Applying Convex Optimal Power Flow to Combined Economic and Emission Dispatch

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1. Introduction

Electricity generation from fossil fuel based power plants is one major source of greenhouse gas emissions, [1] [2]. According to reference [1], 37.5% of total CO_{2} emissions are from the global electricity supply sector. The current approaches to mitigate CO_{2} emissions include more efficient fossil fuel conversion, switching to low- carbon energy resources, decarbonisation of fuels and nuclear power, [1]. It is critical to analyze cost and performance of different CO_{2} reduction or capture approaches, [3]. The combined economic and emission dispatch (CEED) considered in this paper can be categorized to efficient management or market arrangement to reduce CO_{2} emissions. Reference [4] formulate the combined economic and emission dispatch (CEED) by converting the bi-objective optimization problem to weighted single objective optimization. The Gravitational Search Algorithm (GSA) is used to solve the nonconvex CEED model in [5]. Reference [6] proposes to apply nondominated sorting genetic algorithm-II to find pareto-optimal solutions of the formulated dynamic economic emission dispatch problem in one single run. Reference [7] deploys Particle Swarm Optimization (RPSO) to solve CEED in distribution system. The core mathematical model in the CEED is optimal power flow (OPF). OPF solves a system level objective constrained by power network physical and operational conditions, [7-12]. Current methods to solve optimal power flow problem include using DC OPF model and nonconvex AC OPF model. DC OPF assumes voltage magnitude of all buses or nodes in the networks are fixed to 1 per unit value and the voltage phase angle difference across each line is small enough so that, [7]. DC OPF neglect resistance parameter of power transmission lines. These assumptions are valid for most high voltage power transmission networks. But in low voltage power distribution networks, DC OPF is not valid, [8]. Even for high voltage power transmission networks, it is still possible that assumptions used in DC OPF fail to meet real world situations, [8-9]. For the nonconvex AC OPF, current nonlinear programming solvers are only able to find local optimal solutions, [10]. There is no guarantee to obtain global optimal by using nonconvex AC OPF. Though branch and bound is promising method to find global optimal solution of nonconvex AC OPF, the computation time of branch and bound method is too long to be implemented in practice especially for large scale networks, [11]. The advantage of convex methods in CCEED is that global optimal solutions can be guaranteed by mature nonlinear programming solvers such as MOSEK, [13].

2. Convex Combined Economic and Emission Dispatch

For ease of illustration, the nonconvex AC optimal power flow model used in the economic dispatch problem is reformulated here as (1)-(11). The formulations in (1)-(11) are based on transmission line sending end power injection variables. The advantage of including voltage phase angle explicitly in the formulation (1)-(11) is that we can obtain voltage phase angle solutions directly by solving this model.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

where is the index of bus or node in the power network. is the index of transmission line or distribution line. in equation (1) is the objective function. Note that the variables in the objective function (1) are only for purpose of illustration. The objective function of OPF can be formulated in any way based on the operation targets. is the active power injection at bus. is the active power load at bus. is the reactive power injection at bus. and are power network operation parameters. is the active power transmitted at the sending end of line. is the reactive power transmitted at the sending end of line. is the network topology matrix with if node is at the sending end of line, if node is at the receiving end of line and if node is not connected with line. is the network topology matrix with if node is at the receiving end of line and if node is at the receiving end of line or not connected with line. and are network topology parameters. is the voltage magnitude square at the sending end of line. is the voltage magnitude square at the receiving end of line. is the voltage magnitude at the sending end of line. is the voltage magnitude at the receiving end of line. is the resistance of line. is the reactance of line. and are network physical parameters. is active power loss of line. is reactive power loss of line. is voltage phase angle difference of line. is the upper bound of active power loss. Equation (2) represents the active power balance for each bus. Equation (3) represents the reactive power balance for each bus. Constraint (6) is active power loss constraint. Constraints (8)-(11) are bounds for OPF variables. Equation (5) and (6) are nonconvex. We approximate equation (5) in the original AC optimal power flow model by equation (12).

(12)

Apparently, the underlined assumptions for Equation (12) to be valid are (a)-(b).

