*3.2. Chance-Constrained Optimization*

In this section, we briefly introduce the CCO, which also underpins the model proposed in this paper.

CCO is an important tool proposed by Charnes and Cooper [16,17] for solving optimization problems under uncertainties. The general form of a CCO problem is expressed as follows:

$$\begin{array}{ll}\min & f(\mathbf{x}, \boldsymbol{\xi})\\\text{s.t.} & g(\mathbf{x}, \boldsymbol{\xi}) = 0\\& & \Pr\{h(\mathbf{x}, \boldsymbol{\xi}) \ge 0\} \ge 1 - \alpha\end{array} \tag{11}$$

where *f*(·) is the objective function, *g*(·) is the equality constraint, *h*(·) is the inequality constraint, *x* is the decision variable, ξ is the uncertainty variable and α is the reliability parameter representing the allowed constraint violation level.

Under the CCO, the inequality constraint is formed as the chance constraint and ensures that the constraint *h*(·) is satisfied with probability 1 −α at least. The original CCO problem is often transformed into an equivalent deterministic form to facilitate the solution [20,24].

#### *3.3. Chance-Constrained PSCOPF Model*

In this section, we present a novel CC-PSCOPF model that considers the uncertainties of power injections and the probability of contingency.

The goal function, power balance constraint and generator output limit of the proposed CC-PSCOPF are the same as those of the traditional PSCOPF model, as Equations (1)–(3) show. The key improvement is the modeling of the line flow inequality constraints.

Instead of line flow constraints under each system topology used in traditional PSCOPF, the overall line flow *Pl* is constrained in CC-PSCOPF. As analyzed in the previous section, *Pl* is a random variable; therefore, the chance constraint is used to place it in a certain range with a high probability, and it is expressed as follows:

$$\Pr\{P\_l \ge \overline{P}\_l\} \le \alpha\_l^+ \quad \forall l \tag{12}$$

$$\Pr\{P\_l \le \underline{P}\_l\} \le \alpha\_l^- \quad \forall l \tag{13}$$

where α<sup>+</sup> *l* and α<sup>−</sup> *l* are predefined violation levels. Considering the low occurrence probability of contingencies, α<sup>+</sup> *l*and α<sup>−</sup> *l*should be carefully defined.

 A comparison of Equations (12) and (13) to Equations (4) and (5) shows that the optimization model proposed in this paper has the following significant advantages:

1. The uncertainty of power injections and contingency occurrence probability are considered through *Pl*. Even with the influence of various uncertainties, the operational state obtained by CC-PSCOPF has a high probability of ensuring that constraints are not violated. Obviously, the operational state is more robust to uncertainties compared to traditional PSCOPF.

2. The violation level for di fferent lines is adjustable; for critical lines, the violation level could be adjusted lower to ensure operational safety, while for noncritical lines, the violation level could be increased to save control costs.

3. The scale of the line flow constraint is significantly reduced, which is only related to the number of lines in the system, allowing the optimization problems to be solved more e fficiently.

4. As the occurrence probabilities of the contingency and chance constraints are used in this model, some contingencies that have quite low probability but high control costs are ignored in the optimization model, which helps to reduce the control costs. The control measures of these low probability contingencies can be solved using a separate accident plan.

However, solving an optimization problem with chance constraints directly is a challenging task. In this paper, chance constraints are transformed into deterministic linear constraints based on the cumulant and Johnson systems in the following section, which ensures that the CC-PSCOPF is tractable and solved e fficiently.

#### *3.4. Deterministic Reformulation of CC-PSCOPF*

The main challenge of solving the proposed model is how to handle the two chance constraints related to overall line flows. In this section, these two constraints are converted into deterministic linear constraints. Through conversion, the optimization model becomes a linear constrained convex optimization problem that is easy to solve.

For convenience, the factors that determine the overall line flow *Pl* are divided into two terms:

$$\begin{array}{lcl}P\_{\text{I,control}} &=&\sum\_{\substack{\mathbf{k}=0 \\ \mathbf{k}=0 \ \mathbf{G}\mathbf{i}=1}}^{\text{N}\_{\text{k}}}\sum\_{\mathbf{G}\mathbf{i}=1}^{\text{N}\_{\text{G}}}A\_{\text{G}\mathbf{i}}^{\text{k}}P\_{\text{G}\mathbf{i}}\mathbf{P}\_{\text{r}}(\mathbf{s}\_{\mathbf{k}})\\P\_{\text{I,uncoricity}} &=&\sum\_{\substack{\mathbf{k}=0 \ \mathbf{R}\mathbf{i}=1}}^{\text{N}\_{\text{k}}}\sum\_{\mathbf{R}\mathbf{i}}^{\text{N}\_{\text{R}}}P\_{\text{R}\mathbf{i}}P\_{\text{r}}(\mathbf{s}\_{\mathbf{k}}) - \sum\_{\substack{\mathbf{k}=0 \ \mathbf{R}\mathbf{i}=1}}^{\text{N}\_{\text{k}}}\sum\_{\mathbf{D}\mathbf{i}}^{\text{N}\_{\text{R}}}A\_{\text{D}\mathbf{i}}^{\text{k}}P\_{\text{D}\mathbf{i}}\mathbf{P}\_{\text{r}}(\mathbf{s}\_{\mathbf{k}}) \qquad\forall\mathbf{l}\end{array} \tag{14}$$

where *P<sup>k</sup> l*,*control* is the line flow part determined by the control variable, which varies with the output of the generators, and *Pl*,*uncertainty* is the line flow part determined by power injections of the RES and load, which is a random variable.

Substituting Equation (14) into Equation (12) yields:

$$\Pr\{P\_{l,\textit{uncrertainy}} \le \overline{P}\_l - P\_{l,\textit{control}}\} \ge 1 - \alpha\_l^+ \quad \forall l \tag{15}$$

Note that *Pl*,*uncertainty* is the uncertainty part of the overall line flows, and we can obtain:

$$\begin{array}{l} \text{CDF}\_{l,\text{uncertainty}}(\overline{P}\_{l} - P\_{l,\text{control}}) \ge 1 - a\_{l}^{+} \\ \Downarrow \overleftarrow{P}\_{l} - P\_{l,\text{control}} \ge \text{CDF}\_{l,\text{uncertainty}}^{-1}(1 - a\_{l}^{+}) \end{array} \tag{16}$$

where *CDFl*,*uncertainty* and *CDF*−<sup>1</sup> *l*,*uncertainty* are the CDF and inverse CDF of *Pl*,*uncertainty*, respectively. Similar to Equation (16), by substituting Equation (14) into (13), we can obtain:

$$
\underline{P}\_l - P\_{l,control} \le \mathcal{C} D F^{-1}\_{l,stochastic} (a\_l^{-}) \quad \forall l \tag{17}
$$

To replace Equations (12) and (13) with deterministic linear constraints Equations (16) and (17), *CDF*−<sup>1</sup> *l*,*uncertainty* should be obtained. Traditionally, *CDFl*,*uncertainty* is obtained from Monte Carlo simulation (MCS), which performs a numerical search for the *CDF*−<sup>1</sup> *l*,*uncertainty* [20]; however, this is time consuming. Moreover, MCS is difficult to implement in the absence of samples. Here, this procedure is improved by using the cumulant and Johnson system to obtain *CDF*−<sup>1</sup> *l*,*uncertainty*.
