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Battery storage is essential for the future smart grid. The inevitable cell degradation renders the battery lifetime volatile and highly dependent on battery dispatch, and thus incurs opportunity cost. This paper rigorously derives the marginal degradation cost of battery for power system dispatch. The derived optimal marginal degradation cost is time-variant to reflect the time value of money and the functionality fade of battery and takes the form of a constant value divided by a discount factor plus a term related to battery state of health. In case studies, we demonstrate the evolution of the optimal marginal costs of degradation that corresponds to the optimal long-term dispatch outcome. We also show that the optimal marginal cost of degradation depends on the marginal cost of generation in the grid., 11 pages, 4 figures

Transient stability boundary (TSB) is an important tool in power system online security monitoring, but practically it suffers from high computational burden using state-of-the-art methods, such as time-domain simulation (TDS), with numerous scenarios taken into account (e.g., operating points (OPs) and N-1 contingencies). The purpose of this work is to establish a data-driven framework to generate sufficient critical samples close to the boundary within a limited time, covering all critical scenarios in current OP. Therefore, accurate TSB can be periodically refreshed by tracking current OP in time. The idea is to develop a search strategy to obtain more data samples near the stability boundary, while traverse the rest part with fewer samples. To achieve this goal, a specially designed transient index sensitivity based search strategy and critical scenarios selection mechanism are proposed, in order to find out the most representative scenarios and periodically update TSB for online monitoring. Two case studies validate effectiveness of the proposed method.

Although distribution grid customers are obliged to share their consumption data with distribution system operators (DSOs), a possible leakage of this data is often disregarded in operational routines of DSOs. This paper introduces a privacy-preserving optimal power flow (OPF) mechanism for distribution grids that secures customer privacy from unauthorised access to OPF solutions, e.g., current and voltage measurements. The mechanism is based on the framework of differential privacy that allows to control the participation risks of individuals in a dataset by applying a carefully calibrated noise to the output of a computation. Unlike existing private mechanisms, this mechanism does not apply the noise to the optimization parameters or its result. Instead, it optimizes OPF variables as affine functions of the random noise, which weakens the correlation between the grid loads and OPF variables. To ensure feasibility of the randomized OPF solution, the mechanism makes use of chance constraints enforced on the grid limits. The mechanism is further extended to control the optimality loss induced by the random noise, as well as the variance of OPF variables. The paper shows that the differentially private OPF solution does not leak customer loads up to specified parameters.

In the face of extreme events, e.g., hurricanes, the transmission systems, especially the transmission lines, are affected across time and space. To mitigate these impacts on the day-ahead market from a probabilistic perspective, a resilient unit commitment (UC) problem is formulated as a two-stage distributionally robust and robust optimization (DR&RO) problem. In the first stage, the commitment, energy, and reserves of generators are pre-scheduled to minimize the operational cost, responding to the worst load forecasting and line failure scenario in the operating day. The operating status of transmission lines are depicted by a novel uncertainty set with a distributionally chance constraint considering the repair of failed lines. This chance constraint is reformulated to its deterministic equivalence. Using both load shedding and generation curtailment, recourse problems are formulated in the second stage considering the time-varying operating status of transmission lines. The formulated DR&RO problem is solved using a hybrid Benders decomposition and column-and-constraint generation scheme. Simulations are conducted on the modified IEEE reliability test system (RTS) and two-area IEEE RTS-96 under hurricanes. Results verify the effectiveness of the proposed method, in comparison with prevalent two-stage stochastic and robust optimization methods.

The electrical grid is evolving from a network consisting of mostly synchronous machines to a mixture of synchronous machines and inverter-based resources such as wind, solar, and energy storage. This transformation has led to a decrease in mechanical inertia, which necessitate a need for the new resources to provide frequency responses through their inverter interfaces. In this paper we proposed a new strategy based on model predictive control to determine the optimal active-power set-point for inverters in the event of a disturbance in the system. Our framework explicitly takes the hard constraints in power and energy into account, and we show that it is robust to measurement noise, limited communications and delay by using an observer to estimate the model mismatches in real-time. We demonstrate the proposed controller significantly outperforms an optimally tuned virtual synchronous machine on a standard 39-bus system under a number of scenarios. In turn, this implies optimized inverter-based resources can provide better frequency responses compared to conventional synchronous machines., Submitted to IEEE Transactions on Power Systems

This paper presents a novel optimization framework of formulating the three-phase optimal power flow that involves uncertainty. The proposed uncertainty-aware optimization (UaO) framework is: 1) a deterministic framework that is less complex than the existing optimization frameworks involving uncertainty, and 2) convex such that it admits polynomial-time algorithms and mature distributed optimization methods. To construct this UaO framework, a methodology of learning-aided uncertainty-aware modeling, with prediction errors of stochastic variables as the measurement of uncertainty, and a theory of data-driven convexification are proposed. Theoretically, the UaO framework is applicable for modeling general optimization problems under uncertainty., Accepted for pubication in the IEEE Transactions on Power Systems

Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper presents two improvements to these envelopes. The first improvement leverages a polar representation of the branch admittances in addition to the rectangular representation used previously. The second improvement is based on a coordinate transformation via a complex per unit base power normalization that rotates the power flow equations. The trigonometric envelopes resulting from this rotation can be tighter than the corresponding envelopes in previous QC relaxation formulations. Using an empirical analysis with a variety of test cases, this paper suggests an appropriate value for the angle of the complex base power. Comparing the results with a state-of-the-art QC formulation reveals the advantages of the proposed improvements.

