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False data injection (FDI) cyber-attacks on power systems can be prevented by strategically selecting and protecting a sufficiently large measurement subset, which, however, requires adequate cyber-defense resources for measurement protection. With any given cyber-defense resource, this paper proposes a preventive-corrective cyber-defense strategy, which minimizes the FDI attack-induced region in a preventive manner, followed by maximizing the cybersecurity margin in a corrective manner. First, this paper proposes a preventive cyber-defense strategy that minimizes the volume of the FDI attack-induced region via preventive allocation of any given measurement protection resource. Particularly, a sufficient condition for constructing the FDI unattackable lines is proposed, indicating that the FDI cyber-attack could be locally rather than globally prevented. Then, given a non-empty FDI attack-induced region, this paper proposes a corrective cyber-defense strategy that maximizes the cybersecurity margin, leading to a trade-off between the safest-but-expensive operation point (i.e., Euclidean Chebyshev center) and the cheapest-but-dangerous operation point. Simulation results on a modified IEEE 14 bus system verify the effectiveness and cost-effectiveness of the proposed preventive-corrective cyber-defense strategy.

To figure out the stability issues brought by renewable energy sources (RES) with non-Gaussian uncertainties in isolated microgrids, this paper proposes a chance constrained stability constrained optimal power flow (CC-SC-OPF) model. Firstly, we propose a bi-level optimization problem, of which the upper level aims to minimize the expected generation cost without violating the stability chance constraint; the lower level concerns about the stability index given by a semi-definite program (SDP). Secondly, we apply the Gaussian mixture model (GMM) to handle the non-Gaussian RES uncertainties and introduce analytical sensitivity analysis to reformulate chance constraints with respect to stability index and operational variables into linear deter-ministic versions. By incorporating linearized constraints, the bi-level model can be efficiently solved by Benders decomposition-based approach. Thirdly, we design a supplementary corrective countermeasure to compensate the possible control error caused by the linear approximation. Simulation results on the 33-bus microgrid reveal that compared to benchmarking approaches, the proposed model converges 30 times faster with more accurate solutions.

In this letter, we analytically investigate the sensitivity of stability index to its dependent variables in general power systems. Firstly, we give a small-signal model, the stability index is defined as the solution to a semidefinite program (SDP) based on the related Lyapunov equation. In case of stability, the stability index also characterizes the convergence rate of the system after disturbances. Then, by leveraging the duality of SDP, we deduce an analytical formula of the stability sensitivity to any entries of the system Jacobian matrix in terms of the SDP primal and dual variables. Unlike the traditional numerical perturbation method, the proposed sensitivity evaluation method is more accurate with a much lower computational burden. This letter applies a modified microgrid for comparative case studies. The results reveal the significant improvements on the accuracy and computational efficiency of stability sensitivity evaluation.

This paper adopts a two-stage sample robust optimization (SRO) model to address the wind power penetrated unit commitment optimal energy flow (UC-OEF) problem for IEGSs. The two-stage SRO model can be approximately transformed into a computationally efficient form. Specifically, we employ linear decision rules to simplify the proposed UC-OEF model. Moreover, we further enhance the tractability of the simplified model by exploring its structural features and, accordingly, develop a solution method., Comment: 10 pages

Ice storms, known for their severity and predictability, necessitate proactive resilience enhancement in power systems. Traditional approaches often overlook the endogenous uncertainties inherent in human decisions and underutilize predictive information like forecast accuracy and preparation time. To bridge these gaps, we proposed a two-stage risk-informed decision-dependent resilience planning (RIDDRP) for transmission systems against ice storms. The model leverages predictive information to optimize resource allocation, considering decision-dependent line failure uncertainties introduced by planning decisions and exogenous ice storm-related uncertainties. We adopt a dual-objective approach to balance economic efficiency and system resilience across both normal and emergent conditions. The first stage of the RDDIP model makes line hardening decisions, as well as the optimal sitting and sizing of energy storage. The second stage evaluates the risk-informed operation costs, considering both pre-event preparation and emergency operations. Case studies demonstrate the model's ability to leverage predictive information, leading to more judicious investment decisions and optimized utilization of dispatchable resources. We also quantified the impact of different properties of predictive information on resilience enhancement. The RIDDRP model provides grid operators and planners valuable insights for making risk-informed infrastructure investments and operational strategy decisions, thereby improving preparedness and response to future extreme weather events.

