Abstract
In this letter, a new formulation of Lebesgue integration is used to evaluate the probabilistic static security of power system operation with uncertain renewable energy generation. The risk of power flow solutions violating any pre-defined operation security limits is obtained by integrating a semi-algebraic set composed of polynomials. With the high-order moments of historical data of renewable energy generation, the integration is reformulated as a generalized moment problem which is then relaxed to a semi-definite program (SDP). Finally, the effectiveness of the proposed method is verified by numerical examples.
WITH rapid adoption of intermittent renewable energy sources, e.g., wind power and solar generation, uncertainties in renewable energy generation have been threatening the security of power system operation. Probabilistic static security assessment (PSSA) is a well-established way to characterize the influence of those uncertainties on power system operation [
In common practices of power engineering, only partial statistical properties, e.g., the mean or first-order moment, the variance or second-order moment and high-order moment, are available for modeling stochastic renewable energy generation [
This letter aims to perform high-order (more than 2) moment-based distributionally robust probabilistic static security assessment (DR-PSSA) based on nonlinear power flow equations (NPFEs) with uncertain input parameters. The DR-PSSA is formulated as a generalized moment problem [
This section gives a brief introduction of the DR-PSSA problem in terms of a single continuous random parameter , where is the range or set of random parameters.
Mathematically, when uncertain parameters are incorporated into NPFEs, PSSA is aimed at evaluating the probability of normal power flow solutions violating any pre-defined operation security limits. and are defined as the feasible solutions of power flow equations and the insecure solutions, i.e., at least one security limit is violated, pertinent to random variable whose PD is , respectively. Accordingly, PSSA is equivalent to the integration (Lebesgue integration) of over :
(1) |
where is the calculated probability that achieves a value between 0 and 1. Assuming that NPFEs are solvable for any , the integration of over always equals 1.
As NPFEs are usually formulated in rectangle coordinate in the existing literature [
(2) |
where is a unified form of NPFEs; and is the state vector consisting of real and imaginary parts of each bus voltage, i.e., e and f. The probability of any branch flow or bus voltage exceeding any security limit can be evaluated through representing as a set of inequalities in state variables. For example, to evaluate the undervoltage probability of each bus by comparing with its limit , can be stated as:
(3) |
where , and and are the
Next, we focus on how to calculate , i.e., the upper probability over all the admissible PDs with given moments of power flow solutions in . The operation risk can be evaluated by . can be regarded as the optimum in the following semi-infinite linear optimization problem [
(4) |
s.t.
(5) |
(6) |
where is the joint PD of and ; and is the provided
Hence, DR-PSSA maximizes the integration on , i.e., (4), by finding in the worst case supported on , i.e., (5), while respecting the moment constraint, i.e., (6). Moreover, if multiple violations are encountered, the violations can be directly added into the set , which means that the proposed method can deal with the intersection set of multiple violations in the set .
First, define . DR-PSSA can be reformulated as an equivalent generalized moment problem through introducing an auxiliary measure of based on Theorem 3.1 in [
(7) |
s.t.
(8) |
(9) |
where is the PD of random variable whose supporting set is ; and is the set of all possible PDs of . means for any . means for any , . is defined in the same way as .
DR-PSSA can be approximated as by performing d-order SDP relaxations of and , respectively, of which moment sequences correspond to and [
(10) |
s.t.
(11) |
(12) |
(13) |
(14) |
where is one of the decision variables, which denotes the probability or the security risk; is the degree of polynomial in or ; and and are the moment matrix and localizing matrix [
The processes of constructing SDP relaxations (10)-(14) for the generalized moment problem (7)-(9) are detailed as follows. The objective function (10) and linear constraints (11) are derived by substituting moment sequences and into (7) and (8), respectively. The semi-definite constraints (12)-(14) are sufficient conditions for satisfying (9) according to Lemma 2.4 and Theorem 2.2 in [
The proposed SDP relaxations lead to computationally tractable convex optimization problems. For any , is a positive semi-definite matrix in d-order SDP relaxations, where . Although the proposed method would suffer the curse of dimensionality, we can further take advantage of the inherent sparsity of NPFE to facilitate its implementation in large-scale applications [
Numerical experiments are carried out based on the revised 4-bus system in [
In this section, we intend to validate the effectiveness of the proposed method through setting the Monte Carlo simulation as a benchmark. To be specific, we calculate the probability of voltage magnitude at bus 2 (denoted as in

Fig. 1 Probabilities with 2-order SDP relaxations and a uniform distribution.
In addition, all of the three probabilities stay at the same value when the voltage magnitude is lower than 0.85 p.u. or higher than 0.94 p.u., which means the operation interval of voltage magnitudes pertinent to the pre-defined active power variations, i.e., [5 p.u., 5 p.u.], is [0.85 p.u., 0.94 p.u.].
A DR-PSSA method for static security assessment of power system operation based on the partial moments of uncertain parameters is proposed in this letter. The proposed method improves the ability of moment-based distributionally robust optimization in dealing with nonlinear models and high-order moment, making the static security assessment of power system operation more robust to uncertainties. Numerical examples show that distribution uncertainties in power system operation are effectively incorporated in DR-PSSA. As the follow-up study, we will explore more structural information, e.g., symmetry and unimodality, embedded in the PDs of uncertain parameters to improve the computation performance of the proposed method.
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