Continuous-time random disturbances (also called stochastic excitations) due to increasing renewable generation have an increasing impact on power system dynamics; However, except from the Monte Carlo simulation, most existing methods for quantifying this impact are intrusive, meaning they are not based on commercial simulation software and hence are difficult to use for power utility companies. To fill this gap, this paper proposes an efficient and nonintrusive method for quantifying uncertainty in dynamic power systems subject to stochastic excitations. First, the Gaussian or non-Gaussian stochastic excitations are modeled with an It\^{o} process as stochastic differential equations. Then, the It\^{o} process is spectrally represented by independent Gaussian random parameters, which enables the polynomial chaos expansion (PCE) of the system dynamic response to be calculated via an adaptive sparse probabilistic collocation method. Finally, the probability distribution and the high-order moments of the system dynamic response and performance index are accurately and efficiently quantified. The proposed nonintrusive method is based on commercial simulation software such as PSS/E with carefully designed input signals, which ensures ease of use for power utility companies. The proposed method is validated via case studies of IEEE 39-bus and 118-bus test systems., Comment: Accepted by IEEE Transactions on Power Systems

The transmission-constrained unit commitment (TC-UC) problem is one of the most relevant problems solved by independent system operators for the daily operation of power systems. Given its computational complexity, this problem is usually not solved to global optimality for real-size power systems. In this paper, we propose a data-driven method that leverages historical information to screen out network constraints in the TC-UC problem. First, past data on demand and renewable generation throughout the network are used to learn the congestion status of transmission lines. Then, we infer the lines that will not become congested for upcoming operating conditions based on such learning and disregard their capacity constraints. This way, we formulate a reduced TC-UC problem that is easier to solve. Numerical results on a medium- and a large-size power system show that the proposed approach outperforms existing ones by significantly reducing the computational time while obtaining solutions that are equal or close to the one obtained with the original TC-UC problem. Furthermore, the purely data-driven method we propose can be seamlessly complemented with a constraint generation procedure to guarantee that the optimal solution to the original TC-UC problem is eventually recovered.

The distribution network restoration problem is by nature a mixed integer and non-linear optimization problem due to the switching decisions and Optimal Power Flow (OPF) constraints, respectively. The link between these two parts involves logical implications modelled through big-M coefficients. The presence of these coefficients makes the relaxation of the mixed-integer problem using branch-and-bound method very poor in terms of computation burden. Moreover, this link inhibits the use of classical Benders algorithm in decomposing the problem because the resulting cuts will still depend on the big-M coefficients. In this paper, a novel decomposition approach is proposed for the restoration problem named Modified Combinatorial Benders (MCB) . In this regard, the reconfiguration problem and the OPF problem are decomposed into master and sub problems, which are solved through successive iterations. In the case of a large outage area, the numerical results show that the MCB provides, within a short time (after a few iterations), a restoration solution with a quality that is close to the proven optimality when it can be exhibited.

Topology design is a critical task for the reliability, economic operation, and resilience of distribution systems. This paper proposes a distributionally robust optimization (DRO) model for designing the topology of a new distribution system facing random contingencies (e.g., imposed by natural disasters). The proposed DRO model optimally configures the network topology and integrates distributed generation to effectively meet the loads. Moreover, we take into account the uncertainty of contingency. Using the moment information of distribution line failures, we construct an ambiguity set of the contingency probability distribution, and minimize the expected amount of load shedding with regard to the worst-case distribution within the ambiguity set. As compared with a classical robust optimization model, the DRO model explicitly considers the contingency uncertainty and so provides a less conservative configuration, yielding a better out-of-sample performance. We recast the proposed model to facilitate the column-and-constraint generation algorithm. We demonstrate the out-of-sample performance of the proposed approach in numerical case studies.

We present a distributionally robust optimization (DRO) approach for the transmission expansion planning problem, considering both long- and short-term uncertainties on the system demand and non-dispatchable renewable generation. On the long-term level, as is customary in industry applications, we address the deep uncertainties arising from social and economic transformations, political and environmental issues, and technology disruptions by using long-term scenarios devised by experts. In this setting, many exogenous long-term scenarios containing partial information about the random parameters, namely, the average and the support set, can be considered. For each long-term scenario, a conditional ambiguity set models the incomplete knowledge about the probability distribution of the uncertain parameters in the short-term operation. Consequently, the mathematical problem is formulated as a DRO model with multiple conditional ambiguity sets. The resulting infinite-dimensional problem is recast as an exact, although very large, finite mixed-integer linear programming problem. To circumvent scalability issues, we propose a new enhanced-column-and-constraint-generation (ECCG) decomposition approach with an additional Dantzig--Wolfe procedure. In comparison to existing methods, ECCG leads to a better representation of the recourse function and, consequently, tighter bounds. Numerical experiments based on the benchmark IEEE 118-bus system are reported to corroborate the effectiveness of the method.

