DIMACS/EPRI Workshop on Next Generation of Unit Commitment Models

September 27 - 28, 1999

Title: What does the Problem Look Like to an Independent System Operator?

Abstract:

**Fred N. Lee & Art Breipohl**, University of Oklahoma and **Walter Hobbs**, Power Costs, Inc.

Title: Unit Commitment in a Competitive Environment

Abstract: Several years ago, there were all kinds of speculations about what would happen to the "unit commitment" function in a competitive energy market. As the market evolves, the role of "unit commitment" has become clearer. Assets are considered by major market participants as a necessity in a successful trading portfolio. As a result, "unit commitment" optimization has become a very essential function in portfolio optimization. In a competitive energy market, "unit commitment" optimization needs to be expanded to encompass markets and contracts of different products in order to fully capture the impact of assets as an arbitrage tool. This paper will illustrate the effectiveness of such an expansion. The objectives and modeling philosophy will also be discussed in this paper.

Title: Current Experience in Solving Unit Commitment Problems

Abstract:

**Joseph Graves**, PHB Hagler Bailly, University of Oklahoma

Title: Simultaneous Clearing and Optimization of Energy and Ancillary: Service Markets in the Unit Commitment Context - Is Today's MIP: Solution Technology Up to the Task?

Abstract: The notion is that by clearing the markets in a simultaneous optimization, the lowest cost solution can be obtained. Obviously, there are multiple obstacles to be overcome. These include the following. Can the bid formats and market price definitions be structured so that market participants are indifferent between what services they are designated to provide? Can the ability/inability of particular sources to provide different products be adequately represented in the MIP formulation? Can the resulting MIP problem be solved to the optimal solution in the limited timeframes needed for market operations given today's solution algorithms and computing technology?

**Ross Baldick**, University of Texas

Title: Alternative Formulations, Multiplicity of Solutions, and Unpriced Products

In this paper I discuss alternative formulations for the unit commitment problem, considering ways in which the formulation might better fit the behavior of market participants. Alternative formulations may help to resolve the problem of there being many solutions with very similar objective value. One aspect of this is that there are various ``unpriced products'' in the energy marketplace that could be used to distinguish bids and also help to resolve the multiplicity of solutions.

**Robert Bixby**, Rice University and ILOG

Title: MIP: Closing the gap between theory and practice

Abstract: A mixed integer programming (MIP) problem is a linear programming problem in which some or all of the variables are restricted to take integral values.

The MIP modeling paradigm is extremely powerful, finding wide application to real-world planning and scheduling problems. However, as is well known, MIP is NP-hard, and the obvious question is, can we solve MIP problems in practice? Equally important, can we do it without writing special-purpose algorithms for each individual application?

Two ingredients have come together to make the answer to the above questions yes with greater frequency than ever before. First, there have been completely unexpected and quite remarkable improvements in the linear-algebra engine for MIP, the dual simplex method for linear programming. Second, as anyone familiar with the theory of combinatorial optimiziation and integer programming knows, a large backlog of important theoretical and computational advances have accumulated over the last decade, and more. These developments have created a gap between theory and practice. That gap is now closing.

**Sebastian Ceria**, Columbia University and Dash Optimization, Inc.

Title: New Developments in Integer Programming

Abstract: The last fifteen years have been witness to a revolution in the efficient solution of mixed-integer programming problems, with large application to Electrical Power Generation.

The early research from the 70's in branch-and-bound methods that gave birth to the first commercial codes has served as the foundation stone for a new generation of faster and better integer programming solvers. The integer programming software packages available to practitioners today incorporate the research developments from the last fifteen years, which combined with far superior linear programming performance translate into orders of magnitude improvements in solution times.

But the need from practitioners still far exceeds the capabilities of general purpose integer programming solvers. This demand for increased functionality and performance is promoting a new generation of "intelligent" tools that allow the seamless integration of modeling engines and solution algorithms.

We will illustrate how this approach has been successfully applied to solving unit commitment models more effectively.

** Alvaro Baillo, Mariano Ventosa, Michel Rivier, Andres Ramos**, IIT of Universidad Pontificia Comillas

Title: Unit Commitment and Bidding Strategies for Generation Companies in Deregulated Electricity Markets

Abstract: The electricity industry is in the midst of a profound restructuring process in an increasing number of countries. These major changes are intended to bring about competition in some of the electricity business activities so as to promote a higher level of efficiency in the provision of electric services.

