With larger penetrations of wind power, the uncertainty increases in power systems operations. The wind power forecast errors must be accounted for by adapting existing operating tools or designing new ones. A switch from the deterministic framework used today to a probabilistic one has been advocated. This two-part paper presents a framework for risk-basedoperations of power systems. This framework builds on the operating risk defined as the probability of the system to be outside the stable operation domain, given probabilistic forecasts for the uncertainty, load and wind power generation levels. This operating risk can be seen as a probabilistic formulation of the N - 1 criterion. In Part I, the definition of the operating risk and a method to estimate it were presented. A new way of modeling the uncertain wind power injections was presented. In Part II of the paper, the method's accuracy and computational requirements are assessed for both models. It is shown that the new model for wind power introduced in Part I significantly decreases the computation time of the method, which allows for the use of later and more accurate forecasts. The method developed in this paper is able to tackle the two challenges associated with risk-based real-time operations: accurately estimating very low operating risks and doing so in a very limited amount of time.

With larger penetrations of wind power, the uncertainty increases in power systems operations. The wind power forecast errors must be accounted for by adapting existing operating tools or designing new ones. A switch from the deterministic framework used today to a probabilistic one has been advocated. This two-part paper presents a framework for risk-basedoperations of power systems. This framework builds on the operating risk defined as the probability of the system to be outside the stable operation domain, given probabilistic forecasts for the uncertainty (load and wind power generation levels) and outage rates of chosen elements of the system (generators and transmission lines). This operating risk can be seen as a probabilistic formulation of the N - 1 criterion. The stable operation domain is defined by voltage-stability limits, small-signal stability limits, thermal stability limits and other operating limits. In Part I of the paper, a previous method for estimating the operating risk is extended by using a new model for the joint distribution of the uncertainty. This new model allows for a decrease in computation time of the method, which allows for the use of later and more up-to-date forecasts. In Part II, the accuracy and the computation requirements of the method using this new model will be analyzed and compared to the previously used model for the uncertainty. The method developed in this paper is able to tackle the two challenges associated with risk-based real-time operations: accurately estimating very low operating risks and doing so in a very limited amount of time.

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only line flows have been used as security constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. In this paper we aim to extend the stochastic optimal power flow problem to include constraints for voltage stability as well as small-signal stability. This is done by approximating the voltage stability and small-signal stability constraint boundaries with second-order approximations in parameter space. Then we refine methods from mathematical finance to be able to estimate the probability of violating the constraints. In this first part of the paper, we derive second-order approximations of stability boundaries in parameter space. In the second part, the approximations will be used to solve a stochastic optimal power flow problem.

Larger amounts of variable renewable energy sources bring about larger amounts of uncertainty in the form of forecast errors. When taking operational and planning decisions under uncertainty, a trade-off between risk and costs must be made. Today's deterministic operational tools, such as N-1-based methods, cannot directly account for the underlying risk due to uncertainties. Instead, several definitions of operating risks, which are probabilistic indicators, have been proposed in the literature. Estimating these risks require estimating very low probabilities of violations of operating constraints. Crude Monte-Carlo simulations are very computationally demanding for estimating very low probabilities. In this paper, an importance sampling technique from mathematical finance is adapted to estimate very low operating risks in power systems given probabilistic forecasts for the wind power and the load. Case studies in the IEEE 39 and 118 bus systems show a decrease in computational demand of two to three orders of magnitude.

Increasing wind power penetration levels bring about new challenges for power systems operation and planning, because wind power forecast errors increase the uncertainty faced by the different actors. One specific problem is generation re-dispatch during the operation period, a problem in which the system operator seeks the cheapest way of re-dispatching generators while maintaining an acceptable level of system security. Stochastic optimal power flows are re-dispatch algorithms which account for the uncertainty in the optimization problem itself. In this article, an existing stochastic optimal power flow (SOPF) formulation is extended to include the case of non-Gaussian distributed forecast errors. This is an important case when considering wind power, since it has been shown that wind power forecast errors are in general not normally distributed. Approximations are necessary for solving this SOPF formulation. The method is illustrated in a small power system in which the accuracy of these approximations is also assessed for different probability distributions of the load and wind power.