In this paper, the Unit Commitment (UC) problem in a power network with low levels of rotational inertia is studied. Frequency-related constraints, namely the limitation on Rate-of-Change-of-Frequency (RoCoF), frequency nadir and steady-state frequency error, are derived from a uniform system frequency response model that incorporates dynamics and controls of both synchronous generators and grid-forming inverters. These constraints are then included into a stochastic UC formulation that accounts for wind power and equipment contingency uncertainties using a scenario-tree approach. In contrast to the linear RoCoF and steady-state frequency error constraints, the nadir constraint is highly nonlinear. To preserve the mixed-integer linear formulation of the stochastic UC model, we propose a computationally efficient approach that allows to recast the nadir constraint by introducing appropriate bounds on relevant decision variables of the UC model. This method is shown to be generally more accurate and computationally more efficient for medium-sized networks than a piece-wise linearization method adapted from the literature. Simulation results for a modified IEEE RTS-96 system revealed that the inclusion of inertia-related constraints significantly influences the UC decisions and increases total costs, as more synchronous machines are forced to be online to provide inertial response.

Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path from an initial operating point to a desired operating point. Given an initial feasible point, the algorithm solves a sequence of convex quadratically constrained optimization problems over conservative convex inner approximations of the OPF feasible space. In each iteration, we obtain a new, improved operating point and a feasible transition from the operating point in the previous iteration. In addition to computing a feasible path to a known desired operating point, this algorithm can also be used to improve the operating point locally. Extensive numerical studies on a variety of test cases demonstrate the algorithm and the ability to arrive at a high-quality solution in few iterations.

This paper develops an efficient algorithm for the multi-period optimal dispatch of deterministic inverter-interfaced energy storage in an unbalanced distribution feeder with significant solar PV penetration. The three-phase, non-convex loss-minimization problem is formulated as a convex second order cone program (SOCP) for the dispatch of batteries in a receding-horizon fashion in order to counter against the variable renewable net-load generation. The solution of the SOCP is used to initialize a nonlinear program (NLP) in order to ensure a physically realizable solution. The phenomenon of simultaneous charging and discharging of batteries is rigorously analyzed and conditions are derived that guarantee it is avoided. Simulation scenarios are implemented with GridLab-D for the IEEE-13 and IEEE-123node test feeders and illustrate not only AC feasibility of the solution, but also near-optimal performance and solve-times within a minute.

The participation of renewable, energy storage, and resources with limited fuel inventory in electricity markets has created the need for optimal scheduling and pricing across multiple market intervals for resources with intertemporal constraints. In this paper, a new multi-period market model is proposed to enhance the efficiency of markets with such type of resources. It is also the first market design that links a forward market and a spot market through the coordination of schedule and price under the multi-period paradigm, achieving reliability, economic efficiency and dispatch-following incentives simultaneously. The forward market solves a multi-period model with a long look-ahead time horizon whereas the spot market solves a series of multi-period dispatch and pricing problems with a shorter look-ahead time horizon on a rolling basis. By using the forward schedules and opportunity costs of intertemporal constraints as a guideline, the spot market model is able to produce economically efficient dispatch solutions as well as prices that incentivize dispatch following under the perfect forecast condition. The proposed scheme is applied to the dispatch and pricing of energy storage resources. Numerical experiments show that the proposed scheme outperforms the traditional myopic method in terms of economic efficiency, dispatch following and reliability., Add proofs for properties

While energy arbitrage from energy storage can lower power system operating costs, it can also increase greenhouse gas emissions. If power system operations are conducted with the constraint that energy storage operation must not increase emissions, how does this constraint affect energy storage investment decisions? Two bi-level energy storage investment problems are considered, representing ‘philanthropic’ (profitability-constrained) and profit-maximizing storage investors (PhSI, PMSI). A MILP heuristic is developed to obtain good candidate solutions to these inherently MINLP bi-level problems. A case study is conducted on a 30% renewable system, with sensitivity analyses on the price of storage and the price of carbon emissions. Regardless of the emissions-neutrality constraint, a PhSI installs significantly more energy storage than a PMSI, increasing system flexibility. The effect of the emissions-neutrality constraint in the absence of a carbon price is to reduce the quantity of storage purchased and reduce annual system emissions ( ${\scriptstyle\sim}$ 3%), with only minor increases in overall cost ( ${\scriptstyle\sim}$ 0.1%). In cases with a carbon price, storage does not tend to increase emissions and the emissions constraint does not tend to decrease storage investment. The emissions-neutrality constraint is seen to deliver similar emissions reductions even in a system with much higher renewable penetration (46%).

To contribute to a smart grid, the unit commitment and dispatch (UCD) is a very important problem to be revisited in the future microgrid environment. In this paper, the UCD problem is studied by taking account of both traditional thermal units and distributed generation units (DGs) while considering demand response (DR) and carbon emissions trading (CET). Then, the UCD problem is reformulated as the optimal switching problem of a hybrid system. The reformulation allows convenient handling of time-dependent start-up cost for dynamic programming (DP). Afterwards, a closed-loop hierarchical operation (CLHO) algorithm is developed which is capable of producing the optimal schedule in the closed-loop behavior. Theoretical analysis is also presented to show the equivalence of the reformulation and the effectiveness of the CLHO algorithm. Finally, the closed-loop behavior and the effectiveness of the algorithm are further verified by simulation studies. It is also shown that low emission quotas and high emission trading price help reduce the total carbon emissions.

