Handbook of Power Systems II

This article discusses recent advances in mathematical modeling techniques of transmission networks and control devices within the scope of optimal power flow (OPF) implementations. Emphasis is on the newly proposed concept of representing meshed power networks using an extended conic quadratic (ECQ) model and its amenability to solution by using interior-point codes. Modeling of both classical power control devices and modern unified power flow controller (UPFC) technology is described in relation to the ECQ network format. Applications of OPF including economic dispatching, loss minimization, constrained power flow solutions, and transfer capability computation are presented. Numerical examples that can serve as testing benchmarks for future software developments are reported on a sample test network.

This article presents an approach for solving some power systems problems by using optimal dynamic flow problems. The classical optimal flow problems on networks are extended and generalized for the cases of nonlinear cost functions on arcs, multicommodity flows, and time- and flow-dependent transactions on arcs of the network. All parameters of networks are assumed to be dependent on time. The algorithms for solving such kind of problems are developed by using special dynamic programming techniques based on the time-expanded network method together with classical optimization methods.

The wide spread use of power electronic equipment causes serious current harmonics in electrical power system. Harmonic currents that flow in the electrical power system would cause extra copper loss and immature operation of over current protection devices. Voltage distortion due to harmonic voltage drop in the electrical power distribution system impairs the operation of voltage-sensitive equipment. To improve the electrical power quality and reduce energy wastage in the electrical power distribution system, especially under the deregulated environment, the nature of the harmonics must be identified so that the causes and effects of the harmonics would be studied. Moreover, corrective measures cannot be easily implemented without knowing the characteristics of the harmonics existing in the electrical power system. The chapter presents investigation results obtained from the harmonics analysis and modeling of load forecast and consists of three main sections. The first presents an algorithm based on continuous wavelet transform (CWT) to identify harmonics in a power signal. The new algorithm is able to identify the frequencies, amplitudes, and phase information of all distortion harmonic components, including integer harmonics, sub-harmonics, and inter-harmonics. The second section describes a wavelet transform-based algorithm for reconstructing the harmonic waveforms from the complex CWT coefficients. This is useful for identifying the amplitude variations of the nonstationary harmonics over the estimation period. The third section presents wavelet-genetic algorithm-neural network-based hybrid model for accurate prediction of short-term load forecast. Examples and case study will be used to demonstrate the benefits derived from these approaches.

The presence of optionality in the generation and transmission of power means that valuing physical and financial assets requires using option theory, which in turn requires studying stochastic processes appropriate for the description of power prices. Power prices are much more volatile than other commodity prices and exhibit interesting behavior such as regime switching between normal and spiked states. The probability distributions underlying such stochastic process provide an input for price forecasts, which are based on price history. They also provide an input into valuation of transmission and transmission options in cases where the implied market-based measures of volatilities and correlations are lacking. Combining information from this analysis of stochastic processes for power prices with the Black–Scholes framework for option valuation, specifically using that framework to calculate the value of spread options, yields methods for calculating the value of transmission as well as for calculating the value of financial transmission options, which also depend on spread of power prices. The three main techniques for obtaining the option value include analytical approaches, binomial-type trees (finite difference methods), and Monte Carlo simulations. Each of these techniques presented in the paper has its own advantages and disadvantages and is complementary to the other two, providing independent validation and quality control for transmission valuation algorithms.

In this paper, the three main forecasting topics that are currently getting the most attention in electric power systems are addressed: load, wind power and electricity prices. Each of these time series exhibits its own stylized features and is therefore forecasted in a very different manner. The complete set of forecasting models and techniques included in this revision constitute a guided tour in power systems forecasting.

