An Optimal Generation Scheduling Approach Based on Linear Relaxation and Mixed Integer Programming

This paper proposes an optimal generation scheduling approach based on linear relaxation and mixed integer programming, which is used to solve the generation dispatch problem. The quadratic transmission loss constraint of each transmission line is converted into linear constraints by using the linear relaxation and mixed integer programming technique. Consequently, the original optimal generation scheduling problem is formulated as a quadratic programming or mixed integer quadratic programming problem that can be solved by commercial optimization solver. In order to improve the efficiency of algorithm, this paper further analyses the generation scheduling model and deletes the redundant variables and constraints. Three test systems, including IEEE 30-node system, IEEE 118-node system, and Polish 2746-node system, are employed to examine the effectiveness of the proposed method. The comparative results obtained by the proposed method, quadratically constrained quadratic programming method (QCQP), and solving constraint integer programs solver (SCIP) verify the effectiveness of the proposed method in solving the optimal generation scheduling problem.


I. INTRODUCTION
As a major energy consumer, the power industry's slight improvement in generation scheduling will have a significant impact on the national economy and social environment. The energy distribution in China has obvious regional characteristics [1]. High-voltage long-distance transmission plays an important role in the transmission structure, and transmission loss becomes a key factor of generation scheduling. On the other hand, one important direction of renewable energy utilization-power generation and grid connection, has a growing proportion in the traditional power system, which brings great challenges to optimal the generation scheduling problem [2]. The growing penetration of DERs has made it possible for traditional passive consumers to evolve into active prosumers. Compared with traditional consumers, prosumers are capable of managing their energy generation, storage and consumption simultaneously. In China, the centralized The associate editor coordinating the review of this manuscript and approving it for publication was Huai-Zhi Wang . dispatching strategy is still the main dispatching method, when dispatching the generation plan, the dispatching department simply converts network losses into forecasted loads in a certain proportion, but lacks of accurate network loss model [3]. In order to determine an accurate generation scheduling plan, it is necessary to take network losses into account in the generation scheduling model [4], [5].
Many researches have been done on the generation scheduling of traditional power grids considering renewable energy generation and grid connection [6]- [11]. In this paper, the above problems will be classified and explained from two aspects: model and algorithm. In terms of generation scheduling optimization models, the DC optimal power flow model and the AC optimal power flow model are often used. (1) The first scheduling model is focus on reasonably arrange the power generation of different power plant generators on the premise of satisfied the load demand to achieve the effect of allocating electricity and saving energy and reducing emissions [7], [8], [10]. However, this model lacks consideration of the influence of network topology on VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the scheduling scheme. For this scheduling model, the commonly used optimization methods include equal increment rate criterion, dynamic programming, linear and quadratic programming methods [11]. (2) The DC optimal power flow model is a simple and efficient generation scheduling model [12], [13]. This model considers the transmission constraints of each line, but the model is based on a lossless network and lacks consideration of transmission loss constraints of each transmission line on the generation scheduling plan. Therefore, this scheduling plan often has a large deviation.
(3) The AC optimal power flow model is a more comprehensive and widely used generation scheduling model [14], [15], which aims at minimizing total power generation cost, total coal consumption, or network loss, the constraints the transmission capacity and voltage amplitude of each line are taken into consideration. However, it is difficult to accurately predict the reactive load in the day-ahead scheduling of generation plan, which limits the application of the AC optimal power flow model to a certain extent [16]. Moreover, the model often needs to be solved by non-linear programming methods such as the interior point method which the efficiency and robustness still need to be strengthened.
In view of the influence of the network loss of renewable energy generation on the generation scheduling problem, [17] and [18] used the Kron formula to calculate the network loss, but this method can only roughly calculate the value of the network loss, and the transmission loss coefficient of Kron formula needs to be recalculated as the operating conditions of the system change. Reference [4] proposed a dynamic linear segment method to solve the problem of generation scheduling considering network loss. This method linearizes the transmission loss curve of each transmission line in a dynamic way until the mismatch satisfies the convergence conditions. Reference [5] relaxed the transmission line loss constraints with quadratic equality to the quadratic inequality constraints based on the relaxation technique. For the transmission line loss constraints that do not satisfied the equality constraints, the corresponding transmission line loss variables are dealt with by adding the terms of penalty. By employing of the terms of penalty, this method tends to minimize network loss, which changes the nature of the original problem [19].
With the proportion of renewable energy generation increasing, in order to effectively solve the generation scheduling problem considering network losses, this paper proposes an optimal generation scheduling approach based on linear relaxation and mixed integer programming. Three test systems, including IEEE 30-node system, IEEE 118-node system, and Polish 2746-node system, are employed to test the effectiveness of the proposed method. The comparative results obtained by the proposed method, quadratically constrained quadratic programming method (QCQP), and solving constraint integer programs solver (SCIP) verify the effectiveness of the proposed method in solving the optimal generation scheduling problem.

