Decentralized Stochastic Recursive Gradient Method for Fully Decentralized OPF in Multi-Area Power Systems

Abstract

This paper addresses the critical challenge of optimizing power flow in multi-area power systems while maintaining information privacy and decentralized control. The main objective is to develop a novel decentralized stochastic recursive gradient (DSRG) method for solving the optimal power flow (OPF) problem in a fully decentralized manner. Unlike traditional centralized approaches, which require extensive data sharing and centralized control, the DSRG method ensures that each area within the power system can make independent decisions based on local information while still achieving global optimization. Numerical simulations are conducted using MATLAB (Version 1.0.0.1) to evaluate the performance of the DSRG method on a 3-area, 9-bus test system. The results demonstrate that the DSRG method converges significantly faster than other decentralized OPF methods, reducing the overall computation time while maintaining cost efficiency and system stability. These findings highlight the DSRG method’s potential to significantly enhance the efficiency and scalability of decentralized OPF in modern power systems.

1. Introduction

Theoptimal power flow (OPF) is the critical and fundamental concept of the electrical power system. The main objective of OPF is to minimize the distribution line losses, generation, and voltage deviation for a power system. Conventional power systems are constantly converting into microgrids that provide localized and autonomous renewable energy sources (RESs) for power generation and distribution. Optimal operation of RESs-based MG is a challenging task due to its the nonlinear nature and high penetration of distributed RESs [1]. Generally, a classic approach is used for MGs which focuses on optimal operation of single region without considering the interaction of other regions [2]. In classic OPF, mathematical programming and heuristic methods are used for optimal operation. OPF of power systems is employed in the literature by means of mathematical programming methods that include quadratic programming [3], linear programming [4], mixed integer linear programming, and nonlinear programming [5]. Heuristic methods provide better optimal solutions within a reasonable time framework for large-scale networks as compared to the above techniques [6]. Different heuristic methods like the genetic algorithm, particle spam optimization, ant colony, Tabu search, simulated annealing, and swarm intelligence are used to solve the OPF for large-scale networks [7]. Most of these techniques are suitable for centralized MGs and are not suitable for multi-area optimal power flow (MAOPF) [8]. MAOPF has become an attractive field with the surge in the popularity of renewable energy sources. MAOPF has interconnected power systems comprising multiple geographic areas [9].

In MAOPF, the optimal operation of any region is interconnected with other regions, which makes it a more complex and computationally intensive optimization problem [10]. MAOPF faces the issues of data privacy and independent decision making for all areas that are addressed through a decentralized optimization approach. The decentralized approach needs to exchange the boundary data between connected regions and reduce the computational complexity by dividing it into localized subproblems with local decision making [11]. Dual decomposition methods are usually used to address the problems of MAOPF in decentralized manners. Lagrangian relaxation optimization techniques are commonly used to solve MAOPF problems in a decentralized manner by constructing the decomposable dual problem through relaxing the multi-area constraints into an objective function approach [12]. Practically, the convergence of Lagrangian relaxation is slow due to susceptible updating of Lagrangian multipliers [13]. The performance of Lagrangian relaxation is enhanced through augmented Lagrangian relaxation, which enforces the constraints more rigorously by introducing penalty terms into the decomposable dual problem. The analytical target cascading optimization method is presented in [14,15], which divides the MAOPF problem into two hierarchical levels: Level 1 for the entire power system and Level 2 for the individual region. Convergence and efficiency of this process is ensured by deriving Level 1 objectives from the independently optimized results of Level 2. This technique has complexity and convergence challenges during optimization of a highly interconnected system and is suitable for a hierarchical decentralized optimization framework [16]. The auxiliary problem principle is a valuable technique applicable for wide-range MAOPF. It solves the problem by transforming MAOPF problems into unconstrained optimization problems augmented with penalty terms [17].