In this paper, we propose a distributed voltage control in power distribution networks through reactive power compensation. The proposed control can (i) operate in a distributed fashion where each bus makes its decision based on local voltage measurements and communication with neighboring buses, (ii) always satisfy the reactive power capacity constraint, (iii) drive the voltage magnitude into an acceptable range, and (iv) minimize an operational cost. We also perform various numerical case studies to demonstrate the effectiveness and robustness of the controller using the nonlinear power flow model., Comment: 32 pages, 10 figures; Accepted to IEEE Transactions on Power Systems. \c{opyright} 2019 IEEE

Day-ahead scheduling of electricity generation or unit commitment is an important and challenging optimization problem in power systems. Variability in net load arising from the increasing penetration of renewable technologies have motivated study of various classes of stochastic unit commitment models. In two-stage models, the generation schedule for the entire day is fixed while the dispatch is adapted to the uncertainty, whereas in multi-stage models the generation schedule is also allowed to dynamically adapt to the uncertainty realization. Multi-stage models provide more flexibility in the generation schedule, however, they require significantly higher computational effort than two-stage models. To justify this additional computational effort, we provide theoretical and empirical analyses of the value of multi-stage solution for risk-averse multi-stage stochastic unit commitment models. The value of multi-stage solution measures the relative advantage of multi-stage solutions over their two-stage counterparts. Our results indicate that, for unit commitment models, value of multi-stage solution increases with the level of uncertainty and number of periods, and decreases with the degree of risk aversion of the decision maker., 8 pages, 5 figures

Wind power uncertainty poses significant challenges for automatic generation control (AGC) systems. It can enhance control performances to explicitly consider wind power uncertainty distributions within controller design. However, widely accepted wind uncertainties usually follow non-Gaussian distributions which may lead to complicated stochastic AGC modeling and high computational burdens. To overcome the issue, this paper presents a novel Ito-theory-based model for the stochastic control problem (SCP) of AGC systems, which reduces the computational burden of optimization considering non-Gaussian wind power uncertainty to the same scale as that for deterministic control problems. We present an Ito process model to exactly describe non-Gaussian wind power uncertainty, and then propose an SCP based on the concept of stochastic assessment functions (SAFs). Based on a convergent series expansion of the SAF, the SCP is reformulated as a certain deterministic control problem without sacrificing performance under non-Gaussian wind power uncertainty. The reformulated control problem is proven as a convex optimization which can be solved efficiently. A case study demonstrates the efficiency and accuracy of the proposed approach compared with several conventional approaches., 10 pages, 3 figures

Natural disasters, such as hurricanes, earthquakes and large wind or ice storms, typically require the repair of a large number of components in electricity distribution networks. Since power cannot be restored before these repairs have been completed, optimally scheduling the available crews to minimize the cumulative duration of the customer interruptions reduces the harm done to the affected community. Considering the radial network structure of the distribution system, this repair and restoration process can be modeled as a scheduling problem with soft precedence constraints. As a benchmark, we first formulate this problem as a time-indexed ILP with valid inequalities. Three practical methods are then proposed to solve the problem: (i) an LP-based list scheduling algorithm, (ii) a single to multi-crew repair schedule conversion algorithm, and (iii) a dispatch rule based on $\rho$-factors which can be interpreted as Component Importance Measures. We show that the first two algorithms are $2$ and $\left(2 - \frac{1}{m}\right)$ approximations respectively. We also prove that the latter two algorithms are equivalent. Numerical results validate the effectiveness of the proposed methods.

Cascading failures typically occur following a large disturbance in power systems, such as tripping of a generating unit or a transmission line. Such failures can propagate and destabilize the entire power system, potentially leading to widespread outages. One approach to mitigate impending cascading failures is through controlled islanding, in which a set of transmission lines is deliberately tripped to partition the unstable system into several disjoint, internally stable islands. Selecting such a set of transmission lines is inherently a combinatorial optimization problem. Current approaches address this problem in two steps: first, classify coherent generators into groups and then separate generator groups into different islands with minimal load-generation imbalance. These methods, however, are based on computationally expensive heuristics that do not provide optimality guarantees. This paper proposes a novel approach to controlled islanding based on weak submodularity. The new formulation jointly captures the minimal generator non-coherency and minimal load-generation imbalance in one objective function. The islanding problem is then relaxed to a formulation with bounded submodularity ratio and a matroid constraint. An approximation algorithm is proposed which achieves a provable optimality bound on non-coherency and load-generation imbalance. The proposed framework is tested on IEEE 39-bus and 118-bus power systems.

This work considers the task of finding an AC-feasible operating point of a severely damaged transmission network while ensuring that a maximal amount of active power loads can be delivered. This AC Maximal Load Delivery (AC-MLD) task is a nonconvex nonlinear optimization problem that is incredibly challenging to solve on large-scale transmission system datasets. This work demonstrates that convex relaxations of the AC-MLD problem provide a reliable and scalable method for finding high-quality bounds on the amount of active power that can be delivered in the AC-MLD problem. To demonstrate their effectiveness, the solution methods proposed in this work are rigorously evaluated on 1000 N-k scenarios on seven power networks ranging in size from 70 to 6000 buses. The most effective relaxation of the AC-MLD problem converges in less than 20 seconds on commodity computing hardware for all 7000 of the scenarios considered., The problem considered in this work is also known as AC Minimal Load-Shedding