II. GENERAL FRAMEWORK
The real power leaving out of a node by using DC power flow can be shown as (1).
where l ij is the end bus i and j of a transmission line, V i and V j are voltage magnitudes of bus i and j while θ ij = θ i − θ j and θ i and θ j are voltage angles of bus i and j. G ij and B ij are the conductance and susceptance of a transmission line, respectively. However, in real engineering, the injection active power flow of the line is should be calculated as follows: where V ij,N is rated voltage, r ij and x ij are the resistance and reactance of the transmission line, respectively and r ij x ij . However, the real power flow of transmission line l ij can be approximately expressed as the average of the real power difference between the two ends of the transmission line that can be expressed as (3). The transmission loss can be calculate by adding the both ends of a transmission line l ij , which is show as (4). When we introduce a transmission loss variable P l ij into the (4) that can be expressed as (5).

III. PROBLEM STATEMENT A. OBJECTIVE FUNCTIONS
The generator cost curve F i (P i ) are represented by quadratic function and the total fuel cost FT that can be expressed as (6) and (7) F i (P i ) is ith generator cost, a i , b i and c i are the cost coefficients of the ith generator, and P i is the real power output of the ith generator, where N is the number of generators.

B. CONSTRAINTS 1) EQUALITY CONSTRAINTS
These constraints represent typical load flow equations as follows [1].
while (8) and (9) represent the balance limits of real power and the constraint of the DC power flow. G i , L i and D i are the sets of generators, transmission lines and loads. P g is the real power output of the gth generator, P d j is the real power of the kth consumer, P f j and P l j denote the real power flow and the transmission loss of the line. θ l = θ i − θ j , is the phase difference between the voltages at both ends of the transmission line l. N B is the total number of buses whereas the total number of transmission lines is defined by N L . The transmission loss constraint of the transmission line l can be illustrated in (10), where G l is conductance of transmission line l. The reference bus voltage phase angle constraint is presented by (13).

2) INEQUALITY CONSTRAINTS
These constraints represent the system operating constraints as follows.
For secure operation, the transmission line loading P f l is restricted by its upper limit as: The node voltage phase difference constraint at both ends of the transmission line l is shown in the following formula: The ramp-up/down rate of the ith generator limited by its physical characteristics [20]. It must satisfy the following constraint: The ith generator real power output P i is restricted by its lower and upper limits as follows:

IV. PROPOSED APPROACH A. LINEARIZATION RELAXATION FOR TRANSMISSION LOSS CONSTRAINT
Let m l ij is the total number of points that are taken on the network loss curve of the transmission line l ij , which is show in Figure 1. The kth point is linearized to obtain the following relaxed linear inequality: where K k l ij and B k l ij can be calculate as follows: After performing the above-mentioned linear relaxation processing on the transmission line loss constraints of each line, the quadratic equation constraints are relaxed into a series of linear inequality constraints, thereby mathematically transforming the original optimization scheduling problem into a quadratic convex programming problem, which is expressed as: where H and f are the coefficient matrix and coefficient vector of the objective function. A and b represent the coefficient matrix and coefficient vector of the linear inequality constraints. A eq and b eq represent the coefficient matrix and coefficient vector of the linear equality constraints. l b and u b are upper and lower bound of the independent variable, respectively.

