A Linear Programming Method Based Optimal Power Flow Problem for Iraqi Extra High Voltage Grid (EHV)

The objective of an Optimal Power Flow (OPF) algorithm is to find steady state operation point which minimizes generation cost, loss etc. while maintaining an acceptable system performance in terms of limits on generators real and reactive powers, line flow limits etc. The OPF solution includes an objective function. A common objective function concerns the active power generation cost. A Linear programming method is proposed to solve the OPF problem. The Linear Programming (LP) approach transforms the nonlinear optimization problem into an iterative algorithm that in each iteration solves a linear optimization problem resulting from linearization both the objective function and constrains. A computer program, written in MATLAB environment, is developed to represent the proposed method. The adopted program is applied for the first time on Iraqi 24 bus Extra High Voltage (EHV) network (400 kV). The required are data taken from the operation and control office, which belongs to the ministry of electricity


I. INTRODUCTION
Throughout the entire world, the electric power industry has undergone a considerable change in the past decade, and will continue to do so for the next several decades. In the past, the electric power industry has been either a government-controlled or a government-regulated Industry which existed as a monopoly in its service region. All people, businesses, and industries were required to purchase their power from the local monopolistic power company. This was not only a legal requirement, but a physical engineering requirement as well. It just did not appear feasible to duplicate the resources required to connect everyone to the power grid. Over the past decade, however, countries have begun to split up these monopolies in favor of the free market Barkovich 1996, Morgan 1996and Rudnick 1996.
Optimal Power Flow (OPF) solution methods have been developed over the years to meet this very practical requirement of power system operation Acha 2000, El-Hawary 1986,Giacomoni 2010 and Huneault 1991.
The optimal power flow problem has been discussed since its introduction by Carpentier Khaled 2008. Because the OPF is a very large, non-linear mathematical programming problem, it has taken decades to develop efficient algorithms for its solution. Many different mathematical techniques have been employed for its solution. The majority of the techniques discussed in the literature use one of the following five methods The large -scale application of LP -based methods has traditionally been limited to network constrained real and reactive dispatch calculations whose objectives are separable, comprising the sum of convex cost curves. The accuracy of calculation may be lost if the oversimplified approximation is adopted in LP -based OPF. The piecewise linear segmentation of the generator fuel cost curve should be good for avoiding this problem. The piecewise approach can fit an arbitrary curve convexly to any desired accuracy with a sufficient number of segments. Originally, a separable LP variable had to be used for each segment, with the resulting large problems with multi segments cost curve modeling were prohibitively time and storage consuming. The difficulty was alleviated considerably by a separable programming procedure that uses a single variable per cost curve, regardless of the number of the segments. However, the number of segments still affects the solution speed and precision, Jizhong Zho 2009. This paper presents an LP-based OPF for generation cost minimization using as control variables the generator active power and generator voltage. It is intended to overcome the constraints of current LP-based OPF algorithms. The main problem of the current algorithms is the loss of accuracy of the linear approximation of the objective function when the changes of the control variables are not small enough. An attempt to address this issue consists of imposing limits to the deviation of control variables, Alsac 1990 . Although this approach solves the problem, the convergence of the algorithm becomes very slow. The LP based OPF proposed in this paper improves the accuracy of the linear approximation of the objective function. The objective function is approximated by a piecewise linear function determined iteratively by segmented the objective function in each iteration.

PIECEWISE LINEAR APPROXIMATION OF OBJECTIVE FUNCTION
Assuming that the objective function is a quadratic characteristic, the objective function can be linearized by a piecewise linear approach. If the objective function is divided into N linear segments, the real power variable of each generator will also be divided into N variables. Fig. 1 is an objective function with three linear segments. The corresponding slopes are b 1 , b 2 , and b 3 , respectively Jizhong Zho 2009. From Fig. 1, the generator power output variables for each segment can be presented as below: If is selected as the initial generator output power, the incremental generator power outputs for each segment can be expressed as: The piecewise linear objective function becomes Where; : The number of generators : The minimal real power output at generator i :The maximal real power output at generator i

MATHEMATICAL FORMULATION OF THE ALGORITHM
The objective function contains real power generation cost. Mathematically, it is formulated as follows: Where In a high -voltage power network, the value of is very small, and the following approximate equations are easily obtained: Journal of Engineering Volume 20 -2014 Number 4 -April

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In addition, assume that the magnitudes of all bus voltages are the same and equal to 1.0 p.u. Furthermore, suppose the reactance of the branch is much bigger than the resistance of the branch, so that we can neglect the resistance of the branch. Thus, Substituting Eq. (18) to Eq. (21) in to Eq. (17) , we get The above equation can also be written in matrix form, i.e.,