When deploying fast charging stations (FCSs) to support long-distance trips of electric vehicles (EVs), there exist indirect network effects: while the gradual diffusion of EVs directly influences the timing and capacities of FCS allocation, the decisions for FCS allocations, in turn, impact the drivers' willingness to adopt EVs. This interplay, if neglected, can result in uncovered EVs and security issues on the grid side and even hinder the effective diffusion of EVs. In this paper, we explicitly incorporate this interdependence by quantifying EV adoption rates as decision-dependent uncertainties (DDUs) using decision-dependent ambiguity sets (DDASs). Then, a two-stage decision-dependent distributionally robust FCS planning (D$^3$R-FCSP) model is developed for adaptively deploying FCSs with on-site sources and expanding the coupled distribution network. A multi-period capacitated arc cover-path cover (MCACPC) model is incorporated to capture the EVs' recharging patterns to ensure the feasibility of FCS locations and capacities. To resolve the nonlinearity and nonconvexity, the D$^3$R-FCSP model is equivalently reformulated into a single-level mixed-integer linear programming by exploiting its strong duality and applying the McCormick envelope. Finally, case studies highlight the superior out-of-sample performances of our model in terms of security and cost-efficiency. Furthermore, the byproduct of accelerated EV adoption through an implicit positive feedback loop is highlighted.

Cascading failure causes a major risk to society currently. To effectively mitigate the risk, dynamic thermal rating (DTR) technique can be applied as a cost-effective strategy to exploit potential transmission capability. From the perspectives of service life and Braess paradox, it is important and challenging to jointly optimize the DTR placement and operation schedule for changing system state, which is a two-stage combinatorial problem with only discrete variables, suffering from no approximation guarantee and dimension curse only based on traditional models. Thus, the present work proposes a novel two-stage submodular optimization (TSSO) of DTR for risk mitigation considering placement and operation schedule. Specifically, it optimizes DTR placement with proper redundancy in first stage, and then determines the corresponding DTR operation for each system state in second stage. Under the condition of the Markov and submodular features in sub-function of risk mitigation, the submodularity of total objective function of TSSO can be proven for the first time. Based on this, a state-of-the-art efficient solving algorithm is developed that can provide a better approximation guarantee than previous studies by coordinating the separate curvature and error form. The performance of the proposed algorithms is verified by case results.

When performing the resilience enhancement for distribution networks, there are two obstacles to reliably model the uncertain contingencies: 1) decision-dependent uncertainty (DDU) due to various line hardening decisions, and 2) distributional ambiguity due to limited outage information during extreme weather events (EWEs). To address these two challenges, this paper develops scenario-wise decision-dependent ambiguity sets (SWDD-ASs), where the DDU and distributional ambiguity inherent in EWE-induced contingencies are simultaneously captured for each possible EWE scenario. Then, a two-stage trilevel decision-dependent distributionally robust resilient enhancement (DD-DRRE) model is formulated, whose outputs include the optimal line hardening, distributed generation (DG) allocation, and proactive network reconfiguration strategy under the worst-case distributions in SWDD-ASs. Subsequently, the DD-DRRE model is equivalently recast to a mixed-integer linear programming (MILP)-based master problem and multiple scenario-wise subproblems, facilitating the adoption of a customized column-and-constraint generation (C&CG) algorithm. Finally, case studies demonstrate a remarkable improvement in the out-of-sample performance of our model, compared to its prevailing stochastic and robust counterparts. Moreover, the potential values of incorporating the ambiguity and distributional information are quantitatively estimated, providing a useful reference for planners with different budgets and risk-aversion levels.

