Three-phase optimal power flow for networked microgrids based on semidefinite programming convex relaxation

Many autonomous microgrids have extensive penetration of distributed generation (DG). Optimal power flow (OPF) is necessary for the optimal dispatch of networked microgrids (NMGs). Existing convex relaxation methods for three-phase OPF are limited to radial networks. In light of this, we develop a semidefinite programming (SDP) convex relaxation model which can cope with meshed networks and also includes a model of three-phase DG and under-load voltage regulators with different connection types. The SDP model solves the OPF problem of multi-phase meshed network effectively, with satisfactory accuracy, as validated by real 6-bus, 9-bus, and 30-bus NMGs, and the IEEE 123-bus test cases. In the SDP model, the convex symmetric component of the three-phase DG model is demonstrated to be more accurate than a three-phase DG modelled as three single-phase DG units in three-phase unbalanced OPF. The proposed method also has higher accuracy than the existing convex relaxation methods. The resultant optimal control variables obtained from the convex relaxation optimization can be used for both final optimal dispatch strategy and initial value of the non-convex OPF to obtain the globally optimal solution efficiently

Random block coordinate methods for inconsistent convex optimisation problems

We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal gradient method, we establish global convergence of the last iterate as well optimal $O(1/k)$ and $O(1/k^{2})$ complexity rates in the convex and strongly convex case, respectively, $k$ being the iteration count. Motivated by the increased complexity in the control of distribution level electric power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained \revise{as} dual variables in our optimisation framework.Comment: Changed title and revised manuscrip

Distributed Load Control in Multiphase Radial Networks

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.</p

Index heuristics for routing and service control problems within queueing systems

This thesis is naturally broken down into two main problems, one concerning optimal routing control and the other optimal service control. In the routing control problem the arriving customers must be allocated to one of the 'K' possible service stations. We assume that the customers arrive in a single Poisson stream. We take the service at each of the stations to be exponentially distributed, but perhaps with different parameters. The system cost rate is additive across the queues formed at each station. We also have that at each station the holding cost function is increasing convex. Following Whittle's approach to a class of restless bandit problems, we develop a Lagrangian relaxation of the routing control problem which serves to motivate the development of index heuristics. The index by a particular station is characterised as a fair charge for rejecting the arriving customer at that station. We also consider a policy improvement index for comparison to the heuristic. We develop these indices and report an extensive numerical investigation which exhibits strong performance of the index heuristic for both discounted and average costs.The second problem concerns the optimal service control of a multi-class M/G/l queueing system in which customers are served non preemptively. The system cost rate is additive across classes and increasing convex in the numbers present within each class. We again follow the method prescribed by Whittle when considering a class of restless bandits. Hence we develop a Lagrangian relaxation of the service control problem which motivates the development of a class of index heuristics. For a particular customer class the index is characterised as a fair charge for service of that class. These indices are developed and we again report representative results from an extensive numerical study which again implies a strong performance of the index heuristic for both discounted and average costs

Convex relaxation of Optimal Power Flow in Distribution Feeders with embedded solar power

AbstractThere is an increasing interest in using Distributed Energy Resources (DER) directly coupled to end user distribution feeders. This poses an array of challenges because most of today's distribution feeders are designed for unidirectional power flow. Therefore when installing DERs such as solar panels with uncontrolled inverters, the upper limit of installable capacity is quickly reached in many of today's distribution feeders. This problem can often be mitigated by optimally controlling the voltage angles of inverters. However, the optimal power flow problem in its standard form is a large scale non-convex optimization problem, and thus can’t be solved precisely and also is computationally heavy and intractable for large systems. This paper examines the use of a convex relaxation using Semi-definite programming to optimally control solar power inverters in a distribution grid in order to minimize the global line losses of the feeder. The mathematical model is presented in details. Further, case studies are completed with simulations involving a 15-bus radial distribution system. These simulations are run for 24 hour periods, with actual solar data and demand data

AC OPF in Radial Distribution Networks - Parts I,II

The optimal power-flow problem (OPF) has played a key role in the planning and operation of power systems. Due to the non-linear nature of the AC power-flow equations, the OPF problem is known to be non-convex, therefore hard to solve. Most proposed methods for solving the OPF rely on approximations that render the problem convex, but that may yield inexact solutions. Recently, Farivar and Low proposed a method that is claimed to be exact for radial distribution systems, despite no apparent approximations. In our work, we show that it is, in fact, not exact. On one hand, there is a misinterpretation of the physical network model related to the ampacity constraint of the lines' current flows. On the other hand, the proof of the exactness of the proposed relaxation requires unrealistic assumptions related to the unboundedness of specific control variables. We also show that the extension of this approach to account for exact line models might provide physically infeasible solutions. Recently, several contributions have proposed OPF algorithms that rely on the use of the alternating-direction method of multipliers (ADMM). However, as we show in this work, there are cases for which the ADMM-based solution of the non-relaxed OPF problem fails to converge. To overcome the aforementioned limitations, we propose an algorithm for the solution of a non-approximated, non-convex OPF problem in radial distribution systems that is based on the method of multipliers, and on a primal decomposition of the OPF. This work is divided in two parts. In Part I, we specifically discuss the limitations of BFM and ADMM to solve the OPF problem. In Part II, we provide a centralized version and a distributed asynchronous version of the proposed OPF algorithm and we evaluate its performances using both small-scale electrical networks, as well as a modified IEEE 13-node test feeder

A Duality-Based Approach for Distributed Optimization with Coupling Constraints

In this paper we consider a distributed optimization scenario in which a set of agents has to solve a convex optimization problem with separable cost function, local constraint sets and a coupling inequality constraint. We propose a novel distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node solves a local version of the original problem relaxation, and updates suitable dual variables. We prove the algorithm correctness and show its effectiveness via numerical computations