The purpose of electricity price forecasting is to estimate future electricity prices, particularly locational marginal prices (LMP), with consideration to both security and capacity constraints in a grid environment. Electricity price forecasting is vital to both market participants and market operators in wholesale electricity markets. Electricity price forecasts are used to assist the decision making of market participants on bidding submissions, asset allocations, bilateral trades, transmission and distribution planning, and generation construction locations. Electricity price forecasts are also used by market operators to uncover possible market power. The inaccuracy of electricity price forecasting is due to problems associated with volatility of prices, interpretability of explanatory variables, and underlying impacts of power grid security. This study classifies forecasting techniques common in the literature based on their objective, concept, time horizon, input–output specification, and level of accuracy. Thus the state-of-the-art of electricity price forecasting is described in this study. This survey facilitates the validation, comparison, and improvements of specific or combined methods of price forecasting in competitive electricity markets. Moreover, this study demonstrates a hybrid forecasting system, which combines fuzzy inference system and least-squares estimation. The proposed mechanism is applied to the day-ahead electricity price forecasting of an actual security-constrained, wholesale electricity market. This hybrid forecasting system provides both accuracy and transparency to electricity price forecasts. The forecasting information is also interpretable with respect to the fuzzy representations of selected inputs.

This chapter models long-term electricity forward prices with variables that influence the price of electricity. Long-term modelling requires consideration of expected changes in the demand and supply structure. The model combines high-resolution information on fuel costs from financial markets and low-resolution information on the demand/supply structure of the electricity market. We model the latter using consumption and supply capacity and the former with forward prices of fuels, emission allowances and imported electricity. The model is estimated using data from the Nordic electricity market and global long-term forward prices of energy. Owing to a lack of data on consumption and supply capacity, the estimated results provide only the broad influence of these variables on forward prices. Though extrapolation of the prices observed in Nord Pool may suffer from the influence of short-term variables, such as precipitation and temperature, the model yields robust forecasts of the prices of contracts that are not exchange traded.

“Top-down” models, based on observation of market price patterns, may be used to forecast prices in competitive electricity markets, once a reasonable track record is available and provided the market structure is stable. But many studies relate to potential changes in market structure, while prices in hydro-dominated markets are driven by inflow fluctuations and reservoir management strategies, operating over such a long timescale that an adequate track record may not be available for decades, by which time the system itself will be very different. “Bottom-up” analysis can readily model structural change and hydro variation, but must make assumptions about fundamental system data, commercial drivers, and rational optimizing behavior that leave significant unexplained price volatility. Here we describe a technique for fitting a hybrid model, in which a “top-down” approach is used to estimate parameters for a simplified “bottom-up” model of participant behavior, from market data, along with a stochastic process describing residual price volatility. This fitted model is then used to simulate market behavior as fundamental parameters vary. We briefly survey actual and potential applications in other markets, with differing characteristics, but mainly illustrate the application of this hybrid approach to the hydro-dominated New Zealand Electricity Market, where participant behavior can be largely explained by fitted “marginal water value curves.” A second application of a hybrid model, to the Australian National Electricity Market, is also provided.

This paper sheds light on a promising and very active research area for electricity market modeling, that is, agent-based computational economics. The intriguing perspective of such research methodology is to succeed in tackling the complexity of the electricity market structure, thus the fast-growing literature appeared in the last decade on this field. This paper aims to present the state-of-the-art in this field by studying the evolution and by characterizing the heterogeneity of the research issues, of the modeling assumptions and of the computational techniques adopted by the several research publications reviewed.

Within a yearly time framework this chapter describes stochastic programming models to derive the electricity market strategies of producers and retailers. Both a financial futures market and a day-ahead pool are considered. Uncertainties on hourly pool prices and on end-user demands are represented modeling these factors as stochastic processes. Decisions pertaining to the futures market are made at monthly/quarterly intervals while decisions involving the pool are made throughout the year. Risk on profit variability is modeled through the CVaR. The resulting decision-making problems are formulated and characterized as large-scale mixed-integer linear programming problems, which can be solved using commercially available software.

