of all transmission lines. The output is the corresponding minimum damping ratio. As shown in Figure 3, there are three generators (G1,G2,G3), three loads(PL1, PL 2, PL 3) and nine transmission lines. These nine transmission lines respectively are: B2-B7, B7-B8, B8-B9, B9-B3, B7-B5, B5-B4, B9-B6, B6-B4, B4-B1. Considering the active and reactive power, so the number of input neurons should be 30.

Figure 3. Topology diagram of the WSCC-9 bus power system.

Round 2

Reviewer 2 Report

The authors have completed all the revisions. The paper can be accepted in its present form.