ΔPb=B′Δδ
Where the elements of the susceptance matrix B′ are The bus power injection equation can be written as = Since the load demand is constant, the linearization expression of Eq.(26) can be written as below: The above equation can also be written in the following matrix form Eq. (29) stands for the relationship between the incremental generator output power (except for the generator that is taken as slack unit) and the incremental bus voltage angle. Matrix H can also be simplified by using Eq. (18) and Eq. (21). According to Eq. (23) and Eq. (29), we can get the direct linear relationship between the incremental branch power flow and incremental generator output power, i.e., Where It is also called as the linear sensitivity of the branch power flow with respect to the generator power output.
Thus the linear expression of the branch power flow constraints can be written as: The element of the matrix Δ is the incremental power flow limit Δ of the branch ij, i.e., The incremental form of the generator output power constraint is , Jizhong Zho 2009

THE PROPOSED METHOD IMPLEMENTATION
The above mentioned method for solving optimal power by LP uses an iterative technique to obtain the optimal solution, so it is also called a successive linear programming (SLP) method. The solution procedures of SLP for optimal power flow are summarized below: Step1. Select the set of initial control variables Step2.Solve the power flow problem to obtain a feasible solution that satisfies the power balance equality constraint. Step3. Linearize the objective function and inequality constraints around the power flow solution and formulate the LP problem. Then solve the LP problem and obtain the optimal incremental control variables Δ . Step4. Update the control variables : = + Δ Step5. Obtain the power flow solution with updated control variables.
Step6. Check the convergence. If Δ in Step 4.Are bellow the user-defined tolerance, the solution converges. Otherwise, go to Step 3. Fig. 2 shows the flowchart of linear programming method Based optimal power flow problem.

CASE STUDY
The Iraqi 400kV (EHV) network shown in Fig. 3 was chosen to implement the proposed LP algorithm for OPF.
The Iraqi EHV network consists of 24 bus bars, 38 transmission lines and 11 generating stations. Two operational case studies for the Iraqi network were chosen to be studied by this paper for optimal power flow solution. These two case studies are with cheap and expensive international fuel price conditions. All the data for this work was taken from the Iraqi operation and control office that belongs to the ministry of electricity.

Journal of Engineering
Volume 20 -2014 Number 4 -April Table 1 indicates transmission system parameters in p.u. / km (at a base of 100 MVA) for the three types of the transmission lines used in the Iraqi network.

RESULTS
The algorithm described in this paper has been coded in MATLAB (R2008a) language. The performance of the algorithm is illustrated considering for a state of load of the operation of the Iraqi power system. The results obtained from using Linear Programming (LP) method are compared with the results obtained from power flow solution using Newton-Raphson method. It is worth mentioning that the distribution of loads on the power plants identified for Newton-Raphson so that we get less losses. There are four power plants run on two types of fuel to generate electric power, so we compared the results when they operate on the cheap fuel type and expensive fuel type. Fig.4 shows the voltage magnitude in per unit for each bus when cheap fuel price is used to generate power in power plants that operate on two types of fuel and for the two algorithms (Newton-Raphson and Linear Programming). Fig. 5 shows the generation of each plant for Newton-Raphson power flow solution compared with Linear Programming method when cheap fuel price is used, Fig. 6 shows the production cost when cheap fuel price is used Fig.7 shows the voltage magnitude for each bus when expensive fuel price is used to generate power in power plants that operate on two types of fuel and for the two algorithms (Newton-Raphson and Linear Programming). Fig. 8 shows the generation of each plant for Newton-Raphson power flow solution compared with Linear Programming method when expensive fuel price is used. Fig. 9 shows the variation of production cost through optimization using Linear Programming method (LP).

CONCLUSION
The Linear Programming (LP) algorithm is used for the first time on the Iraqi Extra High Voltage (EHV 400kV) Grid for optimal power flow to minimize the active power generation cost. paper has presented a LP based.
The problem constraints are the coupled linearized power flow equations and the system variable limits. A piecewise linear approximation of the objective function is built by adding iteratively a tangent cut in each iteration. It can be also note that the results of the production cost are significantly decreased when using Linear Programming with the results derived in the case of Newton-Raphson. From Table 2 there is about 30.16% decrease in the production costs when using cheap fuel type, whereas there is about 28.2% decrease in production costs when using expensive fuel type as given in Table 3.
The generator fuel cost function Self conductance G ij Mutual conductance Active load power at bus i P Gi Active power generated by unit i Minimum generator active power limit Maximum generator active power limit    .