**4. Case Study**

In this section, we discuss the performances of traditional PSCOPF and CC-PSCOPF, which were tested on two test systems. The optimization problem was solved by Ipopt [37] on a PC with a 2.8-GHz CPU and 16 GB RAM.

#### *4.1. Description of the Test System*

A modified IEEE-30 test system [38] was used here to analyze the characteristics of these two optimization formulations and a modified IEEE-108 test system [38] was used to evaluate the e fficiency of these two methods. The IEEE-30 test system was modified to add two wind power generators at bus 7 and 12, which are representative of a RES power injection. Similarly, the IEEE-108 test system was modified to add four power generators at bus 44, 50, 88 and 98. The rated power of the wind power generators is 80 MW, the forecast outputs of wind power generators are assumed to be 0.8 p.u., and forecast error is assumed to follow a beta distribution: Beta (0.83, 1.82). The forecast error of the load at each bus is assumed to follow a Gaussian distribution, with the means of the power injections equal to those of the base case data and standard deviations equal to 5% of the means.

All N-1 contingencies are included in the contingency set, the occurrence probability of each contingency is assumed to be 0.01, and the violation level α<sup>+</sup> *l*is set at 1% for all lines.

#### *4.2. CDF Approximation Performance of the Proposed Method*

Obtaining an accurate CDF curve, especially in the tail area, is the basis for accurately solving the chance-constrained optimization. This section presents the results to show that the proposed cumulant + Johnson system can accurately and e fficiently approximate a CDF curve.

The proposed method is compared with 10,000 MCS, the commonly used cumulant + Gram-Charlier series proposed in [25], and a CDF curve based on the assumption that wind power forecast error follows a Gaussian distribution.

The CDF curve of line 4-6 flow and line 16-17 flow under the normal state is chosen here as representative and to visually show the approximation ability of the evaluated method. The CDF curves are shown in Figure 1a,b.

**Figure 1.** Cumulative distribution function (CDF) curve for line 4-6 flow (**a**) and line 16-17 flow (**b**).

The average root mean square (ARMS) [18] is also introduced here to quantitatively indicate the accuracy of the CDF approximation. The smaller the ARMS value is, the more accurately the method approximates the CDF curve. Table 1 shows the ARMS values of the evaluated methods.



The results in Figure 1 and Table 1 show that the cumulant + Johnson system approximates the CDF curve best and only a small amount of error exists, while the curve approximated by cumulant + Johnson and Gaussian assumptions shows significant deviation, especially in the tail area.

Moreover, the method proposed in this paper has an advantage over MCS; that is, MCS cannot be implemented when only the moment information of power injections is available, while the method proposed in this paper can provide a reliable CDF curve.

#### *4.3. Solutions of Di*ff*erent Optimization Formulations*

The generation costs for the two optimization approaches are listed in Table 2. The cost of CC-PSCOPF is higher because it considers the uncertainty of power injections and gives a high probabilistic guarantee that line flows limit violations will not happen.



Because the chance constraint is a soft constraint and the Gaussian distribution is an unbounded distribution, there are always extreme values that cause line flow violations. Based on the generator output schemes given by these two problem formulations, we implemented MCS with 10,000 samples to observe the actually probability of line flow violations.

The average violation probability and the maximum violation probability were introduced to illustrate the effectiveness of the proposed method. These two indices are listed in Table 3. It is obvious that the chance constraint works; the maximum violation probability under CC-PSCOPF equals 0.01, which is equal to the preset violation level, while under PSCOPF, the same index significantly exceeds

the violation level. Both the average and maximum violation probability are smaller under CC-PSCOPF, which indicates that the proposed method provides a more robust operational state than PSCOPF.

Moreover, although the goal of the proposed method is to improve the operational reliability by controlling the violation probability of line flows in the overall situation rather than the violation probability under a specific contingency system topology, it is interesting to note that CC-PSCOPF effectively reduces violations under a single contingency system topology. We counted the number of contingencies *NV* where the line flow violation probability exceeds the violation level. For contingencies with violations, the average number of line flows *Nalv* and maximum number of line flows *Nmlv* that exceeds the violation level were also counted. The statistical data are listed in Table 3. Obviously, CC-PSCOPF effectively reduces *NV*, *Nalv* and *Nmlv*, and this indicates that more contingencies are effectively controlled.


**Table 3.** Constraint violation statistics.

The line flow of 15–18 was chosen as representative to visually show the flow probability distributions under these two formulations. The histograms of line flow are shown in Figure 2a,b, respectively. It is clear that more PSCOPF samples fall outside of the line limits.

**Figure 2.** Histograms of line flow 15–18 under preventive security-constrained optimal power flow (PSCOPF) (**a**) and chance-constrained PSCOPF (CC-PSCOPF) (**b**).

#### *4.4. Influence of the Value of Violation Level*

Changing the violation level influences the solution of CC-PSCOPF. Figure 3 shows the change in the generation cost and violation probability of CC-PSCOPF with different violation levels.

**Figure 3.** Generation cost (**a**) and violation probability (**b**) of CC-PSCOPF with different violation levels.

As Figure 3 shows, with increasing violation level, the generation cost of CC-PSCOPF decreases, and inevitably, this causes a larger probability of constraint violations.

Obviously, the contradiction between power generation costs and operational security can be balanced by adjusting the violation level. As different transmission system operators have different requirements for line constraint violation probability, how to determine the optimal violation will be explored in subsequent studies. Note that, theoretically, the violation level can be set at 0 to completely eliminate the constraint violations, but this is not worth the gains. On the one hand, this will cause a surge in control costs, and on the other hand, it is easy to make the problem infeasible, especially when the uncertainty of power injection is large. Moreover, the violation level of different lines can be set to different values. For the critical lines, appropriately reducing the violation level can improve safety, and for non-critical lines, increasing the violation level appropriately can reduce generation costs. Obviously, CC-PSCOPF is more flexible compared with traditional PSCOPF, which can only balance the control costs and security by adjusting the contingency set.

#### *4.5. E*ffi*ciency of the Proposed Method*

The time consumption of PSCOPF and CC-PSCOPF was tested on the IEEE-30 test system and a larger-scale system, the IEEE-118 test system. The constraint numbers and time consumption of these two optimization formulations are listed in Table 4. It is evident that the constraint numbers significantly influence the efficiency of the optimization model. CC-PSCOPF is much faster than PSCOPF because the line limit constraints of all the system topologies are reduced to the same number of lines by the law of total probability.