We propose a novel global solution algorithm for the network-constrained unit commitment problem incorporating a nonlinear alternating current model of the transmission network, which is a nonconvex mixed-integer nonlinear programming (MINLP) problem. Our algorithm is based on the multi-tree global optimization methodology, which iterates between a mixed-integer lower-bounding problem and a nonlinear upper-bounding problem. We exploit the mathematical structure of the unit commitment problem with AC power flow constraints (UC-AC) and leverage optimization-based bounds tightening, second-order cone relaxations, and piecewise outer approximations to guarantee a globally optimal solution at convergence. Numerical results on four benchmark problems illustrate the effectiveness of our algorithm, both in terms of convergence rate and solution quality., 12 pages

This paper studies the optimal transmission switching (OTS) problem for power systems, where certain lines are fixed (uncontrollable) and the remaining ones are controllable via on/off switches. The goal is to identify a topology of the power grid that minimizes the cost of the system operation while satisfying the physical and operational constraints. Most of the existing methods for the problem are based on first converting the OTS into a mixed-integer linear program (MILP) or mixed-integer quadratic program (MIQP), and then iteratively solving a series of its convex relaxations. The performance of these methods depends heavily on the strength of the MILP or MIQP formulations. In this paper, it is shown that finding the strongest variable upper and lower bounds to be used in an MILP or MIQP formulation of the OTS based on the big-$M$ or McCormick inequalities is NP-hard. Furthermore, it is proven that unless P=NP, there is no constant-factor approximation algorithm for constructing these variable bounds. Despite the inherent difficulty of obtaining the strongest bounds in general, a simple bound strengthening method is presented to strengthen the convex relaxation of the problem when there exists a connected spanning subnetwork of the system with fixed lines. The proposed method can be treated as a preprocessing step that is independent of the solver to be later used for numerical calculations and can be carried out offline before initiating the solver. A remarkable speedup in the runtime of the mixed-integer solvers is obtained using the proposed bound strengthening method for medium- and large-scale real-world systems.

This paper proposes a novel method to co-optimize distribution system operation and repair crew routing for outage restoration after extreme weather events. A two-stage stochastic mixed integer linear program is developed. The first stage is to dispatch the repair crews to the damaged components. The second stage is distribution system restoration using distributed generators, and reconfiguration. We consider demand uncertainty in terms of a truncated normal forecast error distribution, and model the uncertainty of the repair time using a lognormal distribution. A new decomposition approach, combined with the Progressive Hedging algorithm, is developed for solving large-scale outage management problems in an effective and timely manner. The proposed method is validated on modified IEEE 34- and 8500-bus distribution test systems., Under review in IEEE Transactions on Power Systems

In this paper, we propose a risk-based data-driven approach to optimal power flow (DROPF) with dynamic line rating. The risk terms, including penalties for load shedding, wind generation curtailment and line overload, are embedded into the objective function. To hedge against the uncertainties on wind generation data and line rating data, we consider a distributionally robust approach. The ambiguity set is based on second-order moment and Wasserstein distance, which captures the correlations between wind generation outputs and line ratings, and is robust to data perturbation. We show that the proposed DROPF model can be reformulated as a conic program. Considering the relatively large number of constraints involved, an approximation of the proposed DROPF model is suggested, which significantly reduces the computational costs. A Wasserstein distance constrained DROPF and its tractable reformulation are also provided for practical large-scale test systems. Simulation results on the 5-bus, the IEEE 118-bus and the Polish 2736-bus test systems validate the effectiveness of the proposed models.

This letter addresses the issue of linearization of quadratic thermal limits of transmission lines for linear optimal power flow formulation. A new irregular polygon based linearization is proposed. The approach is purely based on geometrical concepts and does not introduce any optimization problem. Comparison of the number of constraints is given with different errors for different branch capabilities and systems. Test case analysis is also presented to validate the irregular polygon linearization strategy with optimal value and computational time results.

The paper proposes a new adaptive approach to power system model reduction for fast and accurate time-domain simulation. This new approach is a compromise between linear model reduction for faster simulation and nonlinear model reduction for better accuracy. During the simulation period, the approach adaptively switches among detailed and linearly or nonlinearly reduced models based on variations of the system state: it employs unreduced models for the fault-on period, uses weighted column norms of the admittance matrix to decide which functions to be linearized in power system differential-algebraic equations for large changes of the state, and adopts a linearly reduced model for small changes of the state. Two versions of the adaptive model reduction approach are introduced. The first version uses traditional power system partitioning where the model reduction is applied to a defined large external area in a power system and the other area defined as the study area keeps full detailed models. The second version applies the adaptive model reduction to the whole system. The paper also conducts comprehensive case studies comparing simulation results using the proposed adaptively reduced models with the linearly reduced model on the Northeast Power Coordinating Council 140-bus 48-machine system., Submitted to IEEE Transactions on Power Systems

This paper analyzes different models for evaluating investments in Energy Storage Systems (ESS) in power systems with high penetration of Renewable Energy Sources (RES). First of all, two methodologies proposed in the literature are extended to consider ESS investment: a unit commitment model that uses the System States (SS) method of representing time; and another one that uses a representative periods (RP) method. Besides, this paper proposes two new models that improve the previous ones without a significant increase of computation time. The enhanced models are the System States Reduced Frequency Matrix (SS-RFM) model which addresses short-term energy storage more approximately than the SS method to reduce the number of constraints in the problem, and the Representative Periods with Transition Matrix and Cluster Indices (RP-TM&CI) model which guarantees some continuity between representative periods, e.g. days, and introduces long-term storage into a model originally designed only for the short term. All these models are compared using an hourly unit commitment model as benchmark. While both system state models provide an excellent representation of long-term storage, their representation of short-term storage is frequently unrealistic. The RP-TM&CI model, on the other hand, succeeds in approximating both short- and long-term storage, which leads to almost 10 times lower error in storage investment results in comparison to the other models analyzed.