The alternating direction multiplier method is presented in [18], which combines the decomposability of dual decomposition with the superior convergence properties of the method of multipliers. Optimality condition decomposition is another decentralized method that is suitable for both linear and nonlinear problems. Its performance depends on network partition since it involves decomposing the MAOPF problem using an approximate Newton direction [19]. An approximate dynamic programming-based decentralized solution for MAOPF is presented in [20]. In this method, based on Bellman’s equation, problems of MAOPF are decomposed into different subproblems, and the interaction of subproblems is defined with the approximation of value functions. Piecewise linear functions and Benders cut-based mathematical techniques are used to approximate the value functions. Benders cuts have better performance for decomposition of large-scale MAOPF and enhance convergence over piecewise linear functions [21], but these techniques face difficulties in addressing the nonlinear problems of MAOPF. A parallel dual dynamic programming-based decentralized algorithm is proposed in [22] to enhance the performance of decentralized MGs that perform better convergence to tackle the nonlinearity of MAOPF. The penetration of renewable sources is not considered in this study, which cause the power fluctuations. In [23], an improved Lagrangian and consistency algorithm is presented by considering line security constraints for decentralized OPF. This approach often encounters convergence difficulties in non-convex optimization problems, which are prevalent in power systems [24]. The literature on decentralized optimal power flow (OPF) in multi-area power systems discusses several advanced techniques, including the quadratic Taylor expansion-based ADP algorithm [25], distributed interior point methods [12], SDP-based D-ADMM [26], and multistage distributionally robust optimization [27]. A comparison table of several decentralized OPF methods is presented in

Table 1. Evaluation of different decentralized OPF approaches with emphasis on convergence, scalability, and privacy.

Technique	Advantages	Disadvantages
Auxiliary Problem Principle [28]	Simplifies the MAOPF problem by relaxing inter-area constraints	Slow convergence due to the iterative update of Lagrangian multipliers
Augmented Lagrangian Relaxation [13]	Enhances convergence by adding penalty terms	Higher computational complexity [29]
Analytical Target Cascading [15,16]	Suitable for hierarchical systems and improves convergence via decomposition	Convergence and complexity challenges for highly interconnected systems [30]
Auxiliary Problem Principle  [18,31]	Converts MAOPF into an unconstrained optimization problem with penalties	Computational burden increases with the size and complexity of the system [32]
ADMM [33]	Combines decomposability with superior convergence properties	Still requires significant data exchange between regions
Optimality Condition Decomposition [34]	Effective for both linear and nonlinear problems	Performance highly dependent on network partitioning
ADP [35]	Decomposes MAOPF into subproblems with Bellman’s equation, improving scalability	High complexity in value function approximation, not suitable for non-linear systems
PDDP [23]	Improves performance for nonlinear MAOPF systems	Does not account for renewable energy variability, leading to power fluctuations
Improved Lagrangian Consistency Algorithm [36]	Considers line security constraints for decentralized OPF	Convergence issues with non-convex problems

.

However, these decentralized algorithms often face inefficiencies due to the iterative coordination and communication required between areas. In contrast, centralized techniques, while potentially more efficient, compromise data privacy and limit scalability [37,38]. This paper seeks to overcome these challenges by developing a decentralized stochastic recursive gradient (DSRG) that preserves information privacy, supports independent decision making, and achieves faster convergence compared to traditional decentralized methods. The main contributions of this paper are as follows:

• The proposed DSRG method optimizes power flow in a fully decentralized manner, eliminating the need for extensive data sharing and centralized control.
• It provides a comprehensive mathematical model for the DSRG method, including detailed formulations of the power flow constraints, tie-line constraints, and recursive gradient descent updates.
• The DSRG method is rigorously tested on a 3-area, 9-bus system, where it demonstrates superior convergence speed and cost efficiency.

2. Mathematical Model of MAOPF

The centralized multi-area optimal power flow model involves various mathematical components to ensure the optimal and reliable operation of a power system. Below is a general mathematical model that incorporates the specified aspects.

2.1. Objective Function

Minimize the total generation cost while considering the AC power flow model and incorporating PV generation:

$$min\sum _{\Omega }\sum _G\left(C_{ij}·P_{ij}\right).$$

(1)

In Equation (1) above, $C_{ij}$ and $P_{ij}$ are the coefficient of grid generation cost and output power, respectively, of generator G in area $\Omega$.