Generation companies have traditionally been subject to regulatory policies which guaranteed the full recovery of their costs. In the new framework the generation of electricity is a deregulated activity and firms have to compete to sell the electric energy produced by their facilities. Therefore generation companies are now fully responsible not only for the efficient operation of their units but also for the effective bidding of their output. A higher degree of uncertainty and risk arouses the need for new tools devoted to the maximization of the expected firm's profit in the new competitive environment.

On the one hand generation companies' profits depend on their revenue which is determined by the equilibrium reached in the wholesale energy market. An adequate modeling of this market and the development of sophisticated forecasting techniques are required to estimate the expected evolution of electricity prices and evaluate the effect of different strategies. On the other hand, it is essential that the operation of the generating units is carried out at the least possible cost.

In this paper a strategic unit commitment model which deals with both aspects of the new generation activity will be presented. The type of marketplace considered is a PoolCo system based in a day-ahead market. The model is formulated as a MIP optimization problem. The objective function to be maximized is the firm's total profit for the scope of the model. The cost evaluation techniques and the technical constraints that grant a feasible schedule, usual in traditional unit commitment models, are taken into account. Additionally, new market modeling equations intended to express the relationship between the firm's energy output and the price of energy are incorporated. Both the competitors' and the demand's behavior are modeled by means of an hourly function, namely the hourly residual demand curve, which determines the amount of energy the firm is able to sell at each price. Due to the structural non-linearity of the firm's revenue function an original linearization procedure will be outlined so that the problem can be solved with linear programming algorithms. A detailed mathematical formulation of the model will be included. The strategic unit commitment will be tested in a numerical example.

**Alberto Borghetti**, University of Bologna

Title: Demand-side bidding in unit commitment in a competitive electricity market

Abstract: This presentation focuses on a generalisation of the competitive power pool framework to include demand-side bidding. We cast the competitive power pool into the unit commitment problem formulation framework, in which the supply cost functions are replaced by the bids submitted by the suppliers. To improve competition, customers are allowed to play a proactive role in the price determination process by submitting bids of load reduction in specific periods.

We study the behaviour of demand-side bidding by simulating electricity auctions using a unit commitment tool that we developed. This tool is capable of incorporating the specific characteristics of demand side bidding using the Lagrangian relaxation solution methodology. We present a case study on a small system for a 24-hour auction. In this case study we allow both demand-side bidding and the restoration of the demand due to the inclusion of such bids.

The presentation is based on the work carried out in collaboration, between the University of Illinois at Urbana-Champaign (prof. G. Gross) and the University of Bologna (prof. C. A. Nucci).

**Xiaohong Guan**, Harvard University

Title: Integrated Power Generation Scheduling and Bidding in the Deregulated Electric Power Market

Abstract: The electric power industry worldwide is experiencing unprecedented restructuring. The core of the restructuring is deregulation of the industry and introduction of competition among power suppliers and demanders. Many challenging issues arise under the new competitive market structure. Instead of centralized decision-making in a monopoly environment as in the past, many parties with different goals are now involved and competing in the market. The information available to a party may be limited, regulated, and received with time delay, and decisions made by one party may influence the decision space and well-being of others. These difficulties are compounded by the underlying uncertainties inherent in the system such as the demand for electricity, fuel prices, outages of generators and transmission lines, tactics by certain market participants, etc. Consequently the market is full of uncertainty and risk. The recent experience learned from the California market has shown that MCPs are volatile and often out of bidders' expectation. Key questions to be addressed include how to predict load and market clearing prices, how to consider other parties' decisions in deciding one's own bids, and how to manage uncertainty and risk. Since finding an optimal solution to a traditional unit commitment problem is NP-hard even without considering multiple parties and uncertainties, it may be more practical to know which decision is good with confidence rather than looking for an optimal solution.

