# A convex relaxation approach for power flow problem

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## Abstract

A solution to the power flow problem is imperative for many power system applications and several iterative approaches are employed to achieve this objective. However, the chance of finding a solution is dependent on the choice of the initial point because of the non-convex feasibility region of this problem. In this paper, a non-iterative approach that leverages a convexified relaxed power flow problem is employed to verify the existence of a feasible solution. To ensure the scalability of the proposed convex relaxation, the problem is formulated as a sparse semi-definite programming problem. The variables associated with each maximal clique within the network form several positive semidefinite matrices. Perturbation and network reconfiguration schemes are employed to improve the tightness of the proposed convex relaxation in order to validate the existence of a feasible solution for the original non-convex problem. Multiple case studies including an ill-conditioned power flow problem are examined to show the effectiveness of the proposed approach to find a feasible solution.

## Keywords

Convex relaxation Ill-conditioned power flow Power flow Network reconfiguration## 1 Introduction

Power flow is the underlining problem for power system analysis. Integration of intermittent renewable energy resources and possible network contingencies further highlight the merit of providing an efficient framework to solve this nonlinear problem. The power flow problem is formulated as a set of nonlinear equations. Several iterative approaches including Gauss–Seidel (GS) and Newton–Raphson (NR) were adopted to solve this problem. However, the convergence and stability of these approaches could not be guaranteed. NR is the most popular approach to solve this system of equations as it provides a better convergence rate compared to other techniques. Employing NR to solve the power flow problem leads to the following scenarios: ① a unique solution exists that can be found regardless of the initial point; ② multiple solutions exist and one of the solutions is returned based on the initial point; ③ no solution exists; ④ unique or multiple solutions exist while no solution can be procured because of improper initial point that renders a singular Jacobean matrix in the iterative process. The latter scenario presents an ill-conditioned power flow problem.

Several research works were focused to solve the power flow problem. Lower-upper (LU) factorization is a direct approach to solve NR that is computationally expensive and impractical for large-scale problems. Krylov subspace method [1] is a nonstationary iterative method that converges in at most *N*_{B} iterations, where *N*_{B} is the number of buses in the network. The Newton-generalized minimal residual (GMRES) method is one of the Krylov methods utilized to solve the power flow problem. The adaptive preconditioning schemes to update the preconditioners for the linearized equations of the next iteration and a flexible inner-outer preconditioned GMRES were proposed in [2] and [3], respectively, to improve the convergence of the Krylov methods. Another approach is to utilize incomplete LU factorization as a preconditioner for the Krylov–Newton methods as presented in [4]. The continuous Newton’s method proposed in [5] formulates the power flow problem as a set of ordinary differential equations.

Finding a proper preconditioning for the Krylov methods to procure a feasible solution for the power flow problem is challenging. Although many research works addressed the feasibility of the power flow problem [6, 7, 8], the proposed approaches for finding a feasible solution or providing a certificate for the infeasibility of the problem are not yet effective for ill-conditioned power flow problems. Iterative approaches are incapable of handling the ill-conditioned power flow problems [5, 9].

The analytical approaches (e.g. convex relaxation [10]) can address the singularity issue with the ill-conditioned power flow problems. However, improving the scalability of these approaches is a challenging task [11]. The solution rendered by leveraging the semi-definite programming (SDP) relaxation may not be a tight one that is feasible for the non-convex power flow problem. Therefore, more cutting planes may be needed to obtain a feasible solution with reasonable computational time. A high order of the Lasserre hierarchy could be used to find a feasible solution for the relaxed power flow problem [12]. However, the large computation burden of leveraging higher orders of the Lasserre hierarchy restricts the application of this approach for small-scale power networks. Nevertheless, exploiting the sparsity in power flow equations facilitates large-scale application of the SDP relaxation i.e. first order of Lasserre hierarchy [13].

- 1)
A convex relaxation formulation for the power flow problem is presented, where a set of lifting variables is defined for the nonlinear terms in the power flow formulation. The introduced lifting variables are utilized in the lowest order of moment relaxation to find a solution to a relaxed power flow problem.

- 2)
By exploiting the sparsity in the power network, the size of moment relaxation matrices employed to formulate the relaxed problem is substantially reduced. This reduction in the size of the problem facilitates the scalability of the presented convex relaxation approach.

- 3)
A perturbation scheme is presented to improve the tightness of the presented relaxed problem. Once the original power flow problem is formulated as an optimization problem, the objective is zero. The presented relaxation for this optimization problem might not be tight with the lowest order of the moment relaxation. Presenting a perturbation function to tighten to relaxation does not negatively impact the procured solution to the relaxed problem as the objective of the relaxed optimization problem is zero.