(a)

(b)

Because the operation constraint for voltage magnitude (per unit value) is generally and the voltage phase angle difference is generally small along transmission lines, assumptions (a)-(b) are valid in most situations during power system real operations.

Equation (6) is convexified by rotated cone expressed in (13).

(13)

The method used in (13) to make equation (6) convex is relaxation. In this case, we actually relax quadratic equality constraint to quadratic inequality constraint. The tightness of this relaxation can be guaranteed by implicitly include power loss component in the objective function. This has been proved by numerical results, [13]. Thus, if we use OPF model [(1)-(4), (7)-(13)] instead of [(1)-(11)], we can obtain a convex OPF model. This model can be used to formulate combined economic and emission dispatch problem. Generally the objective function in (1) is the cost of active power generation from all the generators formulated as quadratic function:

(14)

where and are cost coefficients for active power generation. These parameters can be obtained directly from MATPOWER, [14].

To include the CO_{2} emission reduction target to the economic dispatch problem, we formulate the combined economic and emission dispatch by assigning weights to the cost and CO_{2} emission parts in the objective function as follows (15).

(15)

where are weights for corresponding terms in the objective function satisfying. These weights can be adjusted by the system operator depending on the importance of cost and CO_{2} emission. Obviously, when and the CCEED problem is reduced to power system economic dispatch. When and the CCEED problem is reduced to CO2 emission minimization without considering economic cost. in Equation (15) is the CO_{2} emission coefficient for fossil fuel power plant. It is calculated by Equation (16). The results of Equation (16) are listed in Table 1. Emission factor and power plant efficiency data in Table 1 are from reference [15]. The power generation technology for steam coal, fuel oil and diesel oil is assumed to be simple cycle combustion turbine. For natural gas, the combustion turbine is assumed to be combined cycle based technology.

(16)

Table 1. CO_{2} Emission Coefficients for Different Fuel Types.

Table 2. IEEE14 Test Case CCEED Results.

Table 3. IEEE57 Test Case CCEED Results.

3. Test Results

The CCEED model is coded in the General Algebraic Modeling System (GAMS). MOSEK solver in GAMS is used to solve the CCEED model. Generator and IEEE test network parameters in MATPOWER [14] are directly used. To differentiate the CO2 emission parameters for different generators, we assign the four different fuel types evenly to different generators. A laptop running on windows 7 64-bit operating system with Intel i7-2760QM 2.4 GHz CPU and 8G RAM is deployed. The CCEED results for IEEE14, IEEE57 and IEEE118 [16] test cases are listed in Tables 2-4. The performance of CCEED is demonstrated by varying weights parameters () in the objective function (15). The CPU computation time of all test cases in GAMS is less than 0.1 second.

It can be observed from Tables 2-4 that with the increase of weights on the CO_{2} emission term of the objective function in CEEED, the CO_{2} emission is decreasing. The cost of power production increase sharply when and. This means to minimize CO_{2} emissions without considering economic cost is not feasible in reality. As a good compromise, and can be regarded one cost effective arrangement to reduce CO_{2} emissions.

Table 4. IEEE118 Test Case CCEED Results.

4. Conclusion

In this paper, we prove the feasibility of using convex optimal power flow model to solve the combined economic and emission dispatch problem. The original nonconvex optimal power flow model is approximated and relaxed by mathematical techniques. The underlined assumptions of these approximations are explained in detail. The CCEED problem is then formulated by assigning weights to the power generation cost minimization objective and CO_{2} emission minimization objective. The CCEED model is built in GAMS platform and solved by MOSEK. Numerical results from IEEE14, IEEE57 and IEEE118 test cases show that CCEED can be solved efficiently. By adjusting the weights for cost and CO_{2} emissions in the objective of CCEED, a compromise between power generation cost and CO_{2} reduction can be achieved. Instead of nonconvex model, our convex model can guarantee global optimal solutions. Though we demonstrate the usefulness of CCEED by reducing CO_{2} emissions here, the potential applications of CCEED can be extended to reducing other green-house gas emissions or air pollutants.

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