In this paper, we discuss a class of two-stage hierarchical games with multiple leaders and followers, which is called Nash-Stackelberg-Nash (N-S-N) games. Particularly, we consider N-S-N games under decision-dependent uncertainties (DDUs). DDUs refer to the uncertainties that are affected by the strategies of decision-makers and have been rarely addressed in game equilibrium analysis. In this paper, we first formulate the N-S-N games with DDUs of complete ignorance, where the interactions between the players and DDUs are characterized by uncertainty sets that depend parametrically on the players' strategies. Then, a rigorous definition for the equilibrium of the game is established by consolidating generalized Nash equilibrium and Pareto-Nash equilibrium. Afterward, we prove the existence of the equilibrium of N-S-N games under DDUs by applying Kakutani's fixed-point theorem. Finally, an illustrative example is provided to show the impact of DDUs on the equilibrium of N-S-N games.

This paper studies a trilevel profit maximization problem of electric vehicle (EV) aggregator participating in the day-ahead reserve market, considering the uncertain EV connectivity to the grid. At the upper level (UL), the aggregator purchases reserve from individual EVs and trades it in the reserve market. It determines the optimal reserve purchasing price and reserve trading amount to maximize profit. Responding to the reserve purchasing price, at the middle level (ML), each EV owner maximizes his/her own utility by scheduling the battery usage for charging/discharging, reserve or transportation. As the connectivity of EV to the grid is uncertain due to transportation randomness, we characterize the worst-case connectivity at the lower level (LL) such that energy consumption for transportation tasks can be guaranteed. The proposed trilevel optimization problem is challenging because of its multi-level structure and binary variables at ML and LL. Firstly, total unimodularity property, primal-dual and value-function methods are used to convert this problem into a single-level mixed integer nonlinear program (MINLP). Then, a sample-based algorithm is developed to solve the single-level MINLP and the convergence is proved. In addition, an acceleration strategy is proposed to facilitate the computation. Case studies validate the effectiveness of our proposed solution method.

Energy storage is expected to play an increasingly important role in mitigating variations that come along with the growing penetration of renewable energy. In this paper, we study the optimal bidding of an energy storage unit in a semi-centralized market. The energy storage unit offers its available storage capacity and maximum charging/discharging rate to the operator; then the operator clears the real-time market by minimizing the total cost. The energy storage unit is paid/charged at locational marginal price (LMP). The problem casts down to a bilevel optimization problem with a mixed-integer lower-level. An improved surrogate-based method with the combined spatial-temporal entropy term is developed to solve this problem. Numerical examples demonstrate the scalability, efficiency, and accuracy of the proposed method., Comment: 5 pages, 5 figures

This paper investigates the day-ahead optimal reserve management problem of electric vehicle (EV) aggregator. Geographically dispersed EVs are coordinated by the aggregator to participate in the day-ahead reserve market. A bilevel model is proposed to formulate the interaction between the aggregator and the EV owners. In the upper level, the EV aggregator aggregates the reserve capacity provided by the EV owners and then bids in the reserve market. In the lower level, the EV owners decide their optimal energy charging/discharging and reserve capacity based on the reserve price released by the aggregator. Compared with existing works, our proposed bilevel model is more practical. To be specific, we take into account the exclusive right constraint for accessing EV battery, i.e., the battery cannot be simultaneously accessed by the EV owner and the aggregator. This practical model leads to a bilevel mixed integer nonlinear program, which is difficult to solve because the lower level problem incorporates nonconvex integer variables. A novel exact algorithm is developed to solve it and the finite convergence is proved. Comprehensive case studies demonstrate the economic merits of our proposed model to both the aggregator and the EV owners. We also compare the solution optimality with the state-of-the-art approach and thus validate the effectiveness of our proposed exact algorithm. © 2010-2012 IEEE.