- 4)
A network reconfiguration scheme is implemented to improve the tightness of the presented relaxation. The network reconfiguration eliminates the zero-injection buses within the network. The procured voltages with the original topology are the same as those with the reconfigured topology. It is shown that removing the zero-injection buses along with perturbation improves the tightness of the presented sparse convex relaxation for the power flow problem.

- 5)
A specific tightness measure is introduced to evaluate the effectiveness of the presented solution method to obtain a feasible solution to the original power flow problem. Moreover, a recovery process is presented to procure the solution to the original power flow problem from the solution provided by the relaxed problem.

A set of work addressed the optimal power flow problem in the literature [14, 15, 16, 17, 18, 19], using SDP and SOCP relaxations, however, the objective of optimal power flow problem is different from power flow feasibility problem. The focus of the optimal power flow problem is to find the dispatch of generation units with minimum cost. Employing perturbation for the optimal power flow problem will undermine the optimality of the procured solution, while it is not an issue for the power flow problem.

This paper is organized as follows: Section 2 presents the power flow problem formulation. Section 3 presents a solution methodology to evaluate the existence of a feasible solution and find a solution to the power flow problem. Section 4 presents the numerical results to show the effectiveness of the proposed approach, and the conclusions are presented in Section 5.

## 2 Problem formulation

*N*

_{B}buses, there are

*2N*

_{B}known parameters and

*2N*

_{B}unknown variables in the power flow equations. The magnitude and angle of the voltage for the slack/reference bus, the voltage magnitude and real power injection at voltage-controlled buses, and real and reactive power injections at load buses are the known parameters for the power flow problem. The unknown variables are the real and reactive power injections for the slack buses, voltage angle and reactive power injection for the voltage-controlled buses, and the voltage magnitude and voltage angle for the load buses. Real and reactive power injections are not usually enforced for the slack bus as they supposed to compensate for the real and reactive power mismatches in the network. The slack bus is considered as a reference bus to calculate the voltage angle for the load and voltage-controlled buses. The voltage for the slack/reference bus is enforced as shown in (1). The real power balance for the voltage-controlled and load buses is given in (2). The reactive power balance for the load buses is shown in (3). The voltage magnitudes for the voltage-controlled buses are enforced by (4). The reactive power generation limits for the generation units connected to voltage-controlled buses are shown in (5). If the reactive power generation of a unit reaches its limits, the reactive power generation is fixed to the limit and the corresponding bus becomes a load bus.

*i*, respectively; \(V_{ref}^{d}\) and \(V_{ref}^{q}\) are the real and imaginary parts of the voltage phasor at reference bus, respectively; \(\left|V_{i}^{\prime}\right|\) is the voltage magnitude at voltage-controlled bus

*i*; \(P_{i}^{G\prime }\) is the real power generation at bus

*i*; \(P_{i}^{D}\) is the real power demand at bus

*i*;

*B*

_{ij}is the element of the susceptance matrix;

*G*

_{ij}is the element of the conductance matrix;

*PQ*is the set of load buses;

*PV*is the set of voltage-controlled buses; \(Q_{i}^{D}\) is the reactive power demand at bus

*i*; \(Q_{i,\hbox{min} }^{G}\) and \(Q_{i,\hbox{max} }^{G}\) are the minimum and maximum reactive power generations at bus

*i*.

Solving (1)–(5) could be challenging under certain circumstances as discussed earlier and iterative approaches may fail to find a feasible solution. Thus, a non-iterative solution methodology is presented in the next section which tried to address this challenge.

## 3 Solution methodology

To find the solution for (1)–(5), it is reformulated as a convex optimization problem that could render a feasible solution for the power flow problem in polynomial time. Particularly, a sparse SDP relaxation for the problem presented in (1)–(5) is formulated with a suggested perturbation function. A topology reconfiguration scheme is proposed to improve the tightness of the presented convex relaxation. The details of the proposed solution methodology are presented in the following subsections.

### 3.1 Convex relaxation

*O*(

*n*

^{3}), where

*n*is the number of monomials which is twice as the number of buses for the relaxed power flow problem. Exploiting the network sparsity mitigates the computation burden. Several sparse moment matrices associated with each maximal clique are defined. A clique, by definition, is a set of nodes within a graph that are all adjacent to each other. The maximal clique is a clique that its set of nodes is not a subset of any other clique. Here, the problem is reformulated as a first-order moment relaxation problem (SDP problem). The nonlinear terms in (1)–(5) are represented by respective lifting variables in the SDP relaxation matrix as formulated for the relaxed problem in (6)–(19). If all sparse SDP relaxation matrices are near-rank-1, the presented relaxation in (6)–(19) is tight and a feasible solution for the power flow problem in (1)–(5) is procured.