2.2. Generation Constraints

Generation constraints ensure the power generated by each generator in each area is within its specified limits. Mathematically, this can be expressed as:

$$P_{\Omega ,G}^{min}\le P_{\Omega ,G}\le P_{\Omega ,G}^{max}\begin{array}{cc}& \end{array}\forall \Omega ,G.$$

(2)

2.3. Voltage Magnitude Constraints

Voltage magnitude constraints ensure the voltage magnitudes are within acceptable limits. Mathematically, this can be represented as:

$$V_i^{min}\le V_i\le V_i^{max}\begin{array}{cc}& \end{array}\forall i\in \Omega .$$

(3)

In the above equation, $V_i$ is the voltage magnitude, whereas $V_i^{min}$ and $V_i^{max}$ are the allowable minimum and maximum voltage magnitudes at bus i.

2.4. Line Capacity Constraints

Constraints on the capacity of transmission lines are crucial for preventing overloading. Mathematically, this can be expressed as:

$$-S_{ij}^{max}\le S_{ij}\le S_{ij}^{max}\begin{array}{cc}& \end{array}\forall \Omega ,G.$$

(4)

In the above equation, $S_{ij}$ is the complex power flow on the transmission line, and $S_{ij}^{max}$ is the maximum allowable power flow on the line.

2.5. Power Flow Constraints

The power flow equations describe the relationship between the power injections, voltages, and the network impedance. These are nonlinear equations and are often linearized for ease of finding a solution. Mathematically, power flow constraints can be expressed as:

$$P_{ij}-P_{ij}^{prev}-R_{ij}·(V_i·V_j·cos({\theta }_i-{\theta }_j))=0,$$

(5)

$$Q_{ij}-Q_{ij}^{prev}-X_{ij}·(V_i·V_j·sin({\theta }_i-{\theta }_j))=0.$$

(6)

In the above equations, $P_{ij}$ and $Q_{ij}$ are the real and reactive power between the buses i and j, while $R_{ij}$ and $X_{ij}$ are the resistance and reactance of the transmission line. Additionally, $V_i$, $V_j$, and ${\theta }_i$, ${\theta }_j$ are the voltage magnitudes and phase angles at buses i and j. The centralized model of MAOPF can be divided into submodels based on geographical partitioning to solve problems in a decentralized manner. Each area becomes a submodel, and the tie-lines between adjacent areas are considered in the modeling process. The voltages related to these tie-lines become state variables to capture the interaction between adjacent areas.

2.6. Decentralized Tie-Line Constraints

The tie-line constraints and voltage boundary conditions contribute to the area decoupling process. By including the tie-line constraints, the areas remain interconnected, ensuring power balance and coordination between adjacent regions. The voltage boundary conditions capture the interaction between adjacent buses, representing the real and reactive power flows through tie lines, mathematically expressed as:

$$V_i^{boundary}=V_i^{adjacent}.$$

(7)

In the above equation, $V_i^{boundary}$ and $V_i^{adjacent}$ are the voltage magnitude at the boundary and adjacent bus, respectively.

2.7. Decentralized Power Balance Constraints

Decentralized power balance constraints ensure that the total real power injected into an area through local power generation and tie-lines must be equal to the total real power consumed within an area by the loads. The mathematical model can be expressed as:

$$\sum _{j\in \Omega }{P}_{ij}-\sum _{k\in \Omega }{P}_{ki}^{tie}=D_i.$$

(8)

In the above equation, $D_i$ represents the total load demand in area ${\Omega }_i$, $P_{ij}$ is the power injected or withdrawn from bus i to j within area ${\Omega }_i$, and $P_{ki}^{tie}$ is the power exchange through the tie-lines from area ${\Omega }_k$ to ${\Omega }_i$. Bellman’s equation decomposes the optimization problem into a series of subproblems, allowing for a more systematic and efficient solution process. The decentralized Bellman’s equation can be formulated as follows:

$$V_{\Omega }^t={min}_{\alpha }\left[\sum _G\left(C_{i,G}^t·P_{i,G}^t+\beta ·V_{\Omega }^{t+1}\right)\right]\begin{array}{cc}& \forall i\in \Omega \end{array}.$$

(9)

In the above equation, $V_{\Omega }^t$ is the value function for the optimal cumulative cost, and $\alpha$ is the set of decision variables at stage t for area $\Omega$. Additionally, $\beta$ is the discount factor that reflects the importance of future values relative to present values. This equation essentially states that the optimal value is the minimum cost achievable at stage t plus the discounted optimal value at the next stage $t+1$ for area ${\Omega }_i$. The optimization process involves minimizing $\alpha$ with respect to the decision variables, ensuring the decisions made in area ${\Omega }_i$ at stage t lead to the minimum overall cost of that area.

3. Decentralized Stochastic Recursive Gradient Algorithm

The DSRG algorithm is designed for optimizing non-convex problems in multi-agent systems. It features decentralized collaboration, stochastic recursive gradient updates, and gradient tracking with extra mixing. This algorithm achieves optimal incremental first-order oracle complexity, enhancing communication efficiency compared to existing decentralized methods [39]. The DSRG algorithm demonstrates significant advantages in computational efficiency when applied to large-scale power systems. Its decentralized structure, faster convergence rates, lower algorithmic complexity, enhanced resource utilization, and robustness under varying conditions collectively position it as a more suitable option compared to the traditional decentralized algorithm in optimizing power flow. DSRG is a powerful optimization technique designed for fully decentralized OPF in multi-area power systems. It uses a stochastic gradient descent method to iteratively update decision variables, such as power generation levels, in order to minimize the local objective function in each area. The algorithm performs local optimization within each area while exchanging minimal necessary information with neighboring areas to maintain overall system balance. Over multiple iterations, the DSRG algorithm converges to an optimal solution, ensuring efficient, scalable, and decentralized operation of the power system. A flow chart for DSRG is shown in Figure 1. The step-by-step decentralized solution algorithm for solving the AC-OPF problem in multi-area power systems using DSRG methods is explained Algorithm 1.

Algorithm 1 DSRG Algorithm for Fully Decentralized OPF in Multi-Area Power Systems

1: Input:  2: Step 1: Initialization  3: Set iteration count $k=0$.  4: Initialize local variables $P_{ij}^{\left(0\right)},Q_{ij}^{\left(0\right)},V_i^{\left(0\right)}$.  5: Initialize step size $\alpha$ and aggregation parameter ${\gamma }_o$.  6: Identify neighboring areas ${\mathcal{N}}_i$ for each area i.  7: Perform feasibility check to ensure that initial decision variables $(P_i^{\left(0\right)},Q_i^{\left(0\right)},V_i^{\left(0\right)})$ comply with local constraints.  8: Step 2: Local Gradient Computation  9: For each area i, compute the gradient of the local objective function: $$\nabla f_i(P_{ij}^{\left(k\right)},Q_{ij}^{\left(k\right)},V_i^{\left(k\right)}).$$ 10: Compute gradients of local constraints to ensure updated decision variables comply with operational limits. 11: Step 3: Stochastic Update 12: Update decision variables for each area i using the stochastic gradient descent approach: $$P_{ij}^{(k+1)}=P_{ij}^{\left(k\right)}-\alpha ·\nabla h_i^k(P_{ij}^{\left(k\right)},Q_{ij}^{\left(k\right)},V_i^{\left(k\right)}).$$ 13: Update output power for each area using recursive gradient descent: $$P_i^{(k+1)}=P_{ij}^{\left(k\right)}-\alpha ·(\nabla h_i^k\left(P_{ij}^{\left(k\right)}\right)+{\eta }_i^{\left(k\right)}),$$ where ${\eta }_i^{\left(k\right)}$ is a random perturbation to introduce stochasticity. 14: Step 4: Recursive Aggregation 15: Perform recursive aggregation for each area using aggregated information from neighboring areas: $$P_i^{(k+1)}=\sum _{j\in {\mathcal{N}}_i}{w}_{ij}{P}_j^{\left(k\right)},$$ where $w_{ij}$ represents the influence of neighboring area j on area i. 16: Step 5: Consensus Algorithm 17: Apply consensus algorithm to ensure all areas agree on shared variables, particularly tie-line power flows and inter-area exchanges: $$C_i^{(k+1)}=\sum _{j\in {\mathcal{N}}_i}{w}_{ij}{C}_j^{\left(k\right)}.$$ 18: Step 6: Convergence Check 19: Calculate the difference between the current and previous iteration values: $$\Delta P_{ij}=\parallel P_{ij}^{(k+1)}-P_{ij}^{\left(k\right)}\parallel .$$ 20: if $\Delta P_i<ϵ$ for all areas i then 21:     Convergence achieved; terminate the algorithm. 22: else 23:     Increment k and return to Step 3. 24: end if 25: Output the final values: $P_{ij}^{*}=P_{ij}^{\left(k\right)}$, $Q_{ij}^{*}=Q_{ij}^{\left(k\right)}$, $V_i^{*}=V_i^{\left(k\right)}$.

The first step of the proposed algorithm is the initialization, in which each area i begins by setting up its local variables $x_{ij}^0$, such as $P_{ij},Q_{ij}$, and voltage $V_i,V_j$. The step size ${\alpha }_o$ and the aggregation parameter ${\gamma }_o$ are initialized, and the iteration counter is set to $k=0$. Each area then identifies its neighboring areas, denoted as $N_i$, to facilitate the exchange of information during the optimization process. Following this setup, a feasibility check will ensure that the initial decision variables $x_{ij}^0$ comply with all local constraints, including generation limits, voltage magnitudes, line capacities, power flows, tie-line regulations, and power balance requirements. The second step is local gradient computation, in which each area computes the gradients $(\nabla C_{ij}\left(x_i\right))$ of its local objective function $\left(C_{ij}\left(x_i\right)\right)$ for its decision variables $x_i^k$ for bus k. The mathematical expression can be expressed as:

$$\nabla C_{ij}\left(x_i\right)={\displaystyle \frac{\partial C_{ij}\left(x_i^k\right)}{\partial x_i}}.$$

(10)

The gradients of the local constraints are computed to ensure that the updated decision variables comply with all necessary operational limits, including those on generation, voltage, and power flow. The third step is associated with the stochastic update in which decision variables for each area are updated using the stochastic gradient descent approach. The updated decision variables for each area i can be calculated using the following equation:

$$x_i^{k+1}=x_i^k-{\alpha }^k\left(\nabla C_i^k+{\xi }_i^k+{\lambda }_i^k\nabla h_i^k\right).$$

(11)

In the above equation, ${\xi }_i^k$ is a random perturbation that introduces stochasticity and helps the algorithm explore the solution space more effectively. Conversely, ${\lambda }_i^k\nabla h_i^k$ represents the influence of the constraint gradients, which ensure the updated variables stay feasible. The next stage is the calculation of recursive aggregation for each area using aggregated information from its neighboring areas to refine its local solution. Recursive aggregation can be expressed as:

$$R_i^{k+1}=\sum _{j\in N_i}{\omega }_{ij}{x}_j^{k+1}.$$

(12)

In the above equation, weight ${\omega }_{ij}$ represents the influence of neighboring areas and is often chosen as ${\omega }_{ij}=1$, which ensures the aggregated information is a balanced combination of the neighbor’s contributions. This expression also adjusts the tie-line power flows to ensure they meet the decentralized tie-line constraints. After this, a consensus algorithm is applied to ensure that all areas reach an agreement on the shared variables, particularly the tie-line power flows and inter-area exchanges. In this step, each area updates its consensus variables $\left(z_i^{k+1}\right)$ to ensure consistency across the system:

$$x_i^{k+1}=z_i^{k+1}.$$

(13)

In the next step, the algorithm checks for convergence by evaluating whether the change in local variables $\left|x_i^{k+1}-x_i^k\right|$ is below a predefined threshold. If this condition is met, the process is considered to have converged. Alternatively, the algorithm checks if the maximum number of iterations $k_{max}$ has been reached, in which case the process stops regardless of the change in variables. If convergence is achieved, then the algorithm will terminate, and each area i has an optimal solution that minimizes local generation costs. This solution will satisfy all relevant constraints, including generation limits, voltage magnitude, line capacity, power flow, tie-line, and power balance requirements. Moreover, it is consistent with the overall system objectives and constraints, ensuring that the decentralized optimization aligns with the global goals of the entire power system. Otherwise, increase $k=k+1$ and return to the second step. This decentralized stochastic recursive gradient method provides an effective approach to solving the AC-OPF problem in multi-area power systems.

4. Rigorous Convergence Analysis and Theoretical Validation of the DSRG Algorithm

In multi-area power systems, a more thorough convergence investigation is required to confirm the validity of the DSRG algorithm for fully decentralized OPF. The behavior, effectiveness, and resilience of the algorithm under various circumstances would all be theoretically guaranteed by such an analysis.

4.1. Convergence Guarantees

A formal proof of convergence is crucial to establishing that the algorithm achieves a solution that is either globally or locally optimal. This entails demonstrating that the algorithm:

• Minimizes generation cost with each iteration;
• Satisfies all constraints at convergence;
• Converges to an equilibrium where no further updates to the decision variables lead to significant improvements, ensuring system-wide feasibility.

4.2. Theoretical Bounds on Convergence

Theoretical bounds are one of the critical parts of a rigorous analysis that defines convergence rate and behavior. This would typically involve:

• Linear convergence rate bounds under certain convexity assumptions, where the solution gap reduces proportionally with each iteration. Considering linear convergence with a constant $\gamma \in (0,1)$, we demonstrate the rate at which the method approaches the ideal solution $x^{*}$.
• Logarithmic convergence rates for more general, non-convex optimization problems like OPF, where diminishing returns are observed as the algorithm proceeds. The mathematical expression for logarithmic convergence rates is:

  $$\left|x^{k+1}-x^{*}\right|\le \gamma \left|x^k-x^{*}\right|.$$

  (14)
• Upper bounds on the number of iterations required to reach a near-optimal solution within a predefined tolerance level $\epsilon$. If ℘ is the upper bound of a function’s growth rate, the theoretical results can be expressed as:

  $$f\left(x^{k+1}\right)-f^{*}\le \wp \left({\displaystyle \frac{1}{k}}\right).$$

  (15)

5. Results and Discussion

5.1. Numerical Analysis

The verification of the proposed algorithm is implemented on a 3-area, 9-bus system using MATLAB and general algebraic modeling system. Figure 2 illustrates a simple 3-area, 9-bus system along with its network parameters and load data. This system defines the tie-lines as follows: Lines 3–6 connect Area 1 and Area 2, Lines 5–9 connect Area 2 and Area 3, and Lines 1–7 connect Area 1 and Area 3. The generator parameters for testing the proposed algorithm are listed in

Table 2. Generator parameters.

Parameters	${\mathit{G}}_1$	${\mathit{G}}_2$	${\mathit{G}}_3$	${\mathit{G}}_4$	${\mathit{G}}_5$	${\mathit{G}}_6$
$P_{max}$ (MW)	100	150	100	100	150	150
$P_{min}$ (MW)	10	10	10	10	10	50
$Q_{max}$ (MVar)	60	85	60	80	150	180
$Q_{min}$ (MVar)	−30	−40	−30	−40	−75	−90
${\alpha }_1$ ($USD$/WM2)	0.06	0.05	0.04	0.03	0.02	0.01
${\alpha }_2$ ($USD$/WM)	30	40	50	30	15	25
${\alpha }_3$ ($USD$)	200	80	60	100	80	60

. The power flow of the internal and tie-line capacity is set to 100 MW and 150 MW, respectively. The optimized power for all generators using the proposed algorithm is illustrated in Figure 3. This figure shows the distribution of real power and reactive power generated by each generator after the optimization process of the proposed algorithm. The bus voltage magnitudes are constrained within the $0.95$ p.u. to $1.1$ p.u. range to ensure the safe and efficient operation of the power system. Figure 4 depicts the voltage magnitudes of the decentralized 3-area, 9-bus system under the proposed algorithm, all within the specified limits.

