Multi-period vulnerability analysis of power grids under multiple outages: An AC-based bilevel optimization approach
Introduction
Energy is a vital commodity in modern societies and power systems play a crucial role in providing secure and reliable energy. Protection of the power system as a critical infrastructure against different hazards and threats i.e., natural hazards, intentional attacks, and random failures [1] has become a growing concern. Furthermore, the interdependencies between power systems and communication networks in smart grids are introducing new challenges i.e., cyber threats. So, the operators and planners must protect the most vulnerable elements of a system under a variety of attack scenarios in order to improve the system security and deploying a robust and resilient power system [2].
In this context, a key question is which components are critical and must be protected or fortified when the protective and financial resources are limited [3]. To this end, it is fundamental to develop robust methodologies and tools to assess the vulnerability of a power system against external attacks [4]. Hence, the vulnerability analysis and, in particular, prioritizing the vulnerable components result in an effective power system protection with limited resources [5].
Contingency analysis or N-k contingency assessment is a methodology looking for a set of k critical components of the power system whose simultaneous failure would maximize the damage, in terms of the amount of involuntary load shedding in the power system. N-k (k ≥ 2) contingencies are low-probability events but inherently more severe than N-1 contingencies in triggering cascading failures and even blackouts [6]. Therefore, the North American Reliability Council (NERC) suggests power system planners and operators considering N-k contingency analysis in their planning and operation [7]. The difficulty of N-k contingency selection is that it follows the combination formula. It means that for a very modest size power system with N = 1000, there are 1000 N-1 contingencies, 499,500 N-2 contingencies, over 160 million N-3 contingencies, over 40 billion N-4 contingencies and so on. So, the number of possible N-k contingencies, even for small values of k, makes total enumeration approaches computationally impractical in a large-scale interconnected power network [8].
Scientists have been developing innovative methods to determine critical components whose failures lead to the largest system loss. Generally, there are two different lines of work in the literature. Some literature work on the low-order contingencies including single or a small number of component failures that have a very high occurrence probability. For instance, the N-1 security constraint that all of the regulatory agencies in the world enforce the system operators to satisfy it by strict security standards. Within this constraint, the system should normally continue to work after any single failure [9]. Analyzing the loss of two elements consecutively or N-1–1 contingency analysis is another example of this category [10], [11], [12]. It should be noted that the reliability concept that defines the ability of the electric power system to meet the demand with continuity and an acceptable level of quality, comes under the low-order contingencies [13].
The second line of works presents not only the low-order but also the high-order contingencies. The high-order contingencies include a relatively large number of component failures that have a very low occurrence probability but high consequences. The focus of this paper is on this type of assessment i.e. the vulnerability analysis of power systems. The vulnerability can be social, organizational, economic, environmental, territorial, physical, and systemic [14,15]. Most studies focus on physical and systemic vulnerabilities. Physical vulnerability represents the degree of loss of an element due to external pressure such as natural hazards [16]. In contrast, systemic vulnerability considers the degree of redundancy, functionality, and dependency of a system due to the failure of a specific element or an interconnected system [17]. This paper aims to investigate the behavior of the power system i.e. systemic vulnerability to identify the critical components under a worst-case scenario such as an intentional attack.
The works can range from analytical approaches (complex network, flow-based, logical, and functional methods) to Monte Carlo simulations. A detailed comparison of these approaches is recently conducted in [18] (and the references therein). Among them, the optimization-based problem can directly lead to promising results without the need to rank the sets of critical assets. The application of these approaches is considerably increasing in the complex problems thanks to the advent of advanced high-speed multiprocessors with large memory. It makes the problem tractable for a realistic power system [19].
The interdiction model is at the forefront of the models used to identify the worst N-k contingency. It has been developed based on a multilevel optimization problem to assess the vulnerability of power systems [20]. A multilevel optimization is a mathematical program where an optimization problem contains another optimization problem as a constraint [21]. These problems are also known as the hierarchical leader-follower problem or the Stackelberg game [22]. The interdiction model basically includes an upper level whose objective is to identify exactly k components to maximize the damage (load shedding) in the system and a lower level whose objective is mitigating the impacts of attacks and minimizing the damage consequences.
Later, the interdiction model is developed based on two models i.e., bilevel and trilevel interdiction models. For instance, Karush-Kuhn-Tucker (KKT) optimality conditions [23] and duality theory [24] are used to convert a bilevel attacker-defender model to a one-level problem. Arroyo J.M. [25] compared the KKT- and duality-based approaches by introducing minimum and maximum vulnerability models. Brown et al. [26] extended the classical bilevel interdiction model to a general trilevel defender-attacker-defender model to assign limited defensive resources in power systems. Alguacil et al. [27] proposed an approach to allocate the defensive resources in a power system to mitigate the vulnerability. Wu et al. [28] decomposed a planner-attacker-operator model to a master problem and a subproblem using a Benders primal decomposition method. Recently, Fang et al. [29,30] and Che et al. [31] used this approach to identify the vulnerability of power grids exposed to natural hazards and the hidden N-k contingencies, respectively and finally, Nemati et al. [32,33] proposed tri-level transmission expansion planning (TTEP) under physical intentional attacks.