*i*; \(Q_{i}^{G\prime }\) is the reactive power generation at bus

*i*;

*S*is the set of slack/reference buses; \(V_{i,\hbox{min} }^{{}}\) and \(V_{i,\hbox{max} }^{{}}\) are the minimum and maximum voltage magnitudes at load bus

*i*;

*c*is the index for each maximal clique within the network graph; |

*c*| is the number of buses within clique

*c*; \(u_{i,\hbox{max} }^{Q}\) is the auxiliary binary variable, 1 if the reactive power generation at bus

*i*reaches its maximum value, otherwise 0; \(u_{i,\hbox{min} }^{Q}\) is the auxiliary binary variable, 1 if the reactive power generation at bus

*i*reaches its minimum value, otherwise 0;

*ε*is an arbitrary small constant.

The objective of the perturbed convex relaxation is given in (6), where the lifting terms associated with the square of real and imaginary parts of voltage on each bus is employed. The choice of perturbation plays an important role in procuring a near-rank-1 solution. An SDP relaxation for the rank minimization problem is presented in [20], which may not render a feasible solution for the power flow problem. The choice of the perturbation is not unique; however, various functions may lead to various near-rank-1 solutions [21]. This choice depends on the system operator preferences to obtain a particular solution among multiple solutions that may exist for the power flow problem. The system operator does not need to know about the superiority of one of the perturbations over another. However, they might have various preferences and technical considerations to choose a perturbation function. Here, the perturbation matrix is employed in the objective function to determine a feasible solution in which the voltage magnitudes on the buses are close to 1 p.u.. This choice can be the last known voltage of the system, a desired voltage profile, 1 p.u. voltage for all buses, etc. The nonlinear terms in (1)–(5) are presented by their associated lifting variables in (2)–(19), and the SDP matrices associated with each maximal clique given in (19) contain the lifting variables. For example, \(V_{i}^{q} V_{j}^{d}\) is a nonlinear term in (1)–(5) which is replaced by a lifting variable \(\gamma_{{V_{i}^{q} V_{j}^{d} }}\) as defined in the semi-definite matrix constraint (19).

The voltage for the slack/reference bus is enforced by (7). Although, the real power balance for the slack bus is ignored in (1)–(5), enforcing the generation capacity limits for the slack bus will ensure the feasibility of the solution procured by solving the relaxed problem. Enforcing these limits will avoid procuring a solution that is impractical. The power flow problem demonstrates the state of the system, where slack bus cannot provide real and reactive power beyond its generation capacity limits. Thus, the real and reactive power generation capacities of the unit connected to the slack bus is enforced by (8) and (9), respectively. The real power balance for the voltage-controlled and load buses is presented in (10). The reactive power balance for the load buses is shown in (11). The voltage limits for load buses are not usually considered for the power flow problem. However, the power flow problem might have multiple solutions, where some of them are low voltage solution vulnerable to voltage collapse [22]. Thus, to ensure system security and the technical feasibility of the procured solution, the voltage limits for the load buses is presented in (12). For the voltage-controlled buses, the voltage magnitude is enforced by (13) and (14). Once the reactive power of the generation units connected to a voltage-controlled bus reaches its limits, the voltage-controlled bus will transform into a load bus with a fixed reactive power and unknown voltage magnitude. This condition is captured by two auxiliary binary variables for each voltage-controlled bus to check if any of the upper and lower limits for the reactive power generation of the generation units are reached. The reactive power generation of the units connected to voltage-controlled buses is enforced by (15)–(18). Here once the reactive power generation reaches the upper or lower limits, the auxiliary binary variable becomes 1.

*u*, as a continuous variable,

*u*

^{r}, as given in (20). Then enforce a non-convex constraint to ensure the continuous variable takes values 0 and 1 as given in (21). Then, employing a convex relaxation approach to convexify the non-convex feasibility region. To tighten such relaxation regularization linearization technique (RLT) and valid constraints [24] are leveraged. As the convex relaxation problem formulated in the sparse form, relaxing the binary variables would be a better choice for large-scale applications.

### 3.2 Tightness measure

*TR*

_{c}is the tightness measure for each clique

*c*and \(\lambda_{\left| c \right|}^{c}\) is the eigenvalue of the SDP matrix associated with maximal clique

*c*.

A large ratio in (22) indicates that the second eigenvalue is very small compared to the first eigenvalue and can be neglected. Thus, if the ratio is a large number, the rank of the SDP matrix is near-one and the relaxation is tight. Alternatively, the gap between the derived solution and the original solution can be procured using the reciprocal of the presented tightness measure.

### 3.3 Recovering solution to original power flow problem

*c*.