Though distribution system operators have been adding more sensors to their networks, they still often lack an accurate real-time picture of the behavior of distributed energy resources such as demand responsive electric loads and residential solar generation. Such information could improve system reliability, economic efficiency, and environmental impact. Rather than installing additional, costly sensing and communication infrastructure to obtain additional real-time information, it may be possible to use existing sensing capabilities and leverage knowledge about the system to reduce the need for new infrastructure. In this paper, we disaggregate a distribution feeder's demand measurements into: 1) the demand of a population of air conditioners, and 2) the demand of the remaining loads connected to the feeder. We use an online learning algorithm, Dynamic Fixed Share (DFS), that uses the real-time distribution feeder measurements as well as models generated from historical building- and device-level data. We develop two implementations of the algorithm and conduct case studies using real demand data from households and commercial buildings to investigate the effectiveness of the algorithm. The case studies demonstrate that DFS can effectively perform online disaggregation and the choice and construction of models included in the algorithm affects its accuracy, which is comparable to that of a set of Kalman filters., 14 pages, article in press, 2018, IEEE Transactions on Power Systems

A simple characterization of the solvability of power flow equations is of great importance in the monitoring, control, and protection of power systems. In this paper, we introduce a sufficient condition for power flow Jacobian nonsingularity. We show that this condition is second-order conic representable when load powers are fixed. Through the incorporation of the sufficient condition, we propose a voltage stability-constrained optimal power flow (VSC-OPF) formulation as a second-order cone program (SOCP). An approximate model is introduced to improve the scalability of the formulation to larger systems. Extensive computation results on Matpower and NESTA instances confirm the effectiveness and efficiency of the formulation., Accepted for publication in IEEE Transactions on Power Systems

We consider using a battery storage system simultaneously for peak shaving and frequency regulation through a joint optimization framework which captures battery degradation, operational constraints and uncertainties in customer load and regulation signals. Under this framework, using real data we show the electricity bill of users can be reduced by up to 15\%. Furthermore, we demonstrate that the saving from joint optimization is often larger than the sum of the optimal savings when the battery is used for the two individual applications. A simple threshold real-time algorithm is proposed and achieves this super-linear gain. Compared to prior works that focused on using battery storage systems for single applications, our results suggest that batteries can achieve much larger economic benefits than previously thought if they jointly provide multiple services., Comment: To Appear in IEEE Transaction on Power Systems

Higher levels of renewable electricity generation increase uncertainty in power system operation. To ensure secure system operation, new tools that account for this uncertainty are required. In this paper, we adopt a chance-constrained AC optimal power flow formulation, which guarantees that generation, power flows, and voltages remain within their bounds with a predefined probability. We then discuss different chance-constraint reformulations and solution approaches for the problem. We first describe an analytical reformulation based on partial linearization, which enables us to obtain a tractable representation of the optimization problem. We then provide an efficient algorithm based on an iterative solution scheme which alternates between solving a deterministic AC optimal power flow problem and assessing the impact of uncertainty. The flexibility of the iterative scheme enables not only scalable implementations, but also alternative chance-constraint reformulations. In particular, we suggest two sample-based reformulations that do not require any approximation or relaxation of the AC power flow equations. In a case study based on four different IEEE systems, we assess the performance of the method, and demonstrate scalability of the iterative scheme. We further show that the analytical reformulation accurately and efficiently enforces chance constraints in both in- and out-of-sample tests, and that the analytical reformulations outperforms the two alternative, sample-based chance constraint reformulations.

The DC Power Flow approximation has been widely used for decades in both industry and academia due to its computational speed and simplicity, but suffers from inaccuracy, in part due to the assumption of a lossless network. Here we present a natural extension of the DC Power Flow to lossy networks. Our approach is based on reformulating the lossy active power flow equations into a novel fixed-point equation, and iterating this fixed-point mapping to generate a sequence of improving estimates for the active power flow solution. Each iteration requires the solution of a standard DC Power Flow problem with a modified vector of power injections. The first iteration returns the standard DC Power Flow, and one or two additional iterations yields a one or two order-of-magnitude improvement in accuracy. For radial networks, we give explicit conditions on the power flow data which guarantee (i) that the active power flow equations possess a unique solution, and (ii) that our iteration converges exponentially and monotonically to this solution. For meshed networks, we extensively test our results via standard power flow cases., Revision, 9 pages 3 figures. Revised version

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions.Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations., 8 figures, 18 pages

The most common procedure to solve a linear bilevel problem in the PES community is, by far, to transform it into an equivalent single-level problem by replacing the lower level with its KKT optimality conditions. Then, the complementarity conditions are reformulated using additional binary variables and large enough constants (big-Ms) to cast the single-level problem as a mixed-integer linear program that can be solved using optimization software. In most cases, such large constants are tuned by trial and error. We show, through a counterexample, that this widely used trial-and-error approach may lead to highly suboptimal solutions. Then, further research is required to properly select big-M values to solve linear bilevel problems.

Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co- optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system., Comment: Under revision

Energy storage can facilitate the integration of renewable energy resources by providing arbitrage and ancillary services. Jointly optimizing energy and ancillary services in a centralized electricity market reduces the system's operating cost and enhances the profitability of energy storage systems. However, achieving these objectives requires that storage be located and sized properly. We use a bilevel formulation to optimize the location and size of energy storage systems, which perform energy arbitrage and provide regulation services. Our model also ensures the profitability of investments in energy storage by enforcing a rate of return constraint. Computational tractability is achieved through the implementation of a primal decomposition and a subgradient-based cutting- plane method. We test the proposed approach on a 240-bus model of the Western Electricity Coordinating Council system and analyze the effects of different storage technologies, rate of return requirements, and regulation market policies on energy storage participation on the optimal storage investment decisions. We also demonstrate that the proposed approach outperforms exact methods in terms of solution quality and computational performance.

In certain electricity markets, because of non-convexities that arise from their operating characteristics, generators that follow the independent system operator's (ISO's) decisions may fail to recover their cost through sales of energy at locational marginal prices. Discriminatory side payments are made by the ISO to incentivize the compliance of generators. Convex hull pricing is a uniform pricing scheme that minimizes these side payments. The Lagrangian dual problem of the unit commitment problem has been used to determine convex hull prices. However, this approach is computationally expensive. In this paper, we propose a polynomially-solvable primal formulation for the Lagrangian dual problem. This formulation explicitly describes for each generating unit the convex hull of its feasible set and the convex envelope of its cost function. We cast our formulation as a second-order cone program when the cost functions are quadratic, and a linear program when the cost functions are piecewise linear. A 96-period 76-unit transmission-constrained example is solved in less than fifteen seconds on a personal computer.

This paper proposes a methodology to identify and protect vulnerable components of connected gas and electric infrastructures from malicious attacks, and to guarantee a resilient operation by deploying valid corrective actions, while accounting for the interdependence of the gas pipeline network and power transmission network. The proposed mathematical formulation gives rise to to a trilevel optimization problem, where the lower level is a multiperiod economic dispatch of the gas–electric system and seeks operating strategies under available resources and given attack, the middle level distinguishes the most threatening attack on the coupled physical infrastructures, and the upper level provides optimal preventive decisions to reinforce the vulnerable components and increase the system resilience. By reformulating the lower level problem as a mixed integer linear programming, a nested column-and-constraint generation algorithm is developed to solve the min–max–min model. Case studies on two test systems demonstrate the effectiveness and efficiency of the proposed methodology.

Strategic bidding problems in electricity markets are widely studied in power systems, often by formulating complex bi-level optimization problems that are hard to solve. The state-of-the-art approach to solve such problems is to reformulate them as mixed-integer linear programs (MILPs). However, the computational time of such MILP reformulations grows dramatically, once the network size increases, scheduling horizon increases, or randomness is taken into consideration. In this paper, we take a fundamentally different approach and propose effective and customized convex programming tools to solve the strategic bidding problem for producers in nodal electricity markets. Our approach is inspired by the Schmudgen's Positivstellensatz Theorem in semi-algebraic geometry; but then we go through several steps based upon both convex optimization and mixed-integer programming that results in obtaining close to optimal bidding solutions, as evidenced by several numerical case studies, besides having a huge advantage on reducing computation time. While the computation time of the state-of-the-art MILP approach grows exponentially when we increase the scheduling horizon or the number of random scenarios, the computation time of our approach increases rather linearly.

The deepening penetration of renewables in power systems has contributed to the increasing needs for generation scheduling flexibility. Specifically, for short-term operations, flexibility here indicates that sufficient ramp capacities should be reserved to respond to the expected changes in the load and intermittent generation, also covering a certain amount of their uncertainty. To address the growing requirements for flexible ramp capacity, markets for ramp products have been launched in practice such as the ones in California ISO and Midcontinent ISO. Some-times, to guarantee sufficient ramp capacity, expensive fast start units have to be committed in real-time. Moreover, with higher penetration of renewable generation, the flexibility provided by the conventional units might not be enough. Actually, wind power producers are physically capable of offering flexibility, which is sometimes also economically efficient to the entire system. In this paper, we aim to explore the mechanism and possibility of including wind power producers as ramp providers to increase the supply of flexibility. To conduct the anal-yses, a two-stage stochastic real-time unit commitment model considering ramp capacity adequacy is formulated. Case studies indicate that both the system and the wind power producers can benefit if the wind power is allowed to provide flexible ramp products., Submitted to a journal

This paper proposes a necessary condition for power flow insolvability in power distribution systems with distributed generators (DGs). We show that the proposed necessary condition indicates the impending singularity of the Jacobian matrix and the onset of voltage instability. We consider different operation modes of DG inverters, e.g., constant-power and constant-current operations, in the proposed method. A new index based on the presented necessary condition is developed to indicate the distance between the current operating point and the power flow solvability boundary. Compared to existing methods, the operating condition-dependent critical loading factor provided by the proposed condition is less conservative and is closer to the actual power flow solution space boundary. The proposed method only requires the present snapshots of voltage phasors to monitor the power flow insolvability and voltage stability. Hence, it is computationally efficient and suitable to be applied to a power distribution system with volatile DG outputs. The accuracy of the proposed necessary condition and the index is validated by simulations on a distribution test system with different DG penetration levels.