Several parallel restored subsystems exist in the last stage of the build-up restoration process of bulk power systems. The whole system restoration is completed when these subsystems are reconnected and their internal loads are recovered. This paper proposes a robust distributed coordination scheme to achieve reconnection and load recovery of parallel restored subsystems in wind power penetrated transmission systems. First, robust distributed restoration models are constructed considering individual subsystems under uncertain conditions. Then, an inner-outer nested upper-lower (IOUL) iterative algorithm is developed to solve the constructed models. The robust distributed coordination can be realized according to convergence. The proposed algorithm solves the complex non-convex model with uncertain variables by iteratively solving the small-scale mixedinteger linear programming (MILP) problem. The robust distributed restoration scheme allows independent decision-making within subsystems and ensures the feasibility of the distributed restoration strategy under uncertain conditions. The effectiveness of the proposed method is validated using three interconnected 6-bus systems and the IEEE 118-bus system, showing improved computational efficiency and restoration acceleration.

The increasing extreme weather events poses unprecedented challenges on power system operation because of their uncertain and sequential impacts on power systems. This paper proposes the concept of an extended steady-state security region (ESSR), and resilience enhancement for transmission systems based on ESSR in consideration of uncertain varying topology changes caused by the extreme weather events is implemented. ESSR is a ploytope describing a region, in which the operating points are within the operating constraints. In consideration of uncertain varying topology changes with ESSR, the resilience enhancement problem is built as a bilevel programming optimization model, in which the system operators deploy the optimal strategy against the most threatening scenario caused by the extreme weather events. To avoid the curse of dimensionality with regard to system topologies for a large scale system, the Monte Carlo method is used to generate uncertain system topologies, and a recursive McCormick envelope-based approach is proposed to connect generated system topologies to optimization variables. Karush Kuhn Tucker (KKT) conditions are used to transform the suboptimization model in the second level into a group of equivalent constraints in the first level. A simple test system and IEEE 118-bus system are used to validate the proposed.

Autonomous vehicles (AVs) will revolutionarize ground transport and take a substantial role in the future transportation system. Most AVs are likely to be electric vehicles (EVs) and they can participate in the vehicle-to-grid (V2G) system to support various V2G services. Although it is generally infeasible for EVs to dictate their routes, we can design AV travel plans to fulfill certain system-wide objectives. In this paper, we focus on the AVs looking for parking and study how they can be led to appropriate parking facilities to support V2G services. We formulate the Coordinated Parking Problem (CPP), which can be solved by a standard integer linear program solver but requires long computational time. To make it more practical, we develop a distributed algorithm to address CPP based on dual decomposition. We carry out a series of simulations to evaluate the proposed solution methods. Our results show that the distributed algorithm can produce nearly optimal solutions with substantially less computational time. A coarser time scale can improve computational time but degrade the solution quality resulting in possible infeasible solution. Even with communication loss, the distributed algorithm can still perform well and converge with only little degradation in speed., Comment: 10 pages, submitted for publication

This paper analyzes the performances of the Spearman's rho (SR) and Kendall's tau (KT) with respect to samples drawn from bivariate normal and bivariate contaminated normal populations. The exact analytical formulae of the variance of SR and the covariance between SR and KT are obtained based on the Childs's reduction formula for the quadrivariate normal positive orthant probabilities. Close form expressions with respect to the expectations of SR and KT are established under the bivariate contaminated normal models. The bias, mean square error (MSE) and asymptotic relative efficiency (ARE) of the three estimators based on SR and KT to the Pearson's product moment correlation coefficient (PPMCC) are investigated in both the normal and contaminated normal models. Theoretical and simulation results suggest that, contrary to the opinion of equivalence between SR and KT in some literature, the behaviors of SR and KT are strikingly different in the aspects of bias effect, variance, mean square error, and asymptotic relative efficiency. The new findings revealed in this work provide not only deeper insights into the two most widely used rank based correlation coefficients, but also a guidance for choosing which one to use under the circumstances where the PPMCC fails to apply.