This chapter presents a comprehensive decision support system for addressing the generation planning and operation. It is hierarchically divided into three planning horizons: long, medium, and short term. This functional hierarchy requires that decisions taken by the upper level model will be internalized by the lower level model. With this approach, the position of the company is globally optimized. This set of models presented is specially suited for hydrothermal systems. The models described correspond to long-term stochastic market planning, medium-term stochastic hydrothermal coordination, medium-term stochastic hydro simulation, and short-term unit commitment and bidding. In the chapter it is provided a condensed description of each model formulation and their main characteristics regarding modeling detail of each subsystem. The mathematical methods used by these models are mixed complementarity problem, multistage stochastic linear programming, Monte Carlo simulation, and multistage stochastic mixed integer programming. The algorithms used to solve them are Benders decomposition for mixed complementarity problems, stochastic dual dynamic programming, and Benders decomposition for SMIP problems.

Benders’ partitioning method is a classical method for solving mixed integer programming problems. The basic idea is to partition the problem by dividing the variables into complicating variables, normally the variables that are constrained to be integer valued, and easy variables, normally the continuous variables. By fixing the complicating variables, a convex sub-problem is generated. Solving this convex sub-problem and its dual generates cutting planes that are used to create a master problem, which when solved generates new values for the complicating variables, which have a potential to give a better solution. In this work, we assume that the optimal solution is given, and we present a way in which the partitioning idea can be used to generate a valid inequality that supports the optimal solution. By adding some of the continuous variables to the complicating variables, we generate a valid inequality, which is a supporting hyperplane to the convex hull of the mixed integer program. The relaxed original programming problem with this supporting valid inequality added will produce interpretable prices for the original mixed integer programming problem. The method developed can be used to generate economically interpretable prices for markets with non-convexities. This is an important issue in many of the deregulated electricity markets in which the non-convexities comes from large start up costs or block bids. The generated prices are, in the case when the sub-problem generated by fixing the integer variables to their optimal values has the integrality property, also supported by nonlinear price functions that are the basis for integer programming duality.

Medium-term generation planning may be advantageously modeled through market equilibrium representation. There exist several methods to define and solve this kind of equilibrium. We focus on a particular technique based on conjectural variations. It is built on the idea that the equilibrium is equivalent to the solution of a quadratic minimization problem. We also show that this technique is suitable for complex system representation, including stochastic risk factors (i.e., hydro inflows) and network effects. We also elaborate on the use of the computed results for short-term operation.

We present a multi-stage decision model, which serves as a building block for solving various electricity portfolio management problems. The basic setup consists of a portfolio optimization model for a large energy consumer, which has to decide about its mid-term electricity portfolio composition. The given stochastic demand may be fulfilled by buying energy on the spot or futures market, by signing a supply contract, or by producing electricity in a small plant. We formulate the problem in a dynamic risk management-based stochastic optimization framework, whose flexibility allows for extensive parameter studies and comparative analysis of different types of supply contracts. A number of application examples is presented to outline the possibilities of using the basic multi-stage stochastic programming model to address a range of issues related to the design of optimal policies. Apart from the question of an optimal energy policy mix for a large energy consumer, we investigate the pricing problem for flexible supply contracts from the perspective of an energy trader, demonstrating the wide applicability of the framework.

We present recent developments in the field of stochastic programming with regard to application in power management. In particular, we discuss issues of scenario tree modeling, that is, appropriate discrete approximations of the underlying stochastic parameters. Moreover, we suggest risk avoidance strategies via the incorporation of so-called polyhedral risk functionals into stochastic programs. This approach, motivated through tractability of the resulting problems, is a constructive framework providing particular flexibility with respect to the dynamic aspects of risk.

The reliability of power systems is a major concern for network designers and operators. For several reasons, the level of electricity distribution network risk, in the UK as elsewhere, is perceived to be increasing in the short term, and further sources of increased risk can be anticipated in the longer term. An asset management approach seeks to optimise the balance between capital expenditure and the risk to customer supplies. To do this effectively, network risk must first be defined and measured. This chapter describes ways of modelling, evaluating and comparing the levels of risk in different parts of the network and under various operating assumptions. It also enables different strategies for mitigating that risk to be quantified, including maintenance policies, replacement and reinforcement construction projects, network reconfiguration and changes to operating practices. In the longer term, it allows evaluation of the impact on network risk of initiatives such as widespread distributed generation, demand side management and smart grids.