The OPF has several applications in power systems analysis and design [1, 6]. Due to the vast complexity of today’s modern power networks they are more prone to incurring instability due to even smallest undesired changes in their operation. The OPF therefore seems like a reliable tool for devising multiple assessment scenarios implemented to a power system in order to ensure its continued safe operation. For instance, strong AC couplings exist in a power network with AC line/cable interconnections, which in cases of power imbalances, due to a sudden loss of generation or line tripping in one area that causes a change in network operating frequency, are likely to induce frequency deviations to units in other areas which eventually leads to system collapse [2].
Considering the feasibility criteria of OPF, applying an economic dispatch analysis achieved by OPF algorithm to this particular system will guarantee the safe distribution of loads between multiple generating points while maintaining generation at an optimum level that agree well with system’s operational as well as equipments’ physical constraints, hence keeping the balance between demand and
generation at all times and minimising the possibility of equipment failure as well as other undesired dynamic responses. This very fact makes OPF an essential tool for modern network analysis, planning and design.
The OPF by definition is a constrained non-linear convex optimisation problem and therefore it belongs to the category of non-linear programming. Non-linear programming refers to the group of optimisation problems in which the objective function to be minimised (or optimised) or constraints show non-linearity [9-11]. Convexity on the other hand means that the solution space contains at least one global minimum [7, 9, 17]. It is necessary to mention that the OPF solutions carried out throughout this thesis yield the best possible solution, which from practical perspective is the optimum solution. Depending on the types of constraints used, the optimisation problems are categorised into three main groups, namely Equality Constrained Problems (ECP), Inequality Constrained Problems (ICP) and General Programming Problems [8, 10].
A general programming problem refers to those classes of optimisation problems, which contain both equality and inequality constraints. Most of the optimisation problems applied to physical systems (power networks included) are of this type. Within the power systems paradigm, the equality constraints refer to the conditions which must hold if the system is to continue normal steady-state operation, in other words, the operation of a given power system is stable as long as the nodal power balance equations hold for each bus. Moreover, the inequality constraints are the result of implementing network’s realistic operating conditions as well as equipment limits, for instance complex voltage in each node in an inter-connected power system is bounded by its upper and lower margins which are then enforced to ensure system operates within its static stability margins. Several solution methods have been proposed to solve the general programming problems. The most conventional and reliable method is to use numerical solution methods aimed at decreasing the gradient of the problem’s objective function. These are collectively known as gradient-based methods. There are generally three categories of such solution algorithms for a general non-linear programming problem such as the OPF [3, 8- 11].
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II. Exterior Point Methods (e.g. Quadratic Penalty Function)
III. Exact Penalty Function Methods (e.g. Augmented Lagrangian Function)
Lagrangian and penalty function methods share the same mathematical principles that aim to convert the constrained problem of OPF into a single (or a series of) unconstrained problem by penalising the objective function for points outside the feasible solution space (hence the name exterior point method) [9, 10]. The main difference between the two is that in penalty function methods (for instance in quadratic penalty function method), the objective function is penalised directly whereas in Lagrangian type methods, it is the Lagrangian function (formed via the use of Lagrangian multipliers) that is penalised. The latter has considerable numerical advantage over the former approach in that the optimal solution is reached without having to enlarge the penalty parameters of the penalty function to near infinity, a common problem in exterior point methods which introduces ill- conditioning and therefore numerical difficulties [8-10]. The augmented Lagrangian function by comparison is therefore considered as an improvement to the penalty function method, for it is only necessary to form one single unconstrained problem by combining a Lagrangian function (using multipliers) and a quadratic penalty function (using penalty parameters) together, therefore it has a better convergence rate than pure penalty function methods (obviously given the right initial conditions).
Another alternative to exterior point methods is the use of Barrier Functions [8-11, 18-22]. The barrier functions (typically logarithmic) prevent the solution points of the dual problem (unconstrained penalised function) from crossing the feasible space by setting barriers against its boundaries [9]. Because in this method the optimum is reached from within the solution space they are formally called Interior Point Methods.
Over the course of the years, comprehensive research has been carried out in the area of OPF on both methods (exterior or interior point) and there are several publications in open literature that address the problem of the OPF [1, 4-7, 18-35].
These works are normally divided into two categories; the first group pertains to the principal analysis and definition of OPF, which has been developed since the late 60s. One of the most important works done in the area of OPF formulation is the first-order gradient decent approach proposed by Dommel and Tinney in [1]. In this paper published in 1968 the principles of a Jacobian based OPF solution algorithm via Newton’s method is presented, which attempts to minimise a set of linear equations developed from Lagrangian function of the system by directly evaluating a gradient of objective function. Since this method uses Jacobian terms to evaluate the state variables via Newton’s method just like a conventional power flow problem (section 2.2) it gets highly complicated in real multi-node systems, it has also less convergence rate (although it maintains quadratic convergence) than higher order methods such as explicit Hessian-based solutions [4, 5].
On the other hand, applying Newton’s method to explicit Hessian matrix would result in improved convergence rate at the expense of losing the higher degree of sparsity in the matrix of coefficients. The less sparsity of Hessian matrix is a mathematical fact and stems from the definition of the Hessian as being the second order partial derivatives of a function (in case of a power system, nodal powers) with respect to state variables (for instance nodal voltages or phase angles). According to definition of Hessian/Jacobian terms the non-neighbouring partial derivative terms in the Jacobian matrix are always zero but not in the Hessian matrix, which will ultimately yield to a more crowded Hessian matrix for the same system [4]. As an improvement to the Hessian approach a newly defined second order partial derivatives matrix of coefficients is introduced in [6] by direct evaluation of the Lagrangian multipliers in the system of linear equations, thus combining both Hessian and Jacobian terms to achieve better sparsity and yet better convergence rate. One of the difficulties of the method developed in [6] is in the nature of active inequality constraints, however the constraint handling has been improved in [7, 23, 26] with introducing the augmented Lagrangian function by combining multipliers and penalty functions.
From globally convergent algorithm to improvements in interior-point methods, there is a diverse range of different methods to solve the OPF problem. However in this research project the proposed solution algorithm has been the Newton’s method for an augmented Lagrangian function [6, 7] combining the strong attributes of both
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Newton’s method and non-linear programming method of augmented Lagrangian function. The second group of papers relate to the variety of approaches (solutions) and modifications taken regarding the OPF mathematical solution algorithms [18- 20, 22, 28-31, 34-36].
Most recently the trend in developing solution algorithms for OPF problem has been slightly shifted from gradient-based conventional numerical analysis (such as the augmented Lagrangian method) to direct search methods, heuristic approaches and evolutionary programming, and algorithms such as Particle Swarm Optimisation have come to light in the realm of power systems research [24, 27, 32, 33]. These so-called alternative approaches shall be considered in a separate section (section 2.5) at the end of this chapter but it should be mentioned here that analysing various approaches to the problem of optimisation is a purely mathematical argument, which is out of the scope of this research project. In the subsequent paragraphs, however, the basics of the OPF solution algorithms based on Lagrangian methods, has been presented.