For an energy supplier, bidding decisions are coupled with generation resource scheduling or unit commitment since generators' characteristics and how they will be used to meet the accepted bids in the future have to be considered before bids are submitted. For example, if starting up a thermal unit is expected, the associated startup cost should somehow be configured in the bid curves. The decisions, however, can be quite subtle since generally startup costs are not part of a bid. Bidding decisions should therefore be carefully made by considering the anticipated MCP, system demand, generation and startup costs, and competitor's decisions. What further complicates the issue is that some of the information is not available, or will be available but with significant delays

In this talk, a systematic bid selection method based on ordinal optimization and integrated with generation scheduling or unit commitment is introduced to obtain "good enough bidding" strategies for generation suppliers. Ordinal optimization provides a way to obtain reasonable solution with much less effort. The ordinal optimization method was developed by Prof. Y. C. Ho to solve complicated optimization problems possibly with or without uncertainties. Ordinal optimization is based on the following two tenets: (1 it is much easier to determine "order" than "value." To determine whether A is better or worse than B is a simpler task than to determine how much better is A than B (i.e., the value of A-B) especially when uncertainties exist. (2 In stead of asking the "best for sure," we seek the "good enough with high probability." This goal softening makes the optimization problem much easier.

To apply ordinal optimization framework to integrated generation scheduling and bidding, major efforts are needed to build power market simulation models and to devise a strategy to construct a small but good enough select set S. The basic idea of our approach is to use a rough model that describes the influence of bidding strategies on the MCP. This model can be replaced by any model with more accuracy if needed. A nominal bid curve is obtained by solving optimal power generation for a given set of the MCPs with the Lagrangian relaxation framework. Then N bids are generated by perturbing the nominal bid curve. The ordinal optimization method is applied to form a good enough bid set S, which contains some good bids with high probability, by performing rough evaluation. The best bid is then searched and selected over S by solving generation scheduling or unit commitment problems within reasonable computational time. Numerical testing results based on historical MCPs in California market and a generation company with 10 units shown that the ordinal optimization based method is efficient and good bidding strategy is obtained.

**Atif Debs and Charles Hansen**, Decision Systems International;
**Yu-Chi Wu**, National Lien-Ho Institute of Technology & Commerce;
**Miao-Li**, Taiwan, R.O.C.

Title: Development of a Simulation Study Environment for the Electric Energy Market

Abstract: The paper reports on efforts to develop a full realistic simulation study environment (SSE) for the electric energy market. The expected users of the resulting system are:

- Generating Companies (Genco=EDs)
- Regional Transmission Organizations (RTO=EDs) and Independent System Operators (ISO=EDs)
- Energy Traders, Power Exchanges (PX=EDs) and Energy Service Companies= (ESCo=EDs)
- Regulators

(a) A forward market model module which predicts behavior of those markets of interest to the SP. This market model is probabilistic and may encapsulate within it an Auction Market Model (AMM).

(b) An optimization module which is comprised of two major coordinated modules: (1) A Unit Commitment Module, based on EPRI=EDs DYNAMICS and (2) A Security-Constrained Optimal Power Flow (SCOPF) based on DSI=EDs OPF (which uses a direct Interior-Point Method algorithm). For the OPF case, external systems are represented by uncertain network and operational models to establish a form of realism. The results of the optimization module provide the best forward optimal solution, inclusive of possible congestion charges and unit commitment schedules. In the hour-ahead and real-time markets, this module is also useful as will be described in the paper.

(c) An assessment module which provides the SP with the most probable solution based on SP=EDs requirements. For a Genco, this would be the required price bids for the forward and real-time markets. For an RTO/ISO it would be a prediction of market behavior with assessments of needed res ources to insure system reliability and security. For the traders and ESCo =EDs it would provide with positioning strategies as well as bid evaluatior capabilities.

(d) The Overall Market Model (OMM) is modeled as a collection of SP=EDs all serving the same interconnected power system. In order to model the behavior of this market, the decisions of the SP=EDs are sent to an overall Power System Simulator (PSM) which models power system behavior at the following levels of time response

- (a) Automatic Generation Control (AGC) level, 2-4 sec.

- (b) On-line secure operation, 15-30 min

- (c) Resource scheduling (1-168 hours)

In order to realize the SSE, basic industry standards for inter-operability and plug-compatibility are used, including the Control Center Common Information Model (CCAPI) and the Inter-Control Center Communication Protocol (ICCP), both of EPRI. Because of the intensity of some computations, the design of the hardware/software platforms will require careful considerations. The paper will address this issue as well.

