Security Constrained Unit Commitment Based on Modified Line Outage Distribution Factors

The N-1 criterion is universally taken as the security constrains in the security-constrained unit commitment (SCUC) problem. The line outage distribution factors (LODF) is widely used in the N-1 continency for its state independence and computational efficiency. However, the LODF based model fails to consider reactive power and voltage magnitude. This paper closes this gap by deriving modified line outage distribution factors (MLODF) based on a linear power flow model. The proposed model, though state-independent, provides high-quality approximation of voltage magnitudes. Then, we apply this model to formulate the SCUC problem and propose an iterative solution process based on MLODF post-contingency filter to set up the contingency constraints. The simulation results of the benchmark test systems verify the accuracy of proposed MLODF. The six-bus and IEEE 118-bus systems show that the MLODF based model can provide a more accurate and secure generation schedule comparing with the traditional LODF based model while ensuring the AC feasibility for most N-1 continency.


I. INTRODUCTION
Security-constrained unit commitment (SCUC) refers to the economic dispatching of generating units for supplying the system load and satisfying the transmission network security at the normal state [1], which has been widely applied in both vertically integrated utilities and restructured power systems. The efficient solution framework for this problem is to implement iterative procedure between a master problem (unit commitment) and subproblems (network security evaluation). Reference [2] summarized some mainstream methodologies for solving the SCUC master problem and sub-problems in terms of modeling ability, feasibility and optimality, solution stability, computer resource consumption, post-contingency actions and CPU time.
Prevalent SCUC problems are based on the DC power flow formulation owe to its reasonable accuracy and computational efficiency [3]- [4]. Reactive power constraints and bus voltage limits are not considered. However, during the actual operation of power system, a significant voltage drops across long-distance transmission is more likely to occur, which may cause the bus voltage to exceed its limit, especially at load center buses. Besides, the reactive power flow may have a great influence on line current, particularly in power networks with strong coupling between active and reactive power, which may yield uneconomic or insecure solutions. Based on the two reasons above, SCUC problem considering AC power flow constraints is a necessity to improve grid operations. Reference [5] established a SCUC problem with AC constraint and applied the Benders decomposition for separating the unit commitment in the master problem from the network security check in subproblems. The authors of [6] proposed a solution technique based on the outer approximation method to cooptimize real and reactive power scheduling and dispatch subject to both unit commitment constraints and ACOPF constraints. In [7], a novel decomposition method is also proposed for the AC-SCUC problem. By solving the MISOCP model of the master problem, the active power and reactive power are co-optimized based on the conic approximations of the AC power flow equations.
However, the solution cannot guarantee the global optimum because of the non-convexity [8]- [9]. To bridge this gap, several linear power flow models considering the reactive power and voltage profiles are proposed. Reference [10] proposed the decoupled linearized power flow (DLPF) model with respect to voltage magnitude and phase angle.
The model is state independent and the accuracy and robustness are proved in several cases, including radial distribution systems, meshed large-scale transmission systems and ill-conditioned systems. The authors of [11] derived a general linearized power flow model considering tap changers and phase shifters using the logarithmic transform of voltage magnitudes (LTVM). An improved linearized network model based on the Taylor series expansion was derived in [12]. The piece-wise linearization technique method with integer variables is applied to handle the nonconvexity introduced by the quadratic losses. These models provide high-quality approximation of AC power model while maintaining the computation efficiency, having been applied to the optimal power flows [13]- [14], transmission expansion planning [15].
In the contingency security assessment, the line outages are often considered because of their universality. The most classical method is to use the line outage distribution factors (LODF) [16]- [17], which can be directly calculated according to the power transfer distribution factors (PTDF) of the pre-contingency network [18]. PTDF de-scribes how the real power flows change if power injection is shifted from one node to another. Correspondingly, LODF describes how the flows change when one line fails. Awe to its computational efficiency, LODF has been widely used in generation capacity expansion planning problem [19], optimal power flow calculation [20], real-time operation [21] and SCUC problem [22]. Though LODF can straightly determine the post-contingency active power flow, it cannot predict post-outage reactive power flows and voltage. To solve this problem, many researchers also proposed the methods based on the sensitivity coefficients to calculate the post-contingency states [23]- [24]. However, they are related to the system base points, which is difficult to apply in the SCUC problem.
The major contributions claimed in this paper are as follows: • This paper proposes the modified line outage distribution factors (MLODF), which is derived from a state-independent linear power flow model and only related to the network parameters. The proposed factors are distinguished by their high-quality approximation of post-contingency active power flows and voltage magnitudes. • A formulation for SCUC problems based on MLODF for post-contingency states is established and a linearization method is applied to approximate the quadratic branch flow limits. can provide a more secure generation schedule than the traditional LODF based model. The remainder of this paper is organized as follows. Section II describes the derivation of MLODF. Section III introduces the formulation for the SCUC. Section IV presents the solution method based on the MLODF post-contingency filter. Section V contains the case study to verify the proposed model. Finally, Section VI draws the conclusions and suggestions in the future work.