B. MIXED INTEGER PROGRAMMING FOR TRANSMISSION LINE LOSS
In order to deal effectively with the situation that the marginal cost of negative nodes exists in power grid, independent variables, the relaxed linear inequality constraints, which do not satisfied the given threshold after solving by quadratic programming, the following linear processing is further carried out in this paper. The network loss constraints of each transmission line treated is handled by linear approximation, i.e. transmission line loss constraints in the form of quadratic equations is transformed into linear constraints as shown in Figure 2. The linearized transmission line loss constraints are as follows: R l ij represents the number of linear segments of the voltage phase difference between the node i and j at the transmission line l ij . U ij,r and W ij,r are the slope and intercept of the linear segment r, respectively. Z ij,r is the binary variable that corresponding to the continuous sub-variable θ ij,r .
VOLUME 8, 2020 θ lb ij,r and θ ub ij,r represent the lower and upper bounds of the voltage phase difference of linear segment r, respectively.
After linearizing the transmission line loss constraints, since the voltage phase difference between the nodes at both ends of the transmission line can only be located in one of the linear segments, binary variables need to be introduced and the following constraints are added: Constrained by (21), when the voltage phase difference between the two ends of the line is in a linear segment, the continuous sub-variable corresponding to the other linear segments is set to zero as follows: θ lb ij,r Z ij,r ≤ θ ij,r ≤ θ ub ij,r Z ij,r Constrained by (21) and (22), the voltage phase difference between the two ends of the transmission line should be the sum of each continuous sub-variable θ ij,r , which can be expressed as follows: If some transmission line loss constraints solving by the quadratic programming do not satisfy the constraints or exceed the threshold values after relaxing the transmission line loss constraints, we need to modify the transmission line loss constraints into (19), (20) and (22) forms to obtain a mixed integer quadratic programming problem.

C. DELETION METHOD OF MODEL VARIABLES AND CONSTRAINTS
According to (9), the real power flow of each transmission line is expressed as a function of the voltage phase difference between the nodes at both ends of the transmission line. Therefore, the real power flow P f l (l = 1, 2, · · · , N L ) of each transmission line is not taken as an independent variable in the real power optimal scheduling model in this paper. Accordingly, the security transmission capacity constraints (10) of each transmission lines can be deleted. Moreover, the real power flow of each transmission line can be rewritten by correcting the voltage phase difference between the nodes at both ends of each line, which is shown below: By merging the DC power flow constraint (9) into the node power balance constraint (8) to remove the DC power flow constraint, and the following node power balance constraint is obtained as follows: After the above processing, the real power flow variables, the DC power flow constraint of the network and the security transmission capacity constraint are deleted, so the efficiency of the algorithm can be effectively improved.

D. PROCEDURE SOLVING FOR GENERATION SCHEDULING PROBLEM
As a consequence, according to the description stated in the previous section, the procedure for the generation dispatch problem is summarized as follows: • Step1: Simplify the real optimal scheduling model by using the deletion method to reduced model variables and constraints of the network.
• Step2: Linearize the relaxed transmission line loss constraints and use a commercial optimization solver to solve the quadratic convex programming model. If all transmission line loss constraints satisfy the equality constraints or a given threshold valves, and go to step5; otherwise, go to step3.
• Step3: The mixed integer programming technique is used to deal with the transmission line loss constraints that do not satisfy the equality constraints or given thresholds, then employ a commercial optimization solver solve this mixed integer quadratic programming model.
• Step4: If all transmission line loss constraints meet the equality constraints or a given threshold, and go to step5; otherwise, skip to step3; • Step5: Stop and output the results. It is noteworthy that the mixed integer quadratic programming model needs to be handled only when the marginal cost of negative nodes exists in network. In most cases, in the process of optimizing the unit's power generation cost, the transmission loss is also minimized, and the transmission loss constraint usually satisfies the equality constraints or less than a given threshold, so there is only a quadratic programming problem needs to be solved.

V. SIMULATION STUDIES
In this paper, MATLAB R2017b is used and the MIP optimizer of CPLEX software is used to solve the proposed quadratic programming model and mixed integer quadratic programming model [21]. All methods in this paper are implemented on a computer with 4G memory and 3.10-GHz. The examine time is averaged by 30 independent operations.

