Motivation and Proposal

The challenging task of choosing the optimal location and amount of reactive resource has been explored in the literature using many methodologies, but the research in this area is far from being perfect. Most of the methods use time domain simulation as a means to check the performance of voltages during dynamic analysis. Dynamic analysis is not considered explicitly during the process of identifying the optimal locations and amount, mainly because of the computational difficulties involved in such procedure. But time domain simulation with proper representation of appropriate load models plays a crucial role in dynamic VAR planning studies. If system dynamics are not taken into account properly, the proposed control solution may be an expensive over design or an under design which is not capable of mitigating FIDVR problems completely. This brings out the need for the use of dynamic optimization in dynamic VAR planning studies.

Direct simultaneous based dynamic optimization has been used in the literature for dynamic VAR planning. However, this method suffers from the curse of dimensionality as the power system size increase. To overcome this issue, a direct sequential based dynamic optimization formulation has been proposed in this work. The main motivation behind using this method is to separate the dynamics of power system from optimization problem part and utilize commer- cial grade software to solve power system dynamics and optimization. This approach considers the dynamics part during the optimization procedure, but performing dynamics simulations are clearly separated from the optimization procedure. This feature of this approach will help to easily extend this method for large scale power system.

The following section provides background material related to various dynamic optimization methods. .

2.2.1 Dynamic optimization - background information

Dynamic optimization is a class of optimization problems where the optimal solution is ob- tained by considering the dynamics of the system. Mathematically, the dynamic VAR allocation problem can be modeled as a dynamic optimization problem. Dynamic Optimization [27] is an

optimal control problem (OCP), where the optimum values for control and parameters which minimize a certain performance measure are identified. The optimal control should satisfy the dynamics of the system and also the path constraints on system variables. If the objective function has integer variables, then the OCP becomes a mixed integer dynamic optimization (MIDO) problem. In general the OCP is defined as given in (2.1).

minimize tfinal,u,p J = φ(tfinal, u, p, z) + Z tfinal t0 L(x, u, t, p, z)dt Subject to ˙ x = F (x, u, p) 0 = G(x, u, p) ci(x, u, p) ≤ 0 t ∈ [t0, tfinal] , zi∈ [0, 1] x ∈ Rnx_{, u ∈ R}nu_{, p ∈ R}np (2.1)

The objective function of the general OCP problem is given by J and it has two parts - (a) Mayer type, which depends upon final time (tfinal) constraints and (b) Lagrangian type,

which is given by the function L. In OCP, the variables are separated into two classes namely the state variables (x) and control variables (u). The evolution of state variables is dictated by the control variables, via a set of differential and algebraic equations (DAE), which are specified by functions F and G in (2.1). Further, the control and state variables are subjected to performance constraints give by Ci. The parameters and binary variables in the system

are denoted by p and z respectively. The number of state variables, control variables and parameters are given by nx, nu and np respectively.

Numerous analytical and numerical based methods are available to solve the OCP, which are broadly classified into indirect and direct methods. Indirect methods such as Bellman’s optimality principle and Pontryagin’s maximum principle are analytical methods and they are based on the principle of variations[28]. The application of this method to VAR allocation problem is very difficult because of the complexity and size of the problem.

Direct methods are based on the principle of discretizing the optimal control problem and then applying the non-linear programming (NLP) techniques to the resulting optimization problem. These methods take the advantage of the state of the art NLP solvers. Also, these methods can be applied to system described by ODE (ordinary differential equations), DAE and PDAE (partial differential and algebraic equations) models [4]. The three main variants of direct method of solving optimal control problems are direct simultaneous approach, direct sequential approach and multiple shooting methods [29].

Direct simultaneous approach transforms the OCP problem by discretizing the constraint differential and algebraic equations (DAEs) into a set of algebraic equations at every simulating time steps. This method is also referred as state vector parameterization (SVP) or full dis- cretization method or orthogonal collocation method. There are various discretization schemes such as Implicit Euler method, Trapezoidal method and Radau collocation on finite elements to transform the dynamic optimization problem into a non-linear programming problem. The full discretization method tries to satisfy the dynamic equations and optimality conditions together. As the size of the power system increases, the resulting NLP size will increase and obtaining a solution will be difficult due to convergence problems. Besides the curse of dimensionality, due to the close coupling between model and optimization in the discretization process, detailed dynamic models are difficult to embed into dynamic optimization.

The direct sequential method translates the optimal control problem into an NLP by only discretizing the control variables. The system dynamics are still embedded in the NLP problem, but are handled separately by a numerical integrator. This method is also referred as control vector parametrization (CVP). The NLP algorithm adjusts the control variables based on violation of constraints on state variables generated by system dynamics solver and the gradient information. The gradient information provides details on how the objective function and constraints vary for a change in the control variables. With this approach, large scale dynamic optimization problems can be handled in a better way compared to simultaneous approach.