**Karla Hoffman**, George Mason University

Title: AUCTIONS: A new application for combinatorial optimization

Abstract: Combinatorial optimization can be used to assist buyers in a variety of auction settings. We present three different instances of its use in auctions where one wishes to buy a collection of entities that are available either as one large group (in the case of bidding on a large federal contracts), or as bids for individual items when the value is enhanced if more than a single item is won (as was the case in the FCC PCS auctions). In each case, optimization helped the user to better understand the economics of the underlying enterprise and provided fast alternative strategies with measures of the effectiveness of each strategy considered.

**Carlos E. Murillo-Sanchez, Robert J. Thomas**, School of Electrical Engg., Cornell University

Title: Thermal unit commitment with a nonlinear AC power flow network model

Abstract: A formulation of the thermal unit commitment problem that includes nonlinear power flow constraints is presented, thus allowing a more accurate representation of the network than is possible with DC-flow models. This also permits potential VAr production to be used as a criterion for commitment of otherwise expensive generators in strategic locations. A Lagrangian relaxation framework with duplicated variables for each active and reactive source is used, permitting the exploitation of the separation structure of the dual cost. Results for medium systems are in hand. Results from implementing the algorithm in a parallel processing environment will also be presented at the workshop.

**Andras Prekopa**, RUTCOR, Rutgers University

Title: Optimal short term scheduling of power generation by thermal power plants including network constraints

Abstract: A large-scale, nonlinear, mixed-variable mathematical programming model is formulated and solved for the short-term power generation scheduling problem with thermal power plants. An important feature of the model is that we solve the unit commitment problem in such a way that the network conditions are also taken into account. The latters; produce quadratic nonlinearly in the model constraints rendering the problem nonconvex in the continuous variables.

It is assumed that the power demand for the short term is known with a satisfactory precision (simultaneous research in this respect produced a forecasting methodology that works with less than 1 % error).

Time is subdivided into periods, which may be hourly or even shorter ones. In each power plant a number of modes of operation are defined which mean the joint operation or stand still state of groups of generators. One mode of operation allows for the choice of a production level in a given interval.

The variables of the model belong to three groups: 0-1 production mode variables, continuous production variables and voltage variables. The constraints can be grouped as follows: power balance constraints, constraints that connect mode of operation and production level variables, voltage and reactive power constraints (corresponding to some designated nodes), transmission line capacity constraints, period coupling constraints, generator up and down constraints and fuel constraints. The objective function has three parts: power generation costs, generator restart costs and the costs of power losses on transmission lines.

The problem is solved by a combination of sequential linearization, Benders' decomposition and heuristics.

The method has been applied to the Hungarian power system (20 thermal power plants and cca 8,000 MW generation capacity). In this model there are 23 hourly and 4 half-hourly periods. The solution time takes less than 1 minute on a 200 N1Hz computer.

**Jinghai Xu and Richard D. Christie**, University of Washington

Title: Decentralized Unit Commitment in Competitive Energy Markets

Abstract: In a competitive energy market, instead of, or in addition to, a centralized unit commitment, individual generation owners will make independent unit commitment decisions. They will seek to maximize their profits against the predicted market clearing price. Their unit commitment strategy will be expressed in their bids, so that they shut down or start up when market price tells them to. In this paper, a unit commitment based price taking (UCPT) bidding strategy with a simple price prediction mechanism is developed and explored using a market simulator. Simulation results show that an individual generator has higher profits with UCPT bidding than with simple price taking bidding, and that the cost of supplying price-inelastic loads achieved by the market is lower when all generators use UCPT bidding. It appears that UCPT bidding gives results similar to those from a Lagrange relaxation unit commitment (LRUC), without a fix-up step, and has problems with convergence and feasibility similar to LRUC. Cyclic behavior in market prices with UCPT bidding is observed, and is shown to depend on the price prediction mechanism. Alternative price prediction mechanisms can reduce cyclic behavior. Potential strategic behavior and market power arising from unit commitment constraints is conceptually explored.