Uncertainties faced by operators of power systems are expected to increase with increasing amounts of wind power. This paper presents a method to design efficient importance sampling estimators to estimate the operating risk by Monte-Carlo simulations given the joint probability distribution describing the wind power and load forecasts. The operating risk is defined as the probability of violating stability and / or operating constraints. The method relies on an exisiting framework for rare-event simulations but takes into account the peculiarities of power systems. In case studies, it is shown that the number of Monte-Carlo runs needed to achieve a certain accuracy on the estimator can be reduced by up to three orders of magnitude.

Summary form only given. Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only limits on line flows have been used as stability constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. In this article we aim to extend the stochastic optimal power flow problem to include constraints for voltage stability as well as small-signal stability. This is done by approximating the voltage stability and small-signal stability constraint boundaries with second order approximations in parameter space. Then we refine methods from mathematical finance to be able to estimate the probability of violating the constraints. In the first part of the article, we derive second-order approximations of stability boundaries in parameter space. In the second part of the article, we look at how Cornish-Fisher expansion combined with a method of excluding sets that are counted twice, can be used to estimate the probability of violating the stability constraints. We then show in a numerical example how this leads to an efficient solution method for the stochastic optimal power flow problem.

The uncertainty faced in the operation of power systems increases as larger amounts of intermittent sources, such as wind and solar power, are being installed. Traditionally, an optimalgeneration re-dispatch is obtained by solving security-constrained optimal power flows (SCOPF). The resulting system operation is then optimal for given values of the uncertain parameters. New methods have been developed to consider the uncertainty directly in the generation re-dispatch optimization problem. Chance-constrained optimal power flows (CCOPF) are such methods. In this paper, SCOPF and CCOPF are compared and the benefits of using CCOPF for power systems operation under uncertainty are discussed. The discussion is illustrated by a case study in the IEEE 39 bus system, in which the generation re-dispatch obtained by CCOPF is shown to always be cheaper than that obtained by SCOPF.

In this work we show that one can solve a set of multi-modes optimal switching problems with a particular, very natural form of signed costs. The problems are a special type of impulse control problems where the control set is discrete and the system is operated continuously over a finite horizon.

In this work we show that one can solve a finite horizon non-Markovian impulse control problem with control dependent dynamics. This dynamic satisfies certain functional Lipschitz conditions and is path dependent in such a way that the resulting trajectory becomes a flow.

Optimal Power Flow is a central tool for power system operation and planning. Given the substantial rise in intermittent power and shorter time windows in electricity markets, there’s a need for fast and efficient solutions to the Optimal Power Flow problem. With this in consideration, this paper propose an unsupervised deep learning approach to approximate the optimal solution of Optimal Power Flow problems. Once trained, deep learning models benefit from being several orders of magnitude faster during inference compared to conventional non-linear solvers.

We propose a neural network using an unsupervised learning strategy for direct computation of closest saddle- node bifurcations, eliminating the need for labeled training data. Our method not only estimates the worst-case load increase scenarios but also significantly reduces the computational complexity traditionally associated with this task during inference time. Simulation results validate the effectiveness and real-time applicability of our approach, demonstrating its potential as a robust tool for modern power system analysis.

We formulate the day-ahead bidding problem for a hydropower producer having several hydropower plants residing in a river basin. We present a novel approach inspired by Dynamic programming with approximations in value and policy space by neural networks. This allows for more accurate modeling of the problem by avoiding linear approximations of the production function and bidding. Stochastic programming is a method frequently used in literature to solve the hydropower production planning problem. Stochastic programming is used on linearized systems and under assumptions of known distributions of the involved stochastic processes. We test the proposed algorithm on a simplified system, suitable for Stochastic Programming and compare the obtained policy with the results from Stochastic Programming. The results show that the algorithm obtains a policy similar to that of Stochastic Programming.

