Proposed network model and algorithm

We propose a new network model for a nodal OPF to keep communication costs manageable regardless of the system size. For this purpose, the desired properties of the model are as follows:

1. The model must be compatible with PF studies for which Kirchhoff’s laws and voltage magnitudes are well defined.
2. Each node is a short distance away from the rest of the nodes to minimize the communication costs.
3. The voltages at a node and those at the rest of the nodes are linearly related.

In the power flow studies, such as PF, OPF, state estimation, and probabilistic PF, voltages are the variables. The constraints in the studies consist of the power flows and injections, as well as the voltage magnitudes in terms of voltages. In the Cartesian coordinate system, the power flows and injections are quadratic in voltage. For example, the power flow over the line connecting Nodes i and j at i is , and the power injection at Node i is where the quantities sandwiched between voltages are in 2Nb-by-2Nb; where e_i-j is a vector with the cardinality of 2Nl of which the element corresponding to the flow over i-j is 1, and all other elements are zeros, ; and the superscript H is the conjugate transpose. The matrices and S_i have two nonzero rows at i and Nb+i rows due to J e_i.

1. Claim 1: The matrices associated with power flows and power injections are all of rank 4.
2. Claim 2: The matrices associated with the squares of the voltage magnitudes have rank 2.

See S1 Appendix for the proof of the claims.

For a real-valued symmetric rank-4 matrix M_j, a real-valued eigen pair λ and u exist such that . Because M_j is a rank 4 matrix, the number of columns in Ф_j and the number of nonzero diagonal elements in II_j are both 4. Here, the terms and : refer to the matrices associated with the real power injection at Node j ; is associated with reactive power injection at Node j ; is associated with real power flow over the line j-l at the side of Node j ; and is associated with the reactive power flow over the line j-l at the side of Node j . Note that the eigenpairs are dependent solely on the system parameters (not on voltages). With the eigenpairs, the quantities of interest in the PF are the real power injection at Node j p_j, the reactive power injection at Node j q_j, the real power flow over a line j-l at Node j , the reactive power flow over a line j-l at Node j , and the squared voltage’s magnitude at Node j E_j. There are quadratic relationships among them: (1)

Eq (1) expresses Kirchhoff’s laws and the voltage magnitudes. The nodal variables α_j, β_j, γ_j-l,j, δ_j-l,j, and ω_j are defined as follows: the α_j are the voltages projected onto the eigenspaces spanned by the real power injection, and the β_j cover the reactive power injection, the γ_j-l,j cover the real power flow over j-l at Node j, the δ_j-l,j cover the reactive power flow over j-l at Node j, and the ω_j cover the voltage magnitudes at the j^th node in the same manner. Although there is one each of α_j, β_j, and ω_j at each node, the nl_j of γ_j-l,j and δ_j-l,j are defined such that nl_j represents the number of branches that are directly connected to Node j (i.e., l = 1, 2, …, nl_j). It is worth noting that the nodal variables are linearly associated with the voltages (the definitions of the nodal variables in Eq (1)), which indicates that PL = 1. Therefore, the nodal variables satisfy all the desired properties of the new network model.

Next, we let nl_j be the number of lines connected to the j^th node. Then, the dimensions of α, β, γ, δ, and ω are 4, 4, 4nl_j, 4nl_j, and 2, respectively. Note that their dimensions do not depend on the system size. Let μ_j be a nodal variable that integrates local generation variables as follows: where μ_end = 1. The generalized PF formulation is listed in previous work [22]. Note that the cardinality of the variable is fixed—independent of the system size. Therefore, the new network model yields: 1) a fixed number of variables, and 2) a PL of 1.

