Optimal economic operation of electric power systems pdf

This effort includes material published from , when, presumably, power system engineers began to take active interest in the economic allocation of generation among available units. The incremental method was first formally derived by Steinberg and Smith in , even though it was recognized as early as The idea is that the next increment in the load should be picked up by the unit with lowest incremental cost. The classic coordination equations were discovered by Kirchmayer and Stagg in These results form the backbone of our present-day economic operation methodology.

Their development of the first coordination-type equations was associated with studies for the system of the then Hydro-Electric Power Commission of Ontario. The formulation recognizes losses but assumes constant head. The solution of the variable head case in is due to Glimn and Kirchmayer. Attempts to improve upon the loss formula models date back to the late s. This coincided with the appearance of the first load flow solution algorithms. As a result a start toward the optimal load flow was made in terms of the pioneering work of Squires in and Carpentier in Detailed reference to the literature will be made in the text.

The developments cited so far find their mathematical background in the classical optimization results employing classical theory of maxima and minima in the static case and variational calculus in the dynamic case. We will briefly discuss this next. In this regard they are no different from other disciplines of engineering and applied science. The strong influence of newly developed optimization and computational tech- niques in the area can be partly judged by inspection of titles of published work over the past few decades.

Important hydro- thermal scheduling and unit commitment procedures are also based on this powerful technique. Among the early results we have those of Bernholtz and Graham The introduction of the Kuhn-Tucker theorem in to the optimiza- tion literature contributed to the advances in formulating problems including inequality constraints. Indeed, the varied powerful nonlinear programming procedures continue to influence developments in the area of optimal operation of power systems.

As is the case with dynamic, linear, and nonlinear programming, the maximum principle continues to be one of the essential tools of opti- mization used in our type of problem.

The developments in adapting functional analysis concepts to optimiza- tion, leading to the emergence of the abstract minimum norm problems, can be traced back to the late s.

Application of these powerful tech- niques to many optimal operation problems dates to the early part of this decade. The need continues to exist for reliable, fast, and efficient numerical algorithms for digital implementations.

Indeed, the improved con- vergence characteristics of the Newton-Raphson method over the Gauss- Seidel method prompted the adoption of many load flow algorithms. Advances in sparsity oriented procedures continue to improve both speed and cost of implementation. Many investigations into the relative merits of newly reported numerical algorithms continue to be reported in the literature.

The first part represents the back- ground preparation phase and includes Chapters 2 and 3. It presents us, with the tools necessary to clearly define optimal operation problems in terms of modeling and mathematical prerequisites. Part I1 comprises four chapters and provides us with statements, formulations, solutions, and im- plementation procedures for specific problems in optimal operation of power systems.

The format of the chapters in this part is essentially the same. What we shall do now is to review the six following chapters. Chapter 2 provides a general setting for the problems dealt with in this book. Here we discuss modeling of power systems, devices, and associated phenomena from an optimal operation point of view.

The organization of the chapter follows closely the pattern of the recognized subsystems involved. We first present the energy source models for both thermal and hydro plant performance. Emphasis here is on the diversity of models in common use for our type of problem. The next section deals with electric network components modeling.

We outline the basis of generator, transmission line, and transformer models. Our presentation is very brief since detailed analy- ses can be found in many of the available power systems textbooks.

This section concludes with a treatment of the important subject of load mod- eling. Activities in this area can be divided into two categories: short-term load forecasting and modeling of load response. We give a concise summary of advances made in the sister area of transient stability. Load modeling has been gaining more attention recently in the optimal operation area for the potential savings associated with resulting accurate models. The next section discusses electric network models.

We begin with the active power balance models including the active loss formula. We give a simple derivation which serves to motivate the following discussion of the active-reactive power balance. The more rigorous model given by the exact load flow equations also called power flow and some approximations are treated at the end of this section. This may in certain systems be of equal im- portance. We deal with river flow modeling and in this case present a state space approach due to Dahlin and then outline the transport delay approach.

This section concludes with a discussion of aspects of reservoir inflow fore- casting. The material in this section finds application in Chapter 6.