We consider the problem faced by a hydropower plant owner who sells electricity on the spot and balancing power markets. We allow for risk sensitivity and consider the setting of a single power plant connected to a reservoir with a stochastic inflow. Our results indicate that risk-aversion tends to shift trading towards the less risky day-ahead market.

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision maker is risk aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic differenceequations. An example application to hydropower planning is provided

In the day-to-day operation of a power system, the system operator repeatedly solves short-termgeneration planning problems. When formulating these problems theoperators have to weigh the riskof costly failures against increased production costs. The resulting problems are often high-dimensionaland various approximations have been suggested in the literature.In this article we formulate the short-term planning problem as an optimal switching problem withdelayed reaction. Furthermore, we proposed a control variable technique that can be used in MonteCarlo regression to obtain a computationally efficient numerical algorithm.

We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.

We consider a type of optimal switching problems with non-uniform execution delays and ramping. Such problems frequently occur in the operation of economical and engineering systems. We first provide a solution to the problem by applying a probabilistic method. The main contribution is, however, a scheme for approximating the optimal control by limiting the information in the state-feedback. In a numerical example the approximation routine gives a considerable computational performance enhancement when compared to a conventional algorithm.

We consider a type of optimal switching problems with non-uniform execution delays and ramping.Such problems frequently occur in the operation of economical andengineering systems. We first pro-vide a solution to the problem by applying a probabilistic method. The main contribution is, however,a scheme for approximating the optimal control by limiting the information in the state-feedback.In a numerical example the approximation routine gives a considerable computational performanceenhancement, when compared to a conventional algorithm.

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

We consider stochastic optimal switching problems with reaction delays and propose an approximation technique that decreases the computational complexity. In a numerical example the approximation routine gives a considerable computational performance enhancement when compared to a conventional algorithm.

We consider the problem where an operator of n production units acts by turning off and on the units to track a stochastic demand. We investigate the situation when the running cost depends on the time that each unit has been in operation, covering, for example, the case when there are delays in the control response. In this setting, standard methods for numerical solution quickly become intractable as $n$ increases. To resolve this we propose a numerical scheme where delay states are discretized with a different step size than that of the time line. We first show that the value function converges to the optimum as the step size goes to zero. We then show, by a counter-example, that the corresponding control does not necessarily converge to an optimal control for the original problem and propose a perturbation that resolves this issue.

Power system voltage security assessment is generally applied by considering the power system loadability surface. For a large power system, the loadability surface is a complicated hyper-surface in parameter space, and local approximations are a necessity for any analysis. Unfortunately, inequality constraints due to for example generator overexitation limiters, and higher codimension bifurcations, make the loadability surface non-smooth. One situation that is particularly difficult to handle is when a saddle-node bifurcation surface intersects a switching loadability limit surface. In this article we intend to investigate how several local approximations can be combined to obtain an adequate approximation of the loadability surface near such intersections.

Lately, much work in the area of voltage stability assessment has been focused on finding post-contingency corrective controls. In this article a contribution to this area will be presented where we investigate the surface of maximal loadability while allowing for post-contingency corrective controls. This objective is different from the usual, where the aim is to include the post-contingency controls in a security-constrained optimal power flow. Our aim is rather to find approximations of the post-contingency stability boundary, in pre-contingencyparameter space, while including the possibility for post-contingency corrective controls. These approximations can then be used in, for example, a chance-constrained optimal power flow routine.

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional Lp-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.

We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).

We consider an optimal switching problem with random lag and possibility of component failure. The random lag is modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the switching mode. The possibility of failures is modeled by having absorbing components. We show existence of an optimal control for the problem by applying a probabilistic technique based on the concept of Snell envelopes.

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

We consider a system of finite horizon, sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions to our system of RBSDEs by applying a Picard iteration approach. Uniqueness then follows by relating the limit to an auxiliary impulse control problem. Moreover, we show that the solution to our system of RBSDEs is connected to weak solutions of a stochastic differential game where one player implements an impulse control while the opponent plays a continuous control that enters the drift term. As all our arguments are probabilistic and hence hold in a non-markovian framework, we are able to consider the setting where the underlying uncertainty in the game stems from an impulsively and continuously controlled path-dependent stochastic differential equation driven by Brownian motion.

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only steady-state variable limits have been used as security constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. Recently an extension of the stochastic optimal power flow formulation that included constraints for voltage stability as well as small-signal stability was proposed. This was done by approximating the voltage stability and small-signal stability constraint boundaries with second order approximations in parameter space. In this article an alternative solution method to this problem will be proposed. The new improved solution method, which is based on Edgeworth series expansions, is both more efficient and accurate. We also give details on convexity of the problem and discuss some computational issues.

We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of "backward semigroups" we first prove the dynamic programming principle (DPP) for a truncated version of the problem in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Our approach avoids technical constraints imposed in previous works dealing with the same problem and, more importantly, allows us to consider impulse costs that depend on the present value of the state process in addition to unbounded coefficients. Using the dynamic programming principle we deduce that the upper and lower value functions are both solutions (in viscosity sense) to the same Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of solutions to this partial differential inequality we conclude that the game has a value.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

In this paper we show how to build an economically optimal feedback control strategy for the re-dispatch of electricity generation. We assume that the operator steers production in a set of controllable power plants by altering the active power set-point of each generator, within a set of predefined set-points. Theoperators strategy will be based on balancing the operating cost against the expected cost from unserved demand. Important aspects are ramp-rates at which production can approach the set-point, and switching costs arising from increased fuel consumption during ramping and wear and tear on production facilities.

In power systems, the system frequency is a good indicator of the networks resilience to major disturbances. In many deregulated markets, eg the Nordic power market, the system operator controls the system frequency manually by calling off bids handed in to a market, called the regulating market. In this paper, we formulate the problem of optimal bid call-off on the regulating market that the system operator is faced with each operating period, as an optimal starting problem with delays. As general optimal starting problems with delays are computationally cumbersome, we present two alternative approximation schemes. First, we make simplifications to the problem that renders classical solution concepts tractable; then, in a second approach, we define a suboptimal solution scheme, based on limiting the feedback information.

The system frequency of a power systems is a good indicator of the networks resilience to major disturbances. The frequency control is generally a multi-layered control structure with primary, secondary and tertiary control. In a completely deregulated setting, for example in the Nordic power system, the system operator controls the system frequency manually by calling-off bids handed in to a market, called the regulating market. In this paper we formulate the problem of optimal bid call-off on the regulating market, that the system operator is faced with each operating period, as a multi-period optimal switching problem with execution delays. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

The system frequency of a power systems is a good indicator of the networks resilience to major disturbances. In a completely deregulated setting, for example in the Nordic power system, the system operator controls the system frequency manually by calling-off bids handed in to a market, called the regulating market.

In this paper we formulate the problem of optimal bid call-off on the regulating market, that the system operator is faced with each operating period, as an optimal switching problem with execution delays.

As general optimal switching problems with execution delays are computationally cumbersome we resort to a recently developed suboptimal solution scheme, based on limiting the feedback information in the control loop.

In power systems the system frequency is a good indicator of the networks resilience to major disturbances. In many deregulated markets, e.g. the Nordic power market, the system operator controls the system frequency manually by calling off bids handed in to a market, called the balancing power market.

In this paper we consider the problem of optimal bid call-off on the balancing market, that the system operator is faced with each operating period. We formulate the problem as a stochastic optimal control problem of impulse type.

When searching for numerical solutions a complicating factor is the structure of the balancing power market, where the overall marginal price applies to all bids. To retain numerical tractability we propose computationally efficient upper and lower bounds for the value function in the dynamic programming algorithm.

Power system voltage security assessment is generally applied by considering the steady-state stability surface. However, as seen in the literature, random perturbations can drive the system away from stable operation, long before the steady-state stability surface is reached.