Proposed network model: A star grid with two channels

In the definition of the local variable x_j, it is noticeable that there are three variable types: 1) power injection and flow variables α, β, γ, and δ, 2) the voltage magnitude variable ω, and 3) quadratic variables, _Q x_j comprising power flows and power generations (g_j). The first two types of variables are linear, and the third type is quadratic, in terms of the voltages. Once x_j (α, β, γ, and δ) is determined, it is straightforward to find the power generations at Node j with Eq (1) (i.e., x_j defines the feasible region of μ_j). The voltage v_p can be reconstructed from α, β, γ, and δ. The communication path used to collect their values is termed the power channel. Here, ω is directly associated with the voltage magnitudes, and the values are reported through another communication path termed the voltage channel to update voltages v_m. Although the voltages v_p and v_m should be identical, conditions are relaxed so that they can be different.

Fig 2 illustrates an example of the traditional network model (left) and the proposed network model (right) for a modified 4-bus system that has branches connecting 1–2, 1–3, 1–4, 2–4, and 3–4. In the proposed model, all the nodes are connected to the center node (black dot at the center) through the power channel (red lines) and the voltage channel (green lines), and all the nodes are of distance 1 from the center (PL = 1). The network is a complete bipartite graph of order 2 with a maximum-diameter star network. All communications between the variables take place in local nodes (α, β, γ, δ, and ω), and the center node is linear in terms of voltages.

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Fig 2. A star and linear network for a modified IEEE-4 bus system.

The traditional network model is shown on the left (A) and the proposed network model is presented on the right (B). Red lines indicate the power channel, and green lines are the voltage channel.

https://doi.org/10.1371/journal.pone.0251948.g002

Nodal OPF with nodal variables and its convex relaxation

Even though we describe the OPF problem in this paper, the same framework is applied to any other problem, including PF, OPF, state estimation, and probabilistic PF problems, because they have unified formulas [22]. The central OPF formulation as a nonconvex, quadratically constrained quadratic problem is given in Eq (3), its distributed optimization problem (nodal OPF) at each node I is given in Eq (3A), and the nodal OPF is formulated in terms of an ADMM algorithm in Eq (3B). (2) (3A) where and where A is a block matrix to collect an element ; ; ; ; ; ; ;; ; ; ; ; ; and ; , and . (3B) where , ,, or equivalently ; and X_j is the column space of Ф_j. at Node j defined by the constraints listed in 1–5 in (3A).

Note that all A’s are full column rank matrices that collect relevant parts from μ_i, and that Ф_i includes the linear relationships between the local variable x_i and the central variable v and y. Using the definition of the function f_j, the following observations are made: 1) f_j is a nonconvex, but smooth, function that is C¹ on an open set containing X_j (defined by the column space of Ф_j), and ∇f_j is Lipschitz continuous on X_j; 2) X_j is nonempty, closed, and convex; and 3) w_j is coercive.

Even though Eqs (3A) and (3B) have low cardinalities in the decision variables due to the nonconvex nature of the problem, the uniqueness and existence of the solution are not guaranteed. To address the complexity issue, a surrogate function h_j approximating Eq (3B) is defined as follows: (4)

Because u_j is the SDP relaxation of f_j, it is convex, Lipchitz continuous, and continuously differentiable on X_j. Note that u_j relaxes the rank constraint only; therefore, the first derivatives of u_j and of f_j regarding x_j are identical (i.e., ). Because the SDP is convex, the solution at the k^th iteration is uniquely determined by at given y^k and z_j^k. Note that the relaxed nodal OPF is convex and has a small number of variables; therefore, its computational complexity remains manageable.

Properties of u_j, f_j, g_j, h_j, w_j. and

The nodal problem f_j is C^∞ smooth, but nonconvex, and prox-regular at x_j relative to if x_j ∈ dom(f_j), . Further, whenever and , there exists : ; and . Statement (f2) holds by the Descent Lemma [23] with the properties of g_j and of h_j described below. Its SDP relaxation u_j is strongly convex, Lipschitz continuous, and prox-regular, and its first derivative equals that of f_j (i.e.: ; and Whereas the variables considered in f_j and u_j are all local primal variables only, ADMM constraint g_j depends on the central variable y and the multipliers z. With the properties of u_j, f_j, and g_j, w_j is C^∞ smooth, but nonconvex, and prox-regular, whereas h_j is strongly convex and Lipschitz continuous. Because all X_j are bounded sets, w_j and h_j are coercive and lower-bounded over X_j. Because is the optimizer of convex h_j (i.e., ), the following holds:

and .

Proposed algorithm with provable convergence

Proposed algorithm.

We propose the following algorithm to solve w_j in conjunction with h_j in Eq 5:

Distributed, Regulated, Optimally-Homogeneous, and Scalable (DROHS) Algorithm

1. Set k = 0, initialize all parameters, such as Δ^k ∈(0,1), y^k, ρ_j, .

2. If satisfies the termination criteria, terminate the algorithm.

3. Solve the local problems, .

4. Depending on the solution , determine (Fig 3)

• Case 1: is in the real space of convex X_j, i.e., in the column space of Ф_j, .
• Otherwise : Find a feasible projection of onto the feasible region X_j, near ;
• Case 2: the projection is close enough to (i.e., where ); accept the solution , ;
• Case 3: is not sufficiently close; reject the solution , ;
• update , ;
• set the error bound asymptotically to vanish as iteration proceeds (i.e., ).

5. Compute , and update accordingly; and where a ∈ (0.5, 1).

6. Update so that all x-variables are consistent with global variable y (i.e., ).

7. k ← k + 1, and go to 2.

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Fig 3. x-update from the result of optimization with 1) a feasible solution , 2) an infeasible solution, but the distance between the solution and the projection onto the feasible region is close enough , and 3) an infeasible solution in that the projection onto the feasible region is too far .

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Fig 3 illustrates how the x-optimization process determines the described in Rule 4 of the algorithm. If the solution is feasible in X_j, is the solution (Case 1). If the solution is not in X_j but the point projected onto the convex region X_j is sufficiently close, is the projected point (Case 2). For Case 3, where either no solution is identified or the identified solution is not in X_j, and the distance to the projected point is too far away, the solution is rejected, and is the point determined in the previous step. Upon the determination of , the iteration proceeds to a point between the current and the projected point depending on parameter Δ^k.

After the x-optimization is performed, y-optimization follows. The y-optimization at given and z^k is a least square problem–convex problem. It should be noted that the solution to the convex problem is always feasible. Rule 6 is a striking feature of the proposed algorithm that is critically different from the ADMM approaches. Instead of the locally independent update of the primal variables in the ADMM approaches, the nodal variables are updated 1) linearly for the computational efficiency, and 2) with respect to the central variable so that the update is reasonably agreeable among nodal variables. The proof of the convergence of the proposed algorithm is presented in this paper’s S2 Appendix.

Implementation of Rule 4

The local SDP relaxation yields a matrix (known as ) instead of a vector ; therefore, a new criterion should be established to compute , as follows: Because is from SDP, all the eigenvalues are non-negative (i.e., ) where is the m^th column vector, and r is the rank of : (5)

To construct an equivalent criterion to with , we impose to determine whether or not the feasible projection is acceptable. The grouping criterion is simplified to by invoking . Let be the m^th eigenvalue of . Then, the criterion becomes .

The definition of is also modified as using .

Choice of parameters

In contrast to the ADMM approaches, only a few parameters are assigned in the proposed algorithm. Because , and the specific choice of z_j is irrelevant, all z_j‘s are random vectors in the null space of known Φ_j. It is important to hold to guarantee the convergence of the proposed algorithm [24]. Due to the fact that , the maximum τ over all nodes at the k^th iteration is set by The parameters used were as follows: ρ_j = 20 for v_L and 200 for v_M; Δ⁰ = 0.3; and the number of maximum iteration = 100; .

The global variable y is randomly assigned (cold start) to test the algorithm’s robustness. For comparison, a flat start and a warm start are also attempted. Here, a flat start refers a point at which real voltages are unity, and imaginary voltages are within [-0.1, 0.1]; and a warm start [20] is a solution to the PF. It is recognized that a faster convergence is obtained when the initial point is close to the feasible region, which is commonly observed in numerical iterative methods.

Termination criterion

The algorithm is terminated when no further progress is made. The progress is measured in terms of the global variable y, and the local variables x and z. After the solution is identified, the objective function W* is determined according to (3B) where , i.e., . The interim value of W at the k^th iteration is , and the progress measure η^k at the k^th iteration is defined: (6)

The criterion used in Rule 2 terminates the algorithm if the progress measure is less than the tolerance (10⁻⁷) (i.e., η^k ≤ 10⁻⁷).

Simulation environments

The model systems used in the simulations are available from MATPOWER [25] for 3-, 4-, 9-, 14-, 24-, 30-, 39-, 57-, 85-, 118-, 300-, and 2,000-bus systems. All simulations were performed using a Mac pro with 2 × 2.93 GHz 6-core Intel Xeon processors and 6 GB 1333 MHz DDR3 memory. The local SDP problems were solved using the CVX solver [26]. To compare the results from various systems, the lines in the figures are normalized so that all start at zero. The initial points for all the cases are cold starting points, and all the real and imaginary voltages are set to random numbers (not even a flat start). The qualities of the solutions are all numerically identical to those identified using MATPOWER for the model systems tested (all the solutions v* are less than 10⁻⁴ from the MATPOWER solutions (v_MATPOWER), ∥v* − v_MATPOWER∥/∥v*∥ ≤ 10⁻⁴). For comparison, we also attempted a flat start and observed that N_iter decreases at least 30%, but the flat start also finds the same solutions.

Subsystems of the test cases

In the ADMM approaches referred to in the literature [14–20], the grid partitioning is performed, but the details of the partitioning are not well listed. In considering the computational coupling, the partitioning is based on spectral clustering [20, 27] while each subsystem contains at least one generator [14, 20]. The inclusion of a generator in a subsystem seems a natural choice to fulfill the power balance equality constraints (1 in (3)) within each substation. However, the inclusion of a generator does not serve the purpose well, because the generator may not be dispatched if its generation costs are excessively high. The spectral clustering is performed using two different algorithms: 1) unnormalized [27] and 2) normalized [28]. The algorithms yield different grid clustering. For example, the unnormalized algorithm results in two clusters for the IEEE 3-bus systems, while the normalized algorithm finds a single cluster for the same system [28]. Fig 4 illustrates the path length, maximum number of nodes in a subsystem, and the number of subsystems.

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Fig 4. The path length, the maximum number of nodes in a subsystem, and the number of subsystems for the IEEE 3-, 4-, 9-, 14-, 24-, 30-, 39-, 57-, 85-, 118-, 300-, and 2,000-bus systems.

The lines indicate the positive correlations on the size of the system. The red-dotted line, the blue broken line, and the green solid line represent the best-fit curves for the path length, the maximum number of nodes in a subsystem, and the number of subsystems, respectively. Two algorithms are applied for clustering nodes: (A) unnormalized algorithm, and (B) normalized based on the algorithm in [28].

https://doi.org/10.1371/journal.pone.0251948.g004

The lines are best-fit lines, showing the positive correlations of path length (red dotted line path length ∝ Nb^0.30), the maximum number of nodes in a subsystem (green solid line ∝ Nb^0.43), and the number of subsystems (blue broken line ∝ Nb^0.76) based on the unnormalized algorithm (left plot). The results using the algorithm with a different normalization [28] yield path length ∝ Nb^0.34, the maximum number of nodes in a subsystem ∝ Nb^0.45, and the number of subsystems ∝ Nb^0.66 (right plot). Even though the results from the two plots are not exactly the same, the positive correlations with the system size are clear. They affect the computational efficiency of ADMM [14, 20]; path length affects the communication costs and N_iter because the decision at each subsystem should be delivered to the rest of the system; the number of subsystems affects the number of computation cores and the communication costs unless PL equals one; and the maximum number of nodes affects the computation costs at each subproblem. Therefore, the decrease in the computation costs at each subsystem is achieved in exchange for the increased costs of communication among subsystems. Because the path length and number of subsystems increase rapidly with system size, the improvement in computational efficiency of the ADMM approach is questionable.

Maximum cardinality of nodal variables in the proposed algorithm

The nodal variables in the x-optimization μ_j are . Because μ_end is unity, the cardinality of μ_j is 10nl + 2ng + 10 where nl is the number of lines connected to the j^th node, and ng is the number of generators located at the node. In determining the voltages through the power channel, the node with a high value for nl plays a key role. Fig 5 illustrates the maximum cardinality of μ_j with the systems’ sizes. The dashed line indicates . Even though the positive relationship between and Nb is visible among the model systems, the relationship is not necessarily positive. The number of variables in the central OPF is typically 2Nb + 2Ng where Nb and Ng are the numbers of buses and generators in the system, respectively. For small systems, such as 3-, 4-, 9-, 14-, and 24-bus systems, the maximum cardinalities of the nodal variables are higher than the number of variables in the central OPF; therefore, it would be challenging to keep the computational costs of the distributed OPF for the small systems lower than those of the central OPF. It is noteworthy that the communication costs remain manageable, because each node directly communicates with the central node due to the fact that PL equals unity.

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Fig 5. Cardinalities of nodal variables in x-optimization in terms of system size.

https://doi.org/10.1371/journal.pone.0251948.g005

Distributed OPF for 3-, 4-, 24-, 39-, and 85-bus systems

To examine whether the choice of parameters affects the convergences, optimizations were executed with many randomly assigned initial points, and uniform convergences were consistently observed (Fig 6). In general, N_iter depends on the initial guess of the voltages. As the distance of the initial point from the solution becomes closer, N_iter decreases, which is commonly observed in numerical iterative methods. All the solutions identified are numerically the same as the solutions using MATPOWER.

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Fig 6. Convergence of proposed algorithm for selected systems (i.e., IEEE 3-, 4-, 9-, 14-, 24-, 39-, and 85-bus systems).

https://doi.org/10.1371/journal.pone.0251948.g006

A worst-case convergence was observed for the IEEE 39-bus system. The nodal OPF for the system involves large negative eigenvalues (i.e., the nodal OPFs are highly nonconvex). According to Rule 4, a relatively large number of solutions are rejected because they are not close to the feasible regions and, therefore, N_iter becomes large (approximately 40 iterations). However, the solution identified is a local minimizer, and N_iter is still much smaller than the number of iterations for the ADMM approaches. For the visual presentations of the results obtained for various systems, the progress is normalized to the 1^st iteration (i.e., the curves begin at 0 for the 1^st iteration).

Comparison of convergence to ADMM approaches

The convergence measures were compared to those from the ADMM approaches reported in multiple studies [14, 17–19]. It is worth noting that Erseghe [14] and Zhang et al. [17] do not take the flow limits into consideration; the approach of Engelmann et al. [18] involves a high communication cost; and the approach of Madani, Kalbat, and Lavaei [19] may yield a physically infeasible solution. We attempted the ADMM approaches with the flow constraints and feasibility, but our implemented solver failed to converge with any starting points. Instead, we compared the convergence behaviors of the proposed algorithm to those reported in the previous [14, 17–19]. For the visual presentation, the convergence behaviors are normalized so that all the curves begin at the same point (see Fig 7 for the comparison of the convergence).

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Fig 7. Convergence of proposed algorithm for selected small systems (i.e., IEEE 9-, 14-, 30-, 57-, 118-, and 300-bus systems).

The convergences of the ADMM approaches [14, 17–19], are compared.