The behavior of power flow versus line capacities with the presence of internal line and tie-line capacity are illustrated in Figure 5. These visualizations are essential for analyzing and managing congestion in power systems to ensure the power flows remain within safe limits to prevent overloads and maintain system stability. Figure 5a demonstrates the power flow within a single area when congestion occurs. The power flows are compared against the line capacities within that area to understand how close the system is to its limits and to identify potential issues. Figure 5b focuses on the distribution of power across interconnecting lines and how these flows approach the capacities of the inter-area connections. It compares these power flows to the maximum capacities of the transmission lines that connect these areas to understand how close the system is to its limits and to identify potential issues in the inter-area connections. This is crucial for understanding the interaction and power exchange between different areas in the network. Overall, this figure demonstrates the importance of line capacity in managing power flow within a power system, highlighting the differences between internal and tie-lines and how power flow is distributed across different parts of the grid.

The cost analysis versus internal and tie-line capacity are shown in Figure 6. This figure illustrates how different internal and tie-line capacities affect the overall costs of operating the power system and identify optimal capacities that minimize costs. The relationship between line capacity and the associated cost of power flow emphasizes the cost benefits of higher line capacities in both internal and tie-lines within a multi-area power system.

The convergence process of the proposed algorithm is depicted in Figure 7, showing that it takes 24 iterations to find the optimal solution. This figure illustrates how the algorithm progressively reduces costs and stabilizes voltage levels within predefined limits over these 24 iterations. It visually represents the efficiency and effectiveness of the algorithm in optimizing the power system’s performance.

5.2. Comparison Analysis

The validity of the proposed DSRG algorithm is demonstrated through comparison with the centralized interior point method (CIPM), a benchmark and reliable technique for solving OPF problems. Moreover, superiority is verified by comparing with the distributed interior point method (DIPM), approximate dynamic programming (ADP), and parallel dual dynamic programming (PDDP). The optimization results for DIPM are achieved after 40 iterations, ADP converges after 30 iterations, and PDDP reaches its final results after 45 iterations. These outcomes demonstrate that DSRG provides fast optimization results compared to other techniques for fully decentralized OPF in multi-area power systems. Comparisons of active and reactive power distribution among generators with error analysis are depicted in Figure 8 and Figure 9, respectively. These figures provide a clear comparison of how different optimization methods perform in distributing active and reactive power among multiple generators. The error analysis indicates that DSRG consistently allocates power across most generators, demonstrating robust performance. In contrast, other methods exhibit significant deviations and suggest further tuning. ADP provides negative reactive power from G2 that could have several adverse effects on the system’s stability, efficiency, and reliability.

The comparison of the cost of different methods across different internal and tie-line capacities is depicted in Figure 10 and

Table 3. Comparison analysis of total cost.

Technique	Total Cost	Iterations
CIPM	USD 46,684.58	–
DSRG	USD 46,685.69	24
DIPM	USD 46,689.14	45
ADP	USD 46,701.53	30
PDDP	USD 46,762.42	40

. This result shows that the proposed algorithm offers a more cost-effective performance for decentralized OPF in multi-area power systems.

The centralized approach provides superior efficiency and global optimization compared to the proposed algorithm; however, it requires extensive data sharing and centralized control, which can limit scalability and raise privacy concerns. In contrast, decentralized approaches that ensure information privacy and maintenance of independent decision-making processes for each area tend to be less efficient due to the need for iterative coordination and communication. Nevertheless, the proposed decentralized optimization algorithm has demonstrated greater efficiency and faster convergence compared to previous decentralized methods.

6. Conclusions

The proposed DSRG method offers a robust and efficient solution for decentralized OPF in multi-area power systems. By leveraging stochastic gradient descent, the DSRG method achieves fast convergence and optimal cost distribution while maintaining information privacy and decentralized decision making. The method outperforms traditional decentralized algorithms like DIPM, ADP, and PDDP, providing faster and more reliable results. Although centralized methods may offer better global optimization, the DSRG method’s ability to maintain decentralized control and protect data privacy makes it a highly valuable tool for modern power systems. Future work could explore the scalability of the DSRG method to larger and more complex systems, as well as its integration with renewable energy sources and other emerging technologies in the power industry.