*1.2. Contributions*

This paper proposes a novel chance-constrained preventive SCOPF model (CC-PSCOPF) that is an improvement on the traditional preventive SCOPF (PSCOPF) model. The main contributions are as follows:

1. A novel CC-PSCOPF model is proposed to improve the overall operational reliability. The model considers contingency probability and the uncertainty of power injections (including RES and load).

2. Instead of using the large-scale line flow limits of traditional PSCOPF, the probability distribution of the overall line flow is obtained and constrained in the proposed optimization model, which significantly reduces the constraint scale.

3. The cumulant and Johnson systems are combined in this paper to accurately approximate the cumulative distribution function (CDF) of an arbitrary distribution random variable, which only requires the first four orders of moment information.

The remainder of this paper is organized as follows: Section 2 reviews the traditional PSCOPF model. Section 3 describes the formulation of the proposed CC-PSCOPF, and a cumulative distribution function (CDF) approximation method based on cumulants and the Johnson system is also introduced. A case study is presented in Section 4 to test the performance of the proposed model. Section 5 presents our discussion and conclusions.

#### **2. Review of Traditional PSCOPF**

There are two types of SCOPF: PSCOPF and corrective SCOPF (CSCOPF). Using PSCOPF, pre-contingency controls are the only measures allowed to ensure that the system always operates in a state where any single component outage does not lead to constraint violations. This indicates that the operational state determined by PSCOPF simultaneously satisfies the pre- and post-contingency constraints. Di fferent from PSCOPF, CSCOPF determines an operational state that allows post-contingency constraint violations, and it ensures that there are adequate post-contingency control measures, e.g., generator re-dispatch, topology reconfiguration and load shedding to eliminate post-contingency constraint violations. PSCOPF is safer, while CSCOPF is more economical [4].

This paper focuses on improving the traditional PSCOPF, and the proposed optimization model attempts to improve the overall security performance of the system operation through pre-contingency controls.

The DC-based PSCOPF model is reviewed in this section, as it provides the foundation for the optimization model proposed in this paper. DC approximation is used in this paper because it provides a convex guarantee that the optimization problem is tractable [20].

The objective function of DC-based PSCOPF minimizes the system's operational cost in the normal state, and it is expressed as follows:

$$\min \sum\_{Gi=1}^{N\_G} P\_{Gi}^T c\_{2i} P\_{Gi} + c\_{1i}^T P\_{Gi} + c\_{0i} \tag{1}$$

where *NG* is the number of generators; *PGi* is the *i*th generator output in the normal state, which is the control variable of the optimization model; and *c*2*i*, *c*1*i*, and *c*0*i* are the quadratic, linear and constant cost coe fficients, respectively.

The equality and inequality constraints of the PSCOPF model are as follows:

$$\sum\_{Gi=1}^{N\_G} P\_{Gi} + \sum\_{Ri=1}^{N\_R} P\_{Ri} = \sum\_{Di=1}^{N\_D} P\_{Di} \tag{2}$$

$$
\underline{P}\_{\rm Ci} \le P\_{\rm Ci} \le \overline{P}\_{\rm Ci} \qquad \forall i \tag{3}
$$

$$\sum\_{\text{Ci}=1}^{N\_{\text{G}}} A\_{\text{Gi}}^{k} P\_{\text{Gi}} + \sum\_{Ri=1}^{N\_{\text{R}}} A\_{\text{Ri}}^{k} P\_{\text{Ri}} - \sum\_{Di=1}^{N\_{\text{D}}} A\_{Di}^{k} P\_{Di} \le \overline{P}\_{\text{I}} \quad \forall i, \forall k, \forall l \tag{4}$$

$$\underline{P}\_l \le \sum\_{Gi=1}^{N\_G} A\_{Gi}^k P\_{Gi} + \sum\_{Ri=1}^{N\_R} A\_{Ri}^k P\_{Ri} - \sum\_{Di=1}^{N\_D} A\_{Di}^k P\_{Di} \quad \forall i, \forall k, \forall l \tag{5}$$

These include the power balance of the system (2), the generator output limits (3) and the line flow limits (4) and (5). *NR* and *ND* are the number of RES and loads, respectively, in the system; *PRi* and *PDi* are the forecast power injections of the *i*th RES and load; *PGi* and *PGi* are the *i*th generator's minimal output and maximum output, respectively; *Pl* and *Pl* are the lower and upper limits of the *i*th line flows; superscript *k* is the index of system topology; *k* = 0 indicates the normal state system topology, while *k* ≥ 1 indicates the contingency system topology; and *<sup>A</sup>kGi*, *AkRi* and *AkDi* are the power transmission distribution factors (PTDFs) of the *i*th generator, RES and load under system topology *k*, respectively. The PTDF can be obtained from the line susceptance matrix and bus susceptance matrix, and details can be found in [6].

The traditional DC-based PSCOPF optimization model is a typical quadratic programming problem that can be solved by common commercial solvers. As discussed in the introduction, the uncertainties of power injections and contingency probability are not considered in this model; therefore, the operational state obtained by this model is not robust to uncertainty and may be very costly. The constraint number of this model is 1 + *NG* + 2 × *Nl* × *Nk*, where *Nl* is the number of lines and *Nk* is the scale of the contingency set. Obviously, when the system scale is large with a large contingency set, the constraint number of this model is quite high, which significantly increases the calculation burden.

#### **3. Formulation of the Proposed Optimization Model**
