This paper considers the problem of optimally placing fixed and switched type capacitors in a radial distribution network. The aim of this problem is to minimize the costs associated with capacitor banks, peak power, and energy losses whilst satisfying a pre-specified set of physical and technical constraints. The proposed solution is obtained using a two-phase approach. In phase-I, the problem is formulated as a conic program in which all nodes are candidates for placement of capacitor banks whose sizes are considered as continuous variables. A global solution of the phase-I problem is obtained using an interior-point based conic programming solver. Phase-II seeks a practical optimal solution by considering capacitor sizes as discrete variables. The problem in this phase is formulated as a mixed integer linear program based on minimizing the L1-norm of deviations from the phase-I state variable values. The solution to the phase-II problem is obtained using a mixed integer linear programming solver. The proposed method is validated via extensive comparisons with previously published results.
Reactive power flows in a radial distribution network always cause an increase in losses. At heavy loads, the losses due to reactive flows can become very significant. Moreover, these flows result in a line voltage drop that is greater than it would be at unity power factor. Consequently, capacitor banks are commonly installed on distribution lines to compensate for the customer reactive power requirement [1].
The problem of optimal capacitor placement on a radial network has been the subject of research for many decades thus resulting in a myriad of techniques [2], the most promising of which are based on optimization algorithms. Grainger at al. pioneered the application of non-linear programming methods to the problem of optimal capacitor placement [3]. In Ref. [3], both the capacitor locations and sizes were treated as continuous variables. Mixed integer non-linear programming methods were employed by Baran and Wu to select capacitor locations from a set of candidates [4]. Most recently, Aguiar and Cuervo [5] presented a mixed integer linear programming formulation that also accounts for the discrete capacitor sizes. The validity of this approach, however, strongly depends on the accuracy of the loss function approximation which has to be obtained via a properly chosen set of supporting hyper-planes [6]. The optimal capacitor placement problem in which capacitor sizes and locations can only take discrete values is believed to be non-deterministic polynomial (NP) complete [7], i.e., it is almost certain that solving for its global optimum cannot be done efficiently on a computer. According to many experts, a proof that a problem is NP complete is an adequate reason not to devote time and effort to trying to find a global solution [8]. Instead, it is recommended that one searches for a good near optimal solution of the problem. In fact, the published solution methods for the practical capacitor placement problem all adhere to this recommendation. The complexity of the problem has prompted many researchers to consider non-deterministic search techniques. Examples of such techniques which have been reported in the power systems literature include simulated annealing [9], genetic algorithms [10], tabu search [11], immune-based optimization [12], hybrid tabu search including heuristic features [7], hybrid micro-genetic algorithm in conjunction with fuzzy logic [13], ant direction hybrid differential evolution [14], a memtic evolutionary approach [15], and a variable scaling hybrid differential evolution method [16]. Although the above non-deterministic approaches have been validated on sample test systems, their success depends on tuning several algorithmic parameters. In general, these parameters are often system dependent and consequently their optimal setting requires skill on part of the user. A search algorithm which yields an optimal solution to the practical capacitor placement problem is therefore still sought after.
This paper proposes a deterministic two-phase approach to the optimal capacitor placement problem. Phase-I is based on conic programming. In fact, a recent paper by the author demonstrated that the radial load-flow problem could be efficiently solved using conic programming [17]. Phase-I extends the radial load-flow formulation by including candidate reactive sources at all nodes. The conic optimizer allocates reactive power to the sources with the aim of reducing the total system cost. In phase-I, the power allocated to each reactive source is treated as a continuous variable. Consequently, a computationally cheap global solution can be obtained using a path-following interior-point method. Phase-II takes into account the discrete nature of the sizes of the reactive power sources, i.e., the practical capacitor bank kVAr sizes. In essence, phase-II seeks a least absolute value solution having minimum deviation from the phase-I values of the state variables. The discrete nature of the capacitor sizes requires the least absolute value problem to be formulated as a mixed integer linear program, the solution for which can be obtained using branch-and-bound techniques. The main advantage of the phase-I/phase-II approach is that it can make use of existing high-powered software tools such as MOSEK [18] for solving conic and mixed integer linear programming problems. MOSEK [18] includes implementations of an interior-point method for conic programming and a branch-and-bound technique for mixed integer linear programming.
The rest of this paper is organized as follows. Section 2 reviews the formulation of the radial load-flow problem as a second-order cone program. Sections 3 and 4 discuss the phase-I and phase-II approaches, respectively. To simplify the presentation, one load level is initially considered. Section 5 extends the solution algorithm to account for load variations and capacitors of both switched and fixed types. Simulation results are reported in Section 6 and compared with solutions previously published in Refs. [5], [7], [10], [11], [13] and [15]. The paper is concluded in Section 7.
This paper presented a two-phase approach for the optimal capacitor placement problem in a radial network. The approach benefits from the availability of state of the art software packages for conic optimization and mixed integer linear programming. The validity of the obtained solutions was confirmed by comparison with previously published results. The proposed method is deterministic and therefore yields repeatable solutions. This is unlike many of the previously published non-deterministic techniques (e.g. [7], [10], [11], [13] and [15]) that cannot ensure solution repeatability – a fact that may not be favored by many system operators and designers. Moreover, as compared to non-deterministic methods, the proposed approach does not require executing extensive load flow simulations nor does it depend on tuning system dependent algorithmic parameters. The numerical performance of the proposed method improves as the solver technology continues to improve.
