Branch Flow Model: Relaxations and Convexification—Part I

We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

The optimal power flow (OPF) problem is nonconvex and generally hard to solve, see e.g. [1], [2]. In this tutorial, we will provide an overview of two different solution approaches. The first uses the bus injection model, which is the standard model for power flow analysis and optimization. It focuses on nodal variables such as voltages, current and power injections and does not directly deal with power flows on individual branches. A key advantage is the simple linear relationship I = Y V between the nodal current injections I and the bus voltages V through the admittance matrix Y. Recently, it has been observed that this form of OPF can be reformulated as a nonconvex QCQP (quadratic constrained quadratic program), which leads to a standard convex relaxation through semidefinite programming [3]-[5]. For radial networks, different sufficient conditions have been derived under which the semidefinite relaxation turns out to be exact [6]-[8]. The second solution technique employs the branch flow model, which focuses on currents and powers on the branches rather than the nodal variables. The branch flow model has been historically used primarily for modeling distribution circuits, which tend to be radial. It has therefore received far less attention. A branch flow model has recently been proposed for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving OPF that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, both relaxation steps are always exact, provided there are no upper bounds on loads [9]. For mesh networks, the conic relaxation is always exact and we provide a simple way to determine if a relaxed solution is globally optimal. We describe a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints [10]. The tutorial will describe precisely the bus injection model and the semidefinite relaxation of OPF as well as the branch flow model and its associated relaxations. We prove sufficient conditions for exact relaxations and verify our results on simulations of various IEEE test systems. Finally we explain the equivalence between the bus injection model and branch flow model.

We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) problems that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact and we characterize when the angle relaxation may fail. We propose a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Since power networks are sparse, the number of required phase shifters may be relatively small.

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators. Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory. The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.

Optimal power flow (OPF) is an important tool for economic dispatch and other power system applications. Due to the non-convex nature of OPF formulations, convex relaxation serves as a bridge linking OPF formulation variants with efficient polynomial-time algorithms to obtain a global optimum. This paper proposes: 1) an OPF model based on branch flow model in rectangular form, and 2) two convex envelopes for this OPF model. First, by introducing explicit redundant variables for bi-directional power flows, and the products of bus voltages for each branch, OPF is formulated into a non-convex quadratic programming (QP) model. For an n-bus, l-branch power system, the non-convex part of this QP model is collectively expressed by 2n quadratic terms and 4l bilinear terms. Second, primal and dual convex envelopes for these terms are derived to convert the non-convex QP model into a convex one. Third, the tightness of primal and dual convex envelopes is validated via a spatial branch-and-bound framework using IEEE 14-bus system. Numerical studies show that the proposed convex envelopes are tight to get an optimum with small optimality gap and slight bus power mismatches.

A branch flow model has recently been proposed for the analysis and optimization of power flows. In this paper we show that the model is equivalent to the more popular bus injection model. Moreover, we prove the equivalence of various relaxations of these two models.

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable. Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks. This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control. Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations. We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request. Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule. To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks. Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact. To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.

In this paper, we review two convex models for power flow in radial distribution network, namely the bus injection and branch flow models. We start with the fundamental equations of voltage drop, power losses, and line power flows between two buses in a distribution line represented by an impedance. These AC circuit analysis equations contain trigonometric functions such as sine and cosine. We show that we can obtain a new set of equivalent AC circuit analysis without trigonometric functions by defining auxiliary variables and/or using linear combination of original equations. Additionally, by treating squared values of voltages and currents we obtain linear form of AC circuit equation whereas the relation of sine and cosine functions can be equivalently embedded in rotated cones. Consequently, we obtain the bus injection and branch flow models, which are theoretically equivalent. Their numerical performance, however, could be different, due to the numerical ill-conditions that may arise when constructing rotated cones.