A. TEST SYSTEMS DESCRIPTION
Three test systems, including IEEE 30-node system, IEEE 118-node system, and Polish 2746-node system, are employed to verify the effectiveness of the proposed method in this paper, which is shown in Table 1. In order to examine the effectiveness of the proposed approach, the following two representative algorithms are selected for comparison: 1) QCQP: The transmission loss constraint is mathematically relaxed into a quadratic constraint with convex.
In the process, the transmission line loss variables which do not satisfy the equality constraints are added to the objective function by the term of penalty factor, and that will be solved by solving the quadratic constraint quadratic programming problem [5]. 2) SCIP: This is a commercial optimization solver for solving optimization problems with quadratic constraints, and it solver can find a global optimal solution quickly [22].

B. CASE 1: IEEE 30-AND 118-NODE SYSTEMS
In order to verify the effectiveness of the propose approach in this paper, we first tests in IEEE30-node system  and IEEE 118-node system. These system-related datas, IEEE30-node system and IEEE118-node system are come from Matpower5.1. The results obtained by QCQP, SCIP and the proposed approach in this paper is show in Table 2. As shown in this table, from the perspective of solution quality, both QCQP and SCIP have the same global optimum solution on the two test systems. Due to the proposed approach in this paper solving the line loss constraints in a linear relaxation technique, there are little minor differences in the total generator cost and network loss. However, from the point of view of total generation cost and network loss, the simulation comparison with QCQP and SCIP shows that the solution obtained by proposed approach is close to the global optimum within the range of error tolerance.
In terms of the time, for small-scale system, like IEEE30-node test system, QCQP, SCIP and the proposed approach all can obtain satisfactory solutions efficiently, and there is little difference in the calculating time Compared with the IEEE30-node system, the time of the IEEE118-node system of the three methods increases. As a sequence, it can conclude that the calculation time of the proposed approach in this paper is much better that the QCQP and SCIP. Compared with QCQP and SCIP, the advantage of the calculation efficiency of the proposed approach is mainly due to the linear relaxation of transmission line loss constraints. After linear relaxation of secondary linear transmission loss constraints, the original problem is converted into a quadratic programming problem, which can be solved quickly by CPLEX.

C. CASE 2: POLISH 2746-NODE SYSTEMS
To validate the extensibility of the proposed approach, we use the Polish 2746-node systems. The system parameters are derived from the 2746-node system in Matpower 5.1, which corresponds to a load peak on a certain day in winter. Notes that this system is much larger than that of the aforementioned IEEE30-node system and IEEE118-node system, and the difficulty of solving the problem also increases.
The results obtained by QCQP, SCIP and the proposed method are shown in Table 3. From the Table 3, QCQP and SCIP are very close in terms of total generation cost and network loss, respectively. There is a little difference between the proposed method in this paper and the two methods mentioned above. This is mainly due to the small mismatch between the transmission line loss variable and the actual transmission line loss after the linear relaxation of transmission line loss constraints. As the scale of system is enlarged, the error accumulates step by step and which will reach a very larger value. At the same time, in the optimization process, the value of transmission line loss variable is generally close to the relaxed linear constraints, which makes the transmission line loss variable P l j less than or equal to the actual line loss V 2 l,N G l θ 2 l , and it results in the total output of proposed method is slightly less than the QCQP and SCIP. When properly increasing the number of linear slack segments of the line loss curve that will help to reduce the mismatch between the proposed method and QCQP and SCIP in the total power generation cost and network loss, but the calculation time will increase.

VI. CONCLUSION
This paper proposed a generation scheduling algorithm based on linear relaxation and mixed integer programming. This method using linear relaxation and mixed integer programming model is used to deal with the transmission line loss constraints, transforms the original power optimization scheduling problem into a quadratic planning or mixed integer quadratic programming problem, and solved by a commercial optimization solver, CPLEX. IEEE 30-node system, IEEE 118-node system and Polish 2746-node system are used to test the proposed problem. The validity and expansibility of the proposed method are verified, and compared with the two algorithms, QCQP and SCIP. The final results shows that the method proposed in this paper can obtain an optimal scheduling solution efficiently.