**A. Conejo, A. Motto, F.D. Galiana**, McGill University

Title: Decentralized Nodal Prices Pool Scheduling

Abstract: This talk presents a scheme for the scheduling of generation in a power pool context. The model includes network considerations such as losses and transmission congestion in addition to the usual unit commitment 0/1 constraints. The innovative feature of this algorithm (DNA =3D decentralized nodal prices algorithm) is its decentralized nature which allows each individual participant (generator, consumer and network) to self-schedule in response to a sequence of trial nodal prices. Each individual response is based on maximizing its own profit. It is felt that this scheduling independence is important in today's competitive markets in view of the lack of general acceptance of the centralized authority of the ISO (independent system operator). An iterative scheme has been developed to update the nodal prices in response to bus power imbalances that is shown to converge to the optimal solution. The price-updating scheme does not use confidential data but rather information available to each and every participant. The DNA algorithm is simple to implement and interpret, it is highly decentralized thus equitable and less subject to manipulation, and it is optimum and therefore economically efficient.

**Rajesh Rajaraman, Fernando L. Alvarado, Laurence Kirsch, C. Clark**, University of Wisconsin

Title: Reserve Services and Transmission: Responding to Energy and Reserve Prices

Abstract: The key concept of this paper is a description of the self-committment problem in the present of inter-product substitution options (energy versus reserves sales), taking into consideration inter-temporal effects. That is, presnt losses can lead to future profits. The paper considers the impact of price uncertainty in such decisions. The generator models consider minimum and maximum output levels, ramping rate limits, minimum up and down times, incremental energy costs and startup and shutdown costs. The paper describes how one can find the most profitable market-responsive unit committment and dispatch taking into account reserves and inter-temporal constraints. The model can also be used to develop power market bids for energy and reserve services.

**Marceline Madrigal and Victor H. Quintana**, University of Waterloo

Title: An Interior-point/Cutting-plane Method to Solve the Dual Unit Commitment Problem

Abstract: An interior-point/cutting-plane (IP/CP) method for non-differentiable optimization is used to solve the dual to a unit commitment (UC) problem. The IP/CP method has two advantages over previous approaches, such as the sub-gradient and bundle methods: first, it proved to have convergence characteristics in an actual implementation; and second, does not suffer from the parameter-tuning drawback. The results of a performance testing using systems with up to 104 units confirm the superiority of the IP/CP method over previous approaches to solve the dual UC problem. Issues that have influenced the use or not the use of UC models as the auctioning mechanism in electricity markets are revised and discussed. Such issues are duality gap, stopping criteria, cost recovery and the existence of multiple solutions.

**Subir Sen and D.P. Kothari**, Centre for Energy studies, Indian Institute of Technology

Title: Large Scale Thermal Generating Unit Commitment: A Novel Approach

Abstract: One of the most important problems in operational scheduling of electrical power generation is the unit commitment problem. The primary concern of electrical power system operators is to guarantee adequate generation capacity to meet demand continuously. The limited amount of hydro-electric energy stored in the dams and the system reservoirs may not prove to be sufficient to respond to large variation in demands. Therefore, costly thermal generating units are often used to make up for the supply shortage. The scope of the operation scheduling problem will vary largely from utility to utility depending on their mix of units and particular operating constraints. A modern power system consists of large number of thermal generating units. Each unit is of large capacity, normally 200/500 MW. In solving the unit commitment problem of such a large system, the main cause of difficulty is the involvement of large number of units for commitment. The problem can not be solved easily if all units are involved in the search for the optimal solution, since computational facilities could be exhausted. Several approaches were suggested to reduce the requirement for the computational resources to an appropriate limit either by classifying the units into different categories with only the cycling units being included in the search for the lowest cost, or by implementing decomposition technique. Although these methods achieved a remarkable reduction in the computation requirement, further improvements in the quality of the solution is still desired. Very few methods like heuristic, Lagrangian relaxation technique etc. have been developed to facilitate the solution of the unit commitment problem of large power systems. However, no optimal solution to the problem has yet been obtained.

Objective of the paper:

The main objective of this paper is to develop a new efficient solution approach for solving the unit commitment schedule of thermal generating units of a realistic large scale power system. In a very large system having large number of units, all the units do not have different input-output/fuel cost characteristics individually. In fact, many units in the system have same capacities and similar characteristics. Therefore, units having similar characteristics can be grouped together.

The proposed approach of unit commitment problem solution exploits this concept. The approach is based on equivalencing like concept of network reduction which reduces the number of units in the large power system to the lowest possible number according to their fuel/generation cost characteristics. The reduced system is solved using modified dynamic programming technique (a new approach) and from that overall solution to the original unit commitment problem of the entire system is obtaine.