The second-order cone relaxation of the branch flow model (BFM) and bus injection model (BIM) variants of optimal power flow are well-known to be equivalent for radial networks. In this work we show that in meshed networks with parallel lines, BIM dominates BFM, and propose novel constraints to make them equivalent in general. Furthermore, we develop an improvement to the second-order cone relaxations of optimal power flow, adding novel and valid linear constraints on the lifted current expressions. We develop two simple test cases to highlight the advantages of the proposed constraints. These novel constraints tighten the second-order cone relaxation gap on test cases in the ‘PG Lib’ optimal power flow benchmark library, albeit generally in limited fashion.

We compute an optimal day-ahead dispatch plan for distribution networks with stochastic resources and batteries, while accounting for grid and battery losses. We formulate and solve a scenario-based AC Optimal Power Flow (OPF), which is by construction non-convex. We explain why the existing relaxation methods do not apply and we propose a novel iterative scheme, corrected DistFlow (CoDistFlow), to solve the scenario-based AC OPF problem in radial networks. It uses a modified branch flow model for radial networks with angle relaxation that accounts for line shunt capacitances. At each step, it solves a convex problem based on a modified DistFlow OPF with correction terms for line losses and node voltages. Then, it updates the correction terms using the results of a full load flow. We prove that under a mild condition, a fixed point of CoDistFlow provides an exact solution to the full AC power flow equations. We propose treating battery losses similarly to grid losses by using a single-port electrical equivalent instead of battery efficiencies. We evaluate the performance of the proposed scheme in a simple and real electrical networks. We conclude that grid and battery losses affect the feasibility of the day-ahead dispatch plan and show how CoDistFlow can handle them correctly.

Cutting planes based relaxed optimal power flow in active distribution systems

The optimal power flow (OPF) problem is generally nonconvex. Recently a second-order cone relaxation for OPF has been proposed using the branch flow model. In this paper, we provide sufficient conditions under which the relaxation is exact, and demonstrate that these conditions hold for a wide class of practical power distribution systems.

Economic dispatch (ED) is widely studied in power system optimization and is a typical application of optimal power flow (OFF). As more distributed generation resources integrated, e.g. micro-turbines and renewable generators, the AC-constrained ED (ACED) of distribution power networks (a typical radial power networks) is more controllable and the operation optimization of which is essentially a complicated Mixed Integer Nonconvex Nonlinear Programming (MLNNLP) problem with discrete controllable devices, e.g. transformers and compensating capacitors. In this paper, we studied ACED problem of radial power networks based on Branch Flow model. To make this problem tractable, the piecewise linear (PWE) and latest second-order cone (SOC) relaxation techniques are employed to relaxe ACED to a Mixed Integer Second-order Cone Programming (MISOCP) problem which can be efficiently solved by commercial solvers. Besides, we also discussed the exactness of SOC relaxation for ACED problem of radial power networks based on recent research achievements in this area. The modified IEEE 33 bus system is studied which validated the effectiveness and high computation efficiency of the proposed method.

The AC optimal power flow (OPF) problem is essential for the schedule and operation of power systems. Convex relaxation methods have been studied and used extensively to obtain an optimal solution to the OPF problem. When the exactness of convex relaxations is not guaranteed, it is important to recover a feasible solution for the convex relaxation methods. This paper presents an alternative convex optimisation (ACP) approach that can efficiently recover a feasible solution from the result of second‐order cone programming (SOCP) relaxed OPF in mesh networks. The OPF problem is first formulated as a difference‐of‐convex programming (DCP) problem, then efficiently solved by a penalty convex concave procedure (CCP). By using the solution of a tightened SOCP OPF as an initial point, the proposed algorithm is able to find a global or near‐global optimal solution to the AC OPF problem. Numerical tests show that the proposed method outperforms those semi‐definite programming (SDP) and quadratically constrained quadratic programming (QCQP)–based algorithms.

A second-order conic approximation to solving the optimal power flow problem in bipolar DC networks while considering a high penetration of distributed energy resources