Economic Dispatch of Micro-grid Based on Sequential Quadratic Programming — Model and Formulation

. This paper proposes an economic dispatch strategy for micro-grid based on the sequential quadratic programming (SQP) method. In this paper, in the off-grid operating state, according to the load, the kinds of micro sources and the operating cost, the power output minimizing the operating cost is calculated. When the system is connected to the grid, considering the tiered price, generation cost of CHP (Combined heat and power) based on micro-turbine, and generation cost of fuel cell, the paper calculates micro source power output based on the minimum operating cost. The characteristics of the proposed model and its benefits are investigated through several case studies.


Introduction
Compared with the traditional power system, micropower sources in micro-grid generally include intermittent DG and micro gas turbine. With the development and application of renewable energy, micro-grid construction is imperative. How to ensure the safe, stable and economic operation of micro-grid is a hot issue of current research [1][2]. This paper introduces an economic dispatch strategy of micro-grid. The main contribution of this paper is the formulation of the economic dispatch problem based on sequential quadratic programming. The rest of this paper is organized as follows: Section 2 gives the mathematical model of the economic dispatch problem. In Section 3, realization of sequential quadratic programming is presented. Section 4 gives the conclusions.

Establishment of the Mathematical Model of Micro-grid Economic Dispatch
The objective function of micro-grid economic dispatch can be formulated as [3]: where, () ii FP is the fuel consumption function of unit i, Pi is its output power.   The system shown in Figure 1 is built based on Yantai micro-grid, and the distribution network with a voltage level of 380V is adopted as the point of common coupling (PCC). In the system, micro source includes photovoltaic power generation, fuel cell power generation, wind power generation, and micro-gas turbine power generation. Its capacity configuration is as follows: photovoltaic power 5 kW, wind power 20 kW, micro gas turbine 130 kW and fuel cell 15 kW.

Idea of Sequential Quadratic Programming
The optimal search direction of sequential quadratic programming [4][5] (SQP) is determined by the variable scale method, and the scale matrix is used as the symmetric positive definite iteration matrix in quasinewton method. In each iteration, the objective function value is minimized, and the positive definite direction matrix is maintained so that the Hessian matrix can be approximated. Therefore, SQP has the feature of global convergence and maintains local superlinear convergence, which is a very effective method and is recognized as one of the most effective algorithms for solving nonlinear constrained optimization problems. This algorithm can obtain the optimal solution of the original problem by converting the original problem into a series of quadratic programming subproblems, and obtain the quadratic approximation of Lagrange function, so as to provide the approximation degree of quadratic programming subproblem, and can also calculate the optimization problem with strong nonlinear. For a constraint problem With constraints: where, i  is the Lagrange factor. By linearizing the nonlinear constraints, the QP subproblem can be obtained, its objective function is The constraint conditions are: where, x + . The scalar step parameter i  is determined by appropriate linear search process so that an index function is worth enough reduction.

Update of Hessian matrix
where, In each iteration, the BFGS method is used to calculate the positive definite quasi-newton approximation H of Hessian matrix of Lagrangian function. The Hessian matrix remains positive as long as T kk qs is positive and H is initialized to a positive definite matrix.

Solve quadratic programming subproblems
The quadratic programming problem shown in Eq. (15) can be expressed as (a) equality-constrained quadratic programming problem and (b) inequality-constrained quadratic programming problem.
(a) the quadratic programming problem with equality constraints can be expressed as follows: The objective function is The constraint conditions are . Assume that rankA m n =.
(b) the quadratic programming problem with inequality constraints can be expressed as follows: The objective function is The constraint conditions are The basic idea of SQP method is that, at the site of an approximate solution, the original nonlinear programming problem is simplified to a quadratic programming problem, calculate the optimal solution, if any, is believed to be the optimal solution of the original nonlinear programming problem, otherwise, construct a new quadratic programming problem, continue the iteration until the solutions approximate constraints of the original problem.
The solution flow chart is shown in Figure 2.