II. Derivation of MLODF
The line outage model through the incremental injections of buses is simulated in [25]. Figure 1 illustrates the simulation when the line nm is off.
The following linear approximations [10] can be applied to decouple the voltage magnitudes and the phase angles in the power flow equations: (3)

FIGURE 1. Line outage modeling using injections
Based on (2) and (3), we can obtain the linear expression for branch flow as follow: where sh sh sh ij ij ij y g jb =+ are the line shunt admittance including the line-charging susceptance and equivalent admittance of transformers and phase shifters. According to [10], for branches with transformers and phase shifters, We can calculate perturbations for a given operating state by the first-order Taylor expansion of (4) and (5): If we are interested in the changes in voltage magnitude V  and the phase angle   , for given changes in the bus power injections P  , Q  , we can use the following calculation: where , nn mm PQ PQ Collating (4), (5) and (9), we can establish the relationship among the post-outage flowing in the line nm , the incremental injections into buses , nmand the preoutage flowing in the line nm ( * * * * , , , Substituting (1) Through (7) and (11), we can get the increment of bus voltage: Define several sensitivity factors ( , , , as the sensitivity coefficient of pre-outage power flows in the line nm to the incremental voltage at bus i . Therefore, the post-outage voltage can be written as: While the power flowing over the line ij is equal to where

III. Formulation of SCUC based on MLODF
This section provides the detailed formulation of the objective function and constraints relevant to the SCUC model.

A. Objective Function
The total production cost of all units is to be minimized. The operation cost of a thermal unit is defined as a linear function of variables: (  )   TU   TU  SU  SD  NL  , , , , () g t g t g g g t g g g t g p ru P P u P SU x MU , , , , , ,, where , gg RU RD are upward and downward ramping capability of the unit. 4) Logic constraints of unit commitment states:   ,, max 1, 1 , , , where , gt MD is the minimum downtime of the unit. 5) Reactive generation limit constraints: where g Q is the maximum generation reactive output. 6) Pre-Outage network constraints : , i i VV are the emergency voltage magnitude limit of buses.

A. Flowchart of the SCUC Problem
Most of the N-1 contingency constraints are inactive and can be discarded [26]. To reduce the number of N-1 contingency constraints, we propose an iterative filtering process based on the MLODF to set up the contingency constraints. The flowchart of the MLODF post-contingency filter for the SCUC problem is shown in Figure 2.
The solution procedure starts by calculating the precontingency network constrained UC, in which (18)-(32) will be included. The network constrained UC obtains the pre-contingency line flows (

B. Linear Approximation of the Quadratic Constraints
Note that the transmission capacity constraints (32) and (34) are both quadratic items, which makes the scheduling problem nonlinear. To improve the computational efficiency, a linearization method is applied to approximate the quadratic branch flow limits [14]. The feasible region defined by these constraints can be represented by the area enclosed by a circle, which can be approximated by its inscribed n-side polygon, as shown in Figure 3. For a circle of radius r, each secant is related to an arc of 2πr/n length. The area inside the circle, typically defined by 2 2 2 x y r +, can be approximated by n linear represented by:

V. Case Study
Case studies are conducted to demonstrate the performance of the proposed MLODF model. In the first test, several benchmark power systems are applied to analyze the active power flows and voltage magnitudes under N-1 contingencies to verify the accuracy of the proposed MLODF model. In the second experiment, the six-bus system and the IEEE 118-bus system are used to analyze the security performance of the proposed model. All the optimization problems are solved with the commercial solver Gurobi 8.1.1 using MATLAB R2016a.

A. Accuracy verification
A total of 5 benchmark power systems are employed as test examples, the sizes of which range from 6 to 2,383 buses with different loading conditions. All of the test data can be accessed online in the package of MATPOWER without any modification [27].
The following two models are compared: (1) the proposed MLODF, (2) the traditional LODF. Two important results, i.e., branch MW flows and voltage magnitudes are obtained for comparison. With the results of the AC power flow model (ACPF) as the benchmark, the root-mean-square errors of branch MW flows and voltage magnitudes are calculated, as shown in Figure 4 and Figure  5. Obviously, MLODF model performs better than LODF model not only in active power flows but also in voltage magnitudes. Especially for the voltage magnitudes, the root-mean-square errors of MLODF are far smaller than those of LODF. This is because the proposed MLODF considers the influence of voltage and reactive power on the basis of LODF, resulting in a better approximate effect on ACPF. The simulation result also verifies the correctness of the MLODF derivation process.