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In the previous research [14, 17–19], the size of the subsystems increased with that of the system. Therefore, the tradeoff between communication costs amongst the subsystems and the computation costs for each subsystem make it difficult to develop a scalable algorithm. As the system size increases, N_iter increases significantly for the results in two of the previous studies [14, 19]. The performance measure in another study [17] fluctuates consistently with various systems, indicating that the convergence may not be guaranteed. Because the final study [18] requires the information exchange regarding sensitivities as well as the primal variables, the computation and communication costs per iteration increase much more rapidly than do the computation costs of a central OPF solver.

Different from the clustering containing multiple nodes used in the ADMM approach, the proposed algorithm for each subsystem contains only 1 node regardless of the system size; each subsystem directly communicates with the rest of the entire system; and the communication involves only the primal variables. With this difference in mind, the fast convergences observed in the proposed algorithm are similar to those in two of the previous studies [17, 18], which are consistently much faster than those in the other two previous studies [14, 19]. Whereas N_iter are like those reported by Engelmann et al. [18], the convergences of the proposed algorithm occur more uniformly and consistently.

Many approaches using ADMM do not consider the flow limits [14, 17], and/or feasibility [19] and, therefore, it is not appropriate to compare the quality of the solutions. Instead, we compared our solutions with those using a central OPF solver, MATPOWER. For all the cases described above, they yield numerically identical solutions. It is not clear why the proposed algorithm finds the same solution as the central OPF solver. We tested with the central SDP-relaxation for a small-scale system to estimate the “global” solution. Due to the memory issue, our SDP tests are limited up to 118-bus systems. When the relaxation returns the rank-1 solution, the solution is global and physically feasible. For several cases, the test cases are of the global solutions that are also identified by MATPOWER and by the proposed algorithm. However, there are cases where the SDP relaxation returns a physically infeasible solution. For these cases, both MATPOWER and the proposed algorithm identify numerically the same local minimizers that may not be global solutions. MATPOWER finds a point that meets the first-order necessary conditions for optimality [25]. Although it finds a minimizer in most cases, there is no guarantee that the identified solution is a minimizer—it can be either a maximizer or a saddle point. On the other hand, the proposed algorithm identifies a solution that meets the second-order optimality conditions, which guarantees that the solution is a minimizer [29]. From a practical point of view, there are two advantages of the proposed algorithms over MATPOWER: 1) they are numerically stable and, therefore, robust because all the subproblems are convex—no issues regarding the rank-deficient Hessian matrix, and 2) they use distributed computation and, therefore, manageable computation in each subproblem. However, there are disadvantages of the proposed algorithm: 1) the requirement of multiple cores to perform the distributed optimization, and 2) the number of nodal variables can be larger than that of the central OPF; for example, 3-, 4-, 9-, 14-, and 24-bus (See Fig 5).

Large-scale OPF: 2,000-bus system

We tested the algorithm for a large-scale system, the 2,000-bus system that is a synthetic grid on a footprint of Texas. Fig 8 presents a convergence pattern. The solution identified is numerically identical to the one found using MATPOWER. N_iter remains small, as is the case for the small systems (See Figs 6 and 7). Note that the nodal OPF includes only 1 bus, and the communication costs remain small because PL equals unity. If a sufficient number of computation cores are provided, the proposed algorithm is scalable if the computation cost for solving a subproblem does not increase significantly as the system size increases.

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Fig 8. Convergence of proposed algorithm for synthetic 2,000-bus system on a footprint of Texas.

https://doi.org/10.1371/journal.pone.0251948.g008

Different from other algorithms reported in the literature (i.e., [14–18]), the proposed algorithm converges regardless of the initial points, but the number of iterations decreases by at least a factor of 2 if it uses a flat start. Another shortcoming of the algorithms reported in previous research [14–18], is the quality of their solutions. Whereas the existing OPFs find low-voltage [17] or suboptimal [16] solutions, the proposed algorithm finds the same solution as a central OPF solver. One previous study [16] claimed that there might be an optimal number of partitioned subsystems due to the tradeoff between communications and computational costs used to obtain a local solution. The proposed method keeps the PL = 1, and the central computation is a simple addition of x_j through both the power and voltage channels. From the simulations with various parameter values such as ρ_j, Δ^k, , and a, we obtained the same solutions, indicating that the proposed algorithm is robust as well as efficient. In addition, the proposed model does not require any partitioning.