### 3.4 Network reconfiguration

To tackle this challenge, a network reconfiguration is proposed to eliminate the load buses with zero injection from the network. The procured topology is equivalent to the original network topology.

*j*is 1, as shown in Fig. 1, the bus can be removed from the network. The flow of the line connected to this bus is zero, and the voltage of this bus is equal to the bus connected to it. If the set of zero-injection buses with connectivity degree of 1 is

*Ω*

_{1}and \(\left\| \cdot\right\|_{0}\) indicates the number of nonzero elements, the voltage of the zero-injection bus can be further recovered as a function of the adjacent bus voltage using (26) and (27), where \(\gamma_{{V_{i}^{d} }}\) and \(\gamma_{{V_{i}^{q} }}\) are procured from the solution of the reconfigured network.

*j*is 2, as shown in Fig. 2a, the load bus with zero injection can be removed from the network while the two lines connected to this bus will merge into a single line in the reconfigured topology, as shown in Fig. 2b. Here, impedance

*z*

_{ik}=

*z*

_{ij}+

*z*

_{jk}and the Y bus of the configured network is further adjusted. The voltage of this bus can be further recovered as a function of the adjacent bus voltages using (28) and (29), where

*Ω*

_{2}is the set of zero-injection load buses with connectivity degree of 2. Here, two unknowns i.e. \(V_{j}^{d}\) and \(V_{j}^{q}\), could be found once \(\gamma_{{V_{i}^{d} }}\) and \(\gamma_{{V_{i}^{q} }}\) are procured from the power flow solution of the reconfigured network.

*j*is 3, as shown in Fig. 3a, the load bus with zero injection can be removed from the network by changing the network topology to its equivalent shown in Fig. 3b. The branch impedances are procured using Y–△ conversion shown in (30) [26] and Y bus of the reconfigured network is constructed accordingly.

*Ω*

_{3}is the set of zero-injection load buses with connectivity degree of 3. Here, two unknowns, i.e. \(V_{j}^{d}\) and \(V_{j}^{q}\), could be found once \(\gamma_{{V_{i}^{d} }}\) and \(\gamma_{{V_{i}^{q} }}\) are procured from the solution of the reconfigured network.

## 4 Numerical results

To illustrate the effectiveness of the proposed methodology, several case studies are presented. The presented problem formulation is solved using MOSEK [27]. The tightness of the solutions to the power flow problem is compared in the following cases: ① Case 1, relaxation without perturbation and network reconfiguration; ② Case 2, relaxation with perturbation but without network reconfiguration; ③ Case 3, relaxation without perturbation but with network reconfiguration; ④ Case 4, relaxation with perturbation and network reconfiguration.

The comparison among the results for Case 2 and Case 3 with those for Case 4 presents the impact of perturbation and network reconfiguration individually.

### 4.1 Ill-conditioned 13-bus system

The eigenvalues of the procured solution of the clique are 2.2617 × 10^{−9}, 1.2243 × 10^{−8}, 9.1753 × 10^{−8}, 9.7194 × 10^{−8}, and 3.2417. Here, there is only one eigenvalue which is much greater than zero. The tightness ratio for this clique defined in (22) is 7.52. Therefore, the presented solution to the relaxed problem is near-rank-1. The eigenvector associated with the largest eigenvalue is [− 0.5554, − 0.6051, − 0.5696, − 0.0214, 0.0159]. According to (23), the recovered voltages from the dominant eigenvalue and its associated eigenvector are the same as (33) up to seven-digit precision. The real and imaginary parts of the voltage of bus 11 are 1.0896 and 0.0387, respectively. The real and imaginary parts of the voltage of bus 12 are 1.0257 and − 0.0288, respectively. Converting the rectangular form of voltages to polar form returns \(1.0 90 3\angle 2.0 3 2 2^\circ\) and \(1.0 2 6 1\angle 1. 60 7 1{^\circ }\) for bus 11 and bus 12, respectively.

### 4.2 IEEE 30-bus system

### 4.3 IEEE 57-bus system

### 4.4 200-bus system

### 4.5 2383-bus system

## 5 Conclusion

This paper presents a convex relaxation approach to determine the feasibility of the power flow and yields a solution for this problem. The presented approach is scalable for large-scale applications. The near-rank-1 relaxation is procured by using the presented perturbation and network reconfiguration techniques. Leveraging these techniques helps avoiding employment of higher orders of moment relaxation and improves the computation efficiency of the presented relaxation for large networks. The effectiveness of the proposed approach is illustrated for well-conditioned IEEE test cases as well as for an ill-conditioned case.

## Notes

### Acknowledgements

This work was supported by Technology Project of State Grid Corporation of China (No. SGRIJSKJ(2016)800).

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