Equivalencing like concept in unit commitment:

The equivalencing like method for solving the unit commitment problem consider the particular reference to units generation cost(input-output) criterion. The units in the large system are decomposed into various groups according to their fuel cost characteristics so that identical units form one group. Each group is represented by one sample representative unit. Consequently, an equivalent system of units consisting only of representative unit is generated. The basic concept of this method is that large size power system is represented by an equivalent system with fewer number of generating units such that the unit commitment problem of the equivalent system would be easier to handle and solve than, that of the original large size system. When the solution of the unit commitment problem of the equivalent system is obtained, then the solution to the original unit commitment problem can be determined accordingly by uncrunching the equivalent problem solution.

Results and Discussion:

To demonstrate the effectiveness of the proposed model, two large systems comprising 26 and 79 units(Eastern Regional grid of Indian Power System) along with particular daily load demand profile have been considered. In the equivalent system representation, 8 units represent the complete 26 unit system and 19 units represent the 79 unit system. Therefore, it can be seen that the size of the problem has been reduced to great extent. The comparison of the proposed method with other traditional techniques like Lagrangian relaxation, Truncated dynamic programming(DP-TC) etc. have been carried out.

The results show that the proposed equivalencing like concept of the unit commitment solution is able to provide very close to accurate solution , i.e., error within 0.05% and 0.19% respectively for 26 unit systems compared to modified dynamic programming(MDP) and Lagrangian relaxation approach respectively. For 79 unit systems, cost obtained using proposed method is 0.7% more compared to DP-TC based method while it is 1.27% less compared to Lagrangian relaxation based method. Moreover, the solution time in the proposed approach is still less compared to other techniques. This is one of the main requirements of solution of unit commitment problem of large size power system. Conclusions:

The proposed method of unit commitment simplifies unit commitment problem in terms of dimensionality of the problem. As a result, the proposed equivalencing like concept of solving unit commitment problem for large scale system turns out to be a promising approach. The results obtained are also quite comparable with other traditional techniques. It offers good performance.

**C. Richter and G. Sheble**, Iowa State University

Title: Customized GA-UC schedules for the competitive environment

Abstract: Far from being an artifact of the past, the unit commitment (UC) aglorithm is essential to making economical decisions in today's competitive electicity industry. Increases in competition, decreases in the obligations-to-serve, and enhanced futures, forwards, and spot market trading in electricity and other related markets makes the decision of which units to operate more complex than ever before. Simularities in the decentralized auction markets currently being implemented in places like Spain, with UC solutions should encourage people to continue working on finding better and faster solution techniques. UC schedules may be developed for a generation company, a system operator, etc.

The need for many flavors of UC algorithms each considering different inputs and objective functions is growing. Factors such as historical availability of units should be a factor in deciding the UC. Although a particular schedule may result in the lowest cost, or highest profit, it may be dependent on generators that have varying historical availabilities.

In the past, most consumers had very reliable electricity whether they needed it or not. Given a choice in a market-based electricity system, many consumers would choose to pay for a slightly lower level of power availability if it would result in sufficient savings.

As the number of inputs and options grows, the genetic algorithm (GA) becomes an important tool in searching the large solution space. GA solution times-to-solution often scale up linearly with the number of units, or hours being considered. Another benefit of using the GA to generate UC schedules is that an entire population of schedules is developed. In this talk, the authors present the UC-GA, and report on enhancements that use post-processing to compare UC schedules with each other based on their expected monetary value under likely scenarios or contingencies.

**Ruediger Schultz**, Gerhard-Mercator University

Title: A Stochastic Programming Model for Unit Commitment Under Uncertainty in a Hydro-Thermal Power System

Abstract: We develop a two-stage stochastic programming model with integer requirements for solving the unit commitment problem in power generation in the presence of uncertainty of load profiles. The solution methodology rests on a novel scenario decomposition method for stochastic integer programming. This method combines Lagrangian relaxation of non-anticipativity constraints with branch-and-bound.

It can be seen as a decomposition algorithm for large-scale mixed-integer linear programs with block-angular structure. With realistic data from a German utility we validate our model and carry out test runs. Problem sizes go up to 20.000 integer and 150.000 continuous variables together with up to 180.000 constraints.