In this paper, the empirical controllability covariance (ECC), which is calculated around the considered operating condition of a power system, is applied to quantify the degree of controllability of system voltages under specific dynamic var source locations. An optimal dynamic var source placement method addressing fault-induced delayed voltage recovery (FIDVR) issues is further formulated as an optimization problem that maximizes the determinant of ECC. The optimization problem is effectively solved by the NOMAD solver, which implements the Mesh Adaptive Direct Search algorithm. The proposed method is tested on an NPCC 140-bus system and the results show that the proposed method with fault specified ECC can solve the FIDVR issue caused by the most severe contingency with fewer dynamic var sources than the Voltage Sensitivity Index (VSI) based method. The proposed method with fault unspecified ECC does not depend on the settings of the contingency and can address more FIDVR issues than VSI method when placing the same number of SVCs under different fault durations. It is also shown that the proposed method can help mitigate voltage collapse., Comment: Accepted by IEEE Transactions on Power Systems

This paper addresses two vital issues which are barely discussed in the literature on robust unit commitment (RUC): 1) how much the potential operational loss could be if the realization of uncertainty is beyond the prescribed uncertainty set; 2) how large the prescribed uncertainty set should be when it is used for RUC decision making. In this regard, a robust risk-constrained unit commitment (RRUC) formulation is proposed to cope with large-scale volatile and uncertain wind generation. Differing from existing RUC formulations, the wind generation uncertainty set in RRUC is adjustable via choosing diverse levels of operational risk. By optimizing the uncertainty set, RRUC can allocate operational flexibility of power systems over spatial and temporal domains optimally, reducing operational cost in a risk-constrained manner. Moreover, since impact of wind generation realization out of the prescribed uncertainty set on operational risk is taken into account, RRUC outperforms RUC in the case of rare events. Three algorithms based on column and constraint generation (C&CG) are derived to solve the RRUC. As the proposed algorithms are quite general, they can also apply to other RUC models to improve their computational efficiency. Simulations on a modified IEEE 118-bus system demonstrate the effectiveness and efficiency of the proposed methodology, arXiv admin note: text overlap with arXiv:1510.03308

It has been shown that the heating, ventilation, and air conditioning (HVAC) systems of commercial buildings can offer ancillary services to power systems without loss of comfort. In this paper, we propose a new control framework for reliable scheduling and provision of frequency reserves by aggregations of commercial buildings. The framework incorporates energy-constrained frequency signals, which are adopted by several transmission system operators for loads and storage devices. We use a hierarchical approach with three levels: (i) reserve capacities are allocated among buildings (e.g., on a daily basis) using techniques from robust optimization, (ii) a robust model predictive controller optimizes the HVAC system consumption typically every 30 minutes, and (iii) a feedback controller adjusts the consumption to provide reserves in real time. We demonstrate how the framework can be used to estimate the reserve capacities in simulations with typical Swiss office buildings and different reserve product characteristics. Our results show that an aggregation of approximately 100 buildings suffices to meet the 5 MW minimum bid size of the Swiss reserve market., 29 pages, 8 figures, 3 tables, submitted to the IEEE Transactions on Power Systems

Optimal Power Flow (OPF) dispatches controllable generation at minimum cost subject to operational constraints on generation and transmission assets. The uncertainty and variability of intermittent renewable generation is challenging current deterministic OPF approaches. Recent formulations of OPF use chance constraints to limit the risk from renewable generation uncertainty, however, these new approaches typically assume the probability distributions which characterize the uncertainty and variability are known exactly. We formulate a Robust Chance Constrained (RCC) OPF that accounts for uncertainty in the parameters of these probability distributions by allowing them to be within an uncertainty set. The RCC OPF is solved using a cutting-plane algorithm that scales to large power systems. We demonstrate the RRC OPF on a modified model of the Bonneville Power Administration network, which includes 2209 buses and 176 controllable generators. Deterministic, chance constrained (CC), and RCC OPF formulations are compared using several metrics including cost of generation, area control error, ramping of controllable generators, and occurrence of transmission line overloads as well as the respective computational performance.

This paper considers future distribution networks featuring inverter-interfaced photovoltaic (PV) systems, and addresses the synthesis of feedback controllers that seek real- and reactive-power inverter setpoints corresponding to AC optimal power flow (OPF) solutions. The objective is to bridge the temporal gap between long-term system optimization and real-time inverter control, and enable seamless PV-owner participation without compromising system efficiency and stability. The design of the controllers is grounded on a dual epsilon-subgradient method, and semidefinite programming relaxations are advocated to bypass the non-convexity of AC OPF formulations. Global convergence of inverter output powers is analytically established for diminishing stepsize rules and strictly convex OPF costs for cases where: i) computational limits dictate asynchronous updates of the controller signals, and ii) inverter reference inputs may be updated at a faster rate than the power-output settling time. Although the focus is on PV systems, the framework naturally accommodates different types of inverter-interfaced energy resources., Comment: Accepted for publication on IEEE Transactions on Power Systems