Comparison to a heuristic central OPF solver, MATPOWER

In comparing the convergence between the proposed algorithm and the central OPF, the number of variables and N_iter were examined. The maximum cardinality of μ_j increases with the system size in (See Fig 4). Fig 9 illustrates the number of variables in the central OPF as well as the maximum cardinality of the nodal variables. The blue line is the best-fit line for the central OPF (), whereas the red line is the best-fit line for the distributed algorithm. It is clear that increases with Nb in a much faster way than does. The black line is the boundary at which equals . If enough computation cores are available, the proposed algorithm involves reduced computation costs per iteration for systems larger than or equal to IEEE 24-bus. Whereas almost linearly increases with Nb because Ng ≪ Nb in most systems, does not necessarily increase theoretically. A key observation is that the proposed approach can be a scalable algorithm if N_iter does not rapidly increase with the system size.

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Fig 9. Cardinalities of central OPF and proposed algorithm for the test systems.

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Fig 10 presents N_iter for the central OPF (Nit_cent) and of the proposed algorithm (Nit_dist) on the tested systems. A visible increase in Nit_cent is observed with Nb, but the dependence of Nit_dist on Nb is not evident—the solid lines are best-fit curves (log-log plot) that indicate and Nit_dist = Nb^−0.007, respectively. The dotted lines are average N_iter, and the values are 15 and 23, respectively.

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Fig 10. N_iter for central OPF and of the proposed algorithm on the tested systems.

Dotted lines are averages, and solid lines are best-fit curves.

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From the comparisons (the number of variables and N_iter), we conclude that the computation cost of the distributed algorithm increases with Nb at a much slower rate than that of the central OPF if sufficient computational resources are available.

Scalability of the algorithm

The computation efficiency of a distributed computation depends on the number of iterations and the computational cost per iteration. The cost is determined by the local computation with the largest number of variables. Fig 10 shows that the number of iterations does not increase with system size.

Fig 11 illustrates the maximum nodal computation time depending on the maximum cardinality of the nodal variables. This performance dependence on the number of nodal variables may shed light on the claim by Loukarakis, Bialek, and Dent [30] that a larger system does not necessarily imply an inferior convergence performance. The dashed line indicates that the maximum nodal computation time is proportional to the maximum cardinality of the nodal variables with the power of 2.64. The cardinality to the power of 2.64 observed in this study is close to the theoretically estimated 3 for the SDP solver. Note that it is not necessary for the maximum cardinality of the nodal variables to be positively correlated with the system size. Rather, the maximum cardinality depends on the local grid topology of the system. For the tested systems, and , which results in CT ∝ Nb^2.64/4. = Nb^0.66 where CT represents the maximum nodal computation time. The total computation cost is bounded by the product between N_iter and CT. The total computation cost is bounded by CT ∝ ϑ(⌈Nb/N_core⌉Nb^0.66) where N_core is the number of cores. If the computational resource is sufficient, CT ∝ ϑ(Nb^0.66). From this observation, the computation cost increases sub-linearly for a large-scale network.

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Fig 11. Computation times in seconds for nodal OPF of the maximum cardinality of the nodal variables.

Dotted line is the best-fit line with the slope of 2.64.

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If a single core is available for computation, the computation cost is in where p_SDP is the computational complexity for a nodal SDP. For the state-of-art central heuristic OPF solvers, the computation cost is in ϑ(Nb^1.5). Therefore, when the computation resources are highly limited, the proposed algorithm would still be efficient with a convex problem solver that yields p_SDP ≤ 2. A potential improvement in p_SDP of the SDP solver is to explore the sparse structure of the matrices [31] or to utilize a commercial solver such as MOSEK.