The last topic discussed in Chapter 2 concerns objective functionals for optimal operation. Our format includes a concluding section wherein com- ments and reference to the literature are made. Chapter 3 considers the mathematical basis for studies and results to follow. The first important topic reviewed is certain linear algebraic con- cepts.

Here concepts of vectors, matrices, partitioning, and quadratic forms are briefly reviewed. Our purpose is simply to provide essential tools used in the development. Consequently, nothing is proved in the entire chapter. The second major topic is that of static optimization results. Pertinent conclusions for optimizing unconstrained, equality constrained, and in- equality constrained problems are treated.

The next section describes the dynamic programming approach and the principle of optimality. As is the case through- out the chapter we only cite results which provide a background for material in the work of the second part.

We devote the following section to the functional analytic optimization technique. Here we begin by reviewing the rudiments of functional analysis and follow that with a statement of a minimum norm problem and its solution. This is important because this result is used in Chapters 5 , 6 , and 7 to specify optimality conditions for a number of optimal operation problems. With the above coverage of results from optimization theory, we turn our attention to the numerical implementation aspects.

We begin with certain important results from linear dynamic system theory and deal with the con- cept of a state transition matrix. This sets the stage for citing an important result pertaining to nonlinear two-point value problem representation. Such a problem arises naturally in problems of interest in our work. The chapter concludes with a section on iterative solution methods for nonlinear systems.

We refrain from presenting convergence results and restrict our treatment to outlining the salient features of each method discussed. Chapter 4 presents problems of all-thermal scheduling. We begin with dispatch using power balance models. In this section problems are ordered so that the complexity increases as we progress through the development.

This is followed by the case including losses where penalty factors are introduced. The funda- mental problem of active-reactive dispatch is treated next. This section concludes with a discussion of dynamic programming procedure for dealing with valve points based on the cost characteristics.

The chapter concludes with an outline of some optimal load flow representative formulations. We include the classic Carpentier-Siroux treatment and the Dommel-Tinney approach. Due to space limitations we do not attempt to give coverage to many excellent contributions to this important problem.

Instead, adequate reference to the literature is made in the final section of the chapter. In Chapter 5 we treat two major problems of hydro-thermal scheduling in which hydro plants are hydraulically isolated. We begin with the case of fixed-head hydro plants. Here the classical dispatch solution is first developed employing variational calculus principles. This is followed by a dynamic programming approach. We then give a functional analytic solution to a problem with predefined models and cite some computational results.

The second major problem in this chapter concerns systems with variable-head plants. As in the earlier case we offer three alternative methods, beginning with the variational approach and followed this time by a maximum princi- ple approach. Finally a minimum norm solution is offered. In Chapter 6 our interest is focused on hydro-thermal scheduling of systems with hydraulic coupling.

We begin with a system with one set of hydro plants on the same stream. With the detailed river flow model estab- lished in Chapter 2, we give a maximum principle approach to this funda- mental problem. We then use a transport delay approximation and deal with a minimum norm procedure for optimizing the system. The following section generalizes the treatment to systems with multichains of hydro plants. Implementation aspects are discussed and numerical results highlight the performance of some proposed iterative algorithms.

In Chapter 7 we treat two problems in optimal load flow in hydro-thermal systems. The first problem addresses the case of systems with hydraulically isolated plants. The second includes more of the complicating effects of multichain hydro subsystems with a detailed hydro model recognizing effi- ciency and the more realistic trapezoidal form for the reservoir representa- tion.

The problems in this chapter are solved using the minimum norm formulation. Continuing changes in power system operating conditions require re- evaluation of the methods used. Chapter 8 is devoted to a summary and a brief outline of certain directions for future research needs to develop advanced methods for re- ducing the cost of power system operation while retaining acceptable security and control.

Further and detailed reference lists will be given at the end of each chapter. Hydro-thermal economic scheduling. Part 1. Solution by incremental dynamic programming.

AIEE Trans. Carpentier, J. Chandler, W. Dahlin, E. PAS, No. George, E. Glimn, A. Happ, H. Savulescu, ed. New York. Power Appar. PAS, IEEE Standard definitions of terms for automatic generation control on electric power systems, IEEE Trans. Economy-security functions in power System Operations, Publ. IEEE, Piscataway. Kirchmayer, L. Poivrr Appar. Ricard, