Magnus Perninge

A Stochastic Optimal Power Flow Problem With Stability Constraints—Part I: Approximating the Stability Boundary

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.

A Stochastic Optimal Power Flow Problem With Stability Constraints—Part II: The Optimization Problem

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.

On the Validity of Local Approximations of the Power System Loadability Surface

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 makes the loadability surface nonsmooth. This makes the use of local approximations limited and calls for a method for estimating the distance to a nonsmooth part of the surface. This paper suggests a method for calculating the distance from a point on the loadability surface to the closest point of nonsmoothness of the loadability surface.

Risk Estimation of Critical Time to Voltage Instability Induced by Saddle-Node Bifurcation

Prevention of voltage instability in electric power systems is an important objective that the system operators have to meet. Under certain circumstances the operating point of the power system may start drifting towards the set of voltage unstable operating points. If no preventive measures are taken, after some time the operating point may eventually become voltage unstable. It will thus be preferable to have a measure of the risk of voltage collapse in future loading states. This paper presents a novel method for estimation of the probability distribution of the time to voltage instability for a power system with uncertain future loading scenarios. The method uses a distance from the predicted load-path to the set of voltage unstable operating points when finding an estimate of the time to voltage instability. This will reduce the problem to a one-dimensional problem which for large systems decreases the computation time significantly.

Importance Sampling of Injected Powers for Electric Power System Security Analysis

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.

Load modeling using the Ornstein-Uhlenbeck process

In this paper we show how to model the load in an electric power system using the Ornstein-Uhlenbeck process and use the method developed by Lehmann to find the distribution of the maximum of the load process in a bounded time interval. A numerical example showing how to find an upper confidence bound for the maximum of the maximum of the load process in a bounded time interval using the proposed method will also be given.

A Stochastic Control Approach to Manage Operational Risk in Power Systems

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. This will lead to more flexible operating limits, giving a more adequate balance between risk and economic benefit of transmission. The method assumes a set of observable system variables such as transfers through specific transmission corridors, system frequency, or distance to a bifurcation surface. Then impulse control is applied to find an optimal strategy for activation of tertiary reserves, based on the values of the observables.

Irreversible investments with delayed reaction: an application to generation re-dispatch in power system operation

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.

Closure of “applying stochastic optimal power flow to power systems with large amounts of wind power and detailed stability limits”

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.

The value of using chance-constrained optimal power flows for generation re-dispatch under uncertainty with detailed security constraints

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 optimal generation 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.