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only limits on line flows have been used as stability constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation. In this paper we aim to extend the stochastic optimal power flow problem to include constraints for voltage stability as well as small-signal stability. This is done by approximating the voltage stability and small-signal stability constraint surfaces with second-order approximations in parameter space. Then we refine methods from mathematical finance to be able to estimate the probability of violating the constraints. In this, the second part of the paper, we look at how Cornish-Fisher expansion combined with a method of excluding sets that are counted twice, can be used to estimate the probability of violating the stability constraints. We then show in a numerical example how this leads to an efficient solution method for the stochastic optimal power flow problem.

Operating criteria for power systems, such as the (N - 1)-criterion, are often based on evaluating whether the system is vulnerable to a specific set of contingencies. Therefore, a major part of power system security is concerned with establishing regions in parameter space where the system is vulnerable to specific contingencies. In this article we exploit the possibility of using Monte Carlo simulations to build an approximation of the region, in parameter space, where the power system will remain stable following a given contingency.

Power system security analysis is often strongly tied with contingency analysis. To improve Monte Carlo simulation, many different contingency selection techniques have been proposed in the literature. However, with the introduction of more variable generation sources such as wind power and due to fast changing loads, power system security analysis will also have to incorporate sudden changes in injected powers that are not due to generation outages. In this paper, we use importance sampling for injected-power simulation to estimate the probability of system failure given a power system grid state. A comparison to standard crude Monte Carlo simulation is also performed in a numerical example and indicates a major increase in simulation efficiency when using the importance sampling technique proposed in the paper.

In this paper, the novel method operational risk managing optimal power flow (ORMOPF), for minimizing the expected cost of power system operation, is proposed. In contrast to previous research in the area, the proposed method does not use a security criterion. Instead the expected cost of operation includes expected costs of system failures.

Adequate security margins are commonly applied in power systems by keeping predefined transfer limitsthrough certain transmission corridors in the system. These limits are often set to keep the N-1 criterionstating that the system should remain stable after the loss of any component. For many stability criteria suchas, voltage stability, and voltage limits at specific nodes, the distribution of the injected power amongst thenodes of the system will be of vital importance. To incorporate this into the analysis of transfer limits theuncertainties in nodal loading and wind power production will have to be considered. In this article wepropose a new method for generating samples of the power at all nodes given a set of transfers throughspecified corridors of the power system. It is then shown how the method can be used to evaluate the risk ofviolating the system stability limits induced by choosing a specific set of transfer limits. The method can beused in power system operations planning when setting the limits for trading and transfer between thedifferent nodes of the power system.

In this article we consider how the operator of an electric power system should activate bids on the regulating power market in order to minimize the expected operation cost. Important characteristics of the problem are reaction times of actors on the regulating market and ramp-rates for production changes in power plants. Neglecting these will in general lead to major underestimation of the operation cost. Including reaction times and ramp-rates leads to an impulse control problem with delayed reaction. Two numerical schemes to solve this problem are proposed. The first scheme is based on the least-squares Monte Carlo method developed by Longstaff and Schwartz (Rev Financ Stud 14:113–148, 2001). The second scheme which turns out to be more efficient when solving problems with delays, is based on the regression Monte Carlo method developed by Tsitsiklis and van Roy (IEEE Trans Autom Control 44(10):1840–1851, 1999) and (IEEE Trans Neural Netw 12(4):694–703, 2001). The main contribution of the article is the idea of using stochastic control to find an optimal strategy for power system operation and the numerical solution schemes proposed to solve impulse control problems with delayed reaction.

In this article we investigate how to optimally activate regulating bids to handle bottlenecks in power system operation. This will lead to an optimal stopping problem, and activation of a regulating bid is to be performed when the transfer through a specific system bottleneck reaches a certain value. Compared to previous research in the area the work presented in this article includes a more detailed model of the structure of the regulating market, and reaction times of actors on the regulating market is taken into consideration. The emphasis of the presentation will be application to a two area test system. The method is compared to Monte Carlo simulation in a numerical example. The example shows a promising result for the suggested method.