B. Security verification
To illustrate the security of day-ahead scheduling solution obtained by the proposed MLODF model, we carry out a test in a six-bus electric power system. The six-bus system includes 3 generators and 7 transmission lines.  Table I shows the commitment schedules of SCUC based on MLODF model and LODF model, respectively. G3 is not committed at hours 9-24 when using traditional LODF based model due to the following reasons: (1) the traditional LODF based model fails to consider reactive power balance constraints. For hours 9-24, the total reactive load cannot be supplied by G1 alone. (2) the traditional LODF based model ignores the voltage constraints. For hours 9-24, if G1 is the only committed unit, the voltage of bus 2 will drop below its lower limit when the line 1-2 is tripped off. For this reason, the MLODF based model starts up G3 to improve the voltage of bus 2 in this condition. Therefore, the proposed MLODF based model considers the constraints of voltage and reactive power, which can provide a more secure generation schedule.  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  To further demonstrate the accuracy and security of the proposed MLODF model in the large-scale power systems, the IEEE 118-bus system is applied here. The IEEE 118bus system includes 52 generators as well as 186 transmission lines. Figure 7 shows the commitment schedules of SCUC based on MLODF model and LODF model, respectively. To analyze the influence of serious contingency to accuracy, Figures 8-10 visualize the solutions for the worst situation in terms of the maximum voltage magnitude and MW flow errors at hour 17. As shown in Figure 8, the voltage magnitudes of bus 39, 40 and 41 drop below their lower limit. This is because when line 37-39 fails, the reactive load of bus 39, 40 and 41 can't be supplied by the near unit directly. The voltage magnitudes of these buses drop to some extent, which deviates from the assumption on the state-independent linear power flow in [10], thus resulting in a great error. From Figure 9, we can see that the MLODF based model is still accurate in approximating MW flows even in a scenario with a maximum error. Note that the complex flows of line 47-69 (branch number 105) and line 49-69 (branch number 106) are beyond their emergency limits when line 65-68 fails, as shown in Figure  10. This is because line 47-69 and line 49-69 both have a high reactance, which results in a large reactive power loss. However, the linearized power flow model used in this paper neglects the branch losses. Therefore, at this time, the MLODF based model has poor approximation effect on the reactive power of these two transmission lines, which produces larger errors to branch complex flows.   In summary, for few N-1 contingencies, if the bus voltage deviates from the rated value or the reactive power loss of the transmission line increases, the post-contingency voltage magnitudes or power flows may beyond their emergency limits. Under these N-1 contingency scenarios, the actual network power flow deviates from two basic assumptions. One is that all bus voltages of the stateindependent linearized power flow model are assumed near to their rated values. The other is that the branch losses are neglected.
The total operational cost of the MLODF based model is 797,683.7$ while the traditional LODF based model is 794,533.0$. Though the MLODF based model incurs higher cost, it addresses the potential risks of voltage violation and complex flow violation for most N-1 contingency and thus provides a more secure generation schedule. Figure 11 denotes the convergence performance of the proposed solution method. The number of iterations in IEEE 118-bus system is 4 with a relative rap of 0.5%. The total number of limit violations of post-contingency power flows and voltage magnitudes is counted in each iteration. It is clear that the number of violations declines rapidly and the convergence performance of the solution method is acceptable. Therefore, case studies above verify the accuracy and security of the proposed MLODF model and the effectiveness of the solution method based on MLODF post-contingency filter.

VI. Conclusions
In this paper, we propose the MLODF derived from a stateindependent linear power flow model and apply this model to formulate the SCUC problem. The simulation results show that: 1) The proposed MLODF model can calculate postoutage voltage and active power flow through pre-outage line flowing and has a better approximate effect on ACPF than DCPF model. It holds the accuracy as the original state-independent linear power flow model, which also verifies the correctness of the MLODF derivation process.
2) The SCUC problem based on MLODF considers the constraints of voltage and reactive power, which provides a more secure generation schedule than the traditional LODF based model.
3) For most N-1 continencies, the generation schedule obtained by the MLODF based model can make sure that the line flowing and bus voltage are secure within their limits.