**Chung-Li Tseng**, University of Maryland

Title: A Stochastic Model for the Price-Based Unit Commitment Problem

Abstract: With deregulation of the electricity industry a global trend, utilities and power generators must adjust to the new risks of volatile spot prices in the competitive marketplace. Because they are no longer able to rely on embedded cost recovery regulation, these organizations must fundamentally change the way they view power plant operation. For example, the deterministic and cost-based unit commitment (UC) problem that schedules power plants to satisfy demand should be replaced by an optimization problem which, at least, is price-based and takes account of price stochastics. Solving such an optimization problem not only yields the optimal commitment decision, but also reveals the power plant's value over the operating period.

In this paper, the price-based UC is formulated as a multi-stage stochastic problem. The uncertainties characterized are the prices for electricity and the fuel consumed by the generator. Assuming that there are hourly markets for both electricity and fuel and their prices follow some Ito processes, these two price processes are approximated by a price scenario tree. Over a short-term period, the operator must determine a schedule of what units will be used so as to maximize expected profit while satisfying the forecast demand and the operating constraints. In this model, the demand may also be price elastic.

The proposed price-based UC model is solved using a unit decommitment (UD) approach. The UD approach has been proposed to solve various types of deterministic UC problems. It starts with a solution having all available units on-line at all hours in the planning horizon and determines an optimal strategy for decommitting units, one at a time. The UD method has also been reported to be an efficient and reliable approximate method for the LR approach for solving the deterministic UC problem. In this paper, a UD-type method, composed of stochastic programming recursion and economic dispatch, is employed to solve the price-based UC problem. The issue of generating the price scenario tree is discussed. Numerical results will also be presented.

**Samer Takriti**, Enron Corporation

Title: The Stochastic Unit Commitment Problem

Abstract: We present an extension to the unit commitment problem which incorporates uncertainty in electric load, electricity price, fuel price, and fuel constraints. Uncertainty is modeled using a set of scenarios that reflect varying market demand and prices. We solve the resulting stochastic program using progressive hedging, stochastic dynamic programming, Benders decomposition, and Lagrangian relaxation. The resulting solution can be further refined using branch-and-bound. Future extensions, such as incorporating water price, will also be presented.

**Phillip Yoon and Marija Ilic**, MIT

Title: Industry Structure-Dependent Uncertainties in the Unit Commitment Models

Abstract: As the electricity industry goes through the restructuring process, there is an urgent need for examining the current formulation of unit commitment (UC) models. The static and deterministic UC models are only sub-optimal given the uncertainty in supply and demand of electricity due to unbundling of services. The new formulation of UC models must be dynamic and stochastic with properly identified uncertainties arising from various sources.

We first recognized that the industry is composed of three major markets: energy, capacity and transmission. Within each market there is further specialization to incorporate temporal and functional separations. For example, energy market encompasses long-term futures (usually through bilateral contracts), short-term futures (day-ahead and hour-ahead) and spot (real-time) markets. Capacity market includes regulation, spinning reserves, supplemental reserves and backup supply. Transmission market contains long-term firm, long-term non-firm, and short-term non-firm.

If UC models are formulated at individual supplier/distributed level, we need to understand the uncertainty arising from interactions among the markets described above. One of the ways to represent the dynamics of this uncertainty is through constructing price projection models for each market and relating those models to one another through correlation functions.

From various widely available models, we consider the physical characteristics to choose price projection process that best fit each market. Once suitable models are selected, it is a matter of reasonable parameter estimation and optimization based on an appropriate objective function. For individual supplier profit maximizing is a sensible choice. While it is not always possible to solve UC models due to computational complexity, under certain assumptions a solution can be derived. It is worthwhile noting that charges arising from projected prices in transmission markets enter as another uncertain input to UC models while overall system load is assumed to be known.

In optimization of solving UC model at centralized level, an appropriate objective function may be either minimizing cost or minimizing price given physical constraints of individual supplier. The physical constraints include minimum up-time, and minimum down-time. For given physical constraints there is a cost associated with it, i.e. start-up cost and shut-down cost. It is important to recognize that depending on market structure, supplier bids may or may not include the cost information. In case bids do not contain cost information, minimizing price is an appropriate objective function. Otherwise, minimizing cost should be used. The only uncertainty entering the optimization process is the load in this formulation.