Distributed Optimal Control of DC Network Using Convex Relaxation Techniques

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1) In section 4.1, it is proposed to compare the distributed optimal control method with the results obtained using the MATLAB CVX Optimization Toolbox. At the same time, it is not specified which optimization method is used in the MATLAB CVX Optimization Toolbox.

2) What does the phrase "Load mutation" means in Figure 7?

3) Section 4 contains a reference to the IEEE 57 bus system. However, further in the text and in the figures, only the IEEE 10 bus-based system is mentioned. I suggest you fix it.

Author Response

Comments 1: In section 4.1, it is proposed to compare the distributed optimal control method with the results obtained using the MATLAB CVX Optimization Toolbox. At the same time, it is not specified which optimization method is used in the MATLAB CVX Optimization Toolbox.  

Response:  Thanks for your helpful comments.  The optimization solver used in the MATLAB CVX Optimization Toolbox is MOSEK,  whose core algorithm is the interior-point method.  We have clarified it in the revised manuscript (see Section 4.1). 

Comments 2:   What does the phrase "Load mutation" means in Figure 7? 

Response:  Thanks for pointing it out, and we have replaced it with "load disturbance" or "load  variations" . Simulation results show that the proposed method can achieve global optimization under load variation without re-solving the power flow problem. (See Fig.7 and section 4.4)  

Comments 3: Section 4 contains a reference to the IEEE 57 bus system. However, further in the text and in the figures, only the IEEE 10 bus-based system is mentioned. I suggest you fix it.

Response: Thanks for pointing it out. We have modified it according to your comments. 

Reviewer 2 Report

Comments and Suggestions for Authors

The authors propose a novel distributed control strategy for DC microgrids using a convex relaxation method to enhance the system's optimal power flow. The research is very interesting and the approach is well defined. The reviewer has the following comments.

Convex relaxation techniques like Second-Order Cone Programming (SOCP) offer mathematical tractability and global optimality for solving power flow problems. However, they also have several disadvantages. For example, 

1. Where important constraints have been simplified or relaxed, can potentially violate physical and operational limits. This affects the usefulness of the SOCP method for real-time decision-making.

2. SoCP has scalability issues, especially because it presents a computation burden as the size of the testbed/network increases. This makes it impractical for large-scale systems, limiting their real-time applicability.

3. Convex relaxations struggle to handle discrete control variables, such as transformer tap settings and shunt switching. These are essential for certain control actions, necessitating hybrid approaches with mixed-integer programming, which increases complexity and computation time.

Recommendation:

The authors should address these issues and specifically highlight the limitation of the proposed approach in the conclusion part and provide the way forward in this sense.

Author Response

Commets 1: Where important constraints have been simplified or relaxed, can potentially violate physical and operational limits. This affects the usefulness of the SOCP method for real-time decision-making. 

Response: Thank you for your insightful comment. We fully acknowledge that convex relaxation is often achieved by simplifying or relaxing nonlinear constraints, which can potentially lead to solutions that violate physical and operational limits. However, for the optimization problem addressed in this paper, we have proven that the convex relaxation is exact. This means that the optimal solution of the relaxed convex problem will always satisfy the original constraints. As a result, the obtained solution will not violate any physical or operational limits, ensuring the applicability of the method in practical scenarios (This content is in Remark 3 in the revised manuscript).

Comments 2: SoCP has scalability issues, especially because it presents a computation burden as the size of the testbed/network increases. This makes it impractical for large-scale systems, limiting their real-time applicability.

Response: Thank you for your insightful comment. We highly agree with it. Solving large-scale SOCP problems is indeed challenging. For the network with n nodes,  the original problem involves n voltage variables (u_i), but after SOCP relaxation, the number of variables (w_{ij}) increases to n(n+1)/2, which significantly increases the computational burden. However, the core idea of this paper is to derive the KKT conditions of the convex optimization problem and transform them into a consensus form (see equation (20)). By utilizing distributed consensus control, the system automatically operates at the optimal point when the optimal factors x_i are consensus, eliminating the need to solve the optimization problem directly. Therefore, the proposed approach effectively addresses the scalability issue of SOCP. (See remark 4 in the revised manuscript. )

Comments 3: Convex relaxations struggle to handle discrete control variables, such as transformer tap settings and shunt switching. These are essential for certain control actions, necessitating hybrid approaches with mixed-integer programming, which increases complexity and computation time. 

Response: Thanks for your insightful comments. We highly agree with it. We will investigate these topics in the future works.  

Comments 4:  The authors should address these issues and specifically highlight the limitation of the proposed approach in the conclusion part and provide the way forward in this sense. 

Response: Thanks for your helpful comments. We have highlighted the limitation of the proposed approach in the conclusion part as follows. 

The proposed distributed control strategy using convex relaxation techniques offers advantages in mathematical tractability and global optimality. However, it has limitations in handling complex physical constraints, discrete control variables, and scalability for large systems. Two key limitations should be noted: the method assumes linear loads, limiting its application to nonlinear load systems, and it relies on real-time load data, which poses practical challenges. Future work will focus on hybrid algorithms that combine convex relaxation with Mixed-Integer Programming (MIP) to handle discrete variables, and explore distributed optimization and parallel computing to improve scalability. Future research will address these issues by adapting the approach for the optimization model for more practical constraints and nonlinear loads.