The above-surveyed literature uses the simplified formulation of nonlinear AC optimal power flow (ACOPF) i.e., the DC optimal power flow (DCOPF) as the lower level. The DCOPF has some drawbacks. Technically, it cannot provide precise information on the power system since it ignores reactive power, resistance and losses and fixes the voltage values for the buses. Mathematically, restricting the available degrees of freedom (e.g., fixed voltages in DC-based method) makes the solution non-optimal and less accurate [34].
To the best of our knowledge, a few studies considered a real picture of the power grid parameters i.e. both active and reactive powers, losses, and voltage profile to assess the vulnerability of the power system. Kim et al. [35] used the AC power flow equations and the Frank Wolfe algorithm to compute an optimal solution of the problem. However, they assumed that attackers are allowed to increase the impedance of transmission lines in the model. Modeling component removal needs to introduce binary variables that require different and more complicated solution techniques. Recently, a probabilistic N-k model is introduced to analyze a probabilistic generalization of the interdiction model using the cutting-plane algorithm [8]. They use convex relaxations instead of the DC power flow approximation.
Generally speaking, the ACOPF is a non-linear and non-convex optimization problem [19] which is used as the lower level in this paper. Considering the ACOPF for each time period (t) as the lower level converts the problem to a bilevel mixed-integer nonlinear programming (MINLP) problem that is very complicated and challenging to solve. It should be emphasized that employing metaheuristic algorithms or non-linear solvers does not guarantee to have a global optimum solution [36]. The aim of this paper is to tackle the new problem and avoid the probable local solution for each time period (t). The contributions of the proposed model in this paper are threefold:
- (1)
A novel deterministic multi-period AC-based one-level MILP formulation of a bilevel MINLP problem is introduced so as to assess the N-k contingency analysis.
- (2)
The model considers a real picture of the power grid parameters i.e., both active and reactive powers, losses, and voltage profile.
- (3)
The planners can decide the level of accuracy by setting the predefined parameters (i.e., n and M) that are introduced in the linearization process for each time period (t).
The remainder of this paper is organized as follows. Section 2 introduces the multi-period AC-based bilevel MINLP problem. Section 3 proposes the solution approaches to linearize and then, transforming to a one-level MILP problem. Sections 4 and 5 present the test case and the numerical results, respectively. Discussions and concluding remarks are finally provided in Sections 6 and 7, respectively.
Section snippets
The multi-period AC-based bilevel MINLP problem
In this section, the mathematical formulation of multi-period AC-based bilevel MINLP problem is introduced. The model provides the worst-case scenario under multiple outages that is of interest in N-k security assessment. This formulation is based on the following assumptions that are commonly used for vulnerability and contingency assessment of a power system [[23], [24], [25], [26], [27], [28], 37,38]:
- 1-
The rational attacker (the worst-case scenario) is considered trying to maximize the damage
Solution methodology
It should be noted that there will be no guarantee to obtain the global solution due to the non-convexity non-linearity nature of the proposed approach [40] with a non-linear solver or evolutionary approaches [19]. Therefore, we transformed it to one-level MILP problem in two steps. First, the lower-level problem is transformed to a MILP problem to avoid any local solution. Then, the duality theory [41] is used to have a one-level MILP problem in the second step.
Test system
The IEEE Reliability Test System (RTS) and IEEE 57-bus are used in this paper. The IEEE 57-bus test case is only used for comparison. It represents a portion of the American Electric Power system (in the U.S. Midwest) and has 57 buses, 7 generators, and 42 loads [46]. Data availability makes the IEEE RTS an ideal test case for multi-period bulk power system vulnerability analysis. It contains 24 buses, 32 generators, and 38 branches (lines plus transformers) as shown in Fig. 3. The transmission
Numerical results
The proposed model has been successfully applied to the test systems. In this numerical study, the minimum and maximum of the voltage magnitude of buses are assumed to be 0.95 and 1.05 p.u., respectively [48]. The problems are solved on a laptop running with an Intel Core i7, 2.2 GHz processor, and 8 GB RAM. The CPLEX solver which uses the branch and cut algorithm is employed under GAMS (General Algebraic Modeling System) [49]. Furthermore, the ACOPF function of MATPOWER in MATLAB environment
Discussions and future works
The main aim of this paper is to develop an approximation model of the original non-convex bilevel MINLP problem that is NP-hard and computationally challenging. The original problem includes an upper level whose objective is to identify exactly k components to maximize the damage (i.e. load shedding) in the system and a lower level whose objective is mitigating the impacts of attacks and minimizing the damage consequences using ACOPF. We have not done any approximation for the upper-level
Conclusion
A novel multi-period AC-based approach is presented to analyze N-k contingencies in order to enhance the resilience of a bulk power system under multiple outages. This method is based on the Stackelberg game theory which includes an upper level whose objective is to identify exactly k components to maximize the damage (load shedding) in the system and a lower level whose objective is mitigating the impacts of attacks and minimizing the damage consequences. Unlike the literature, in order to
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors thank the anonymous referees for the precious comments, which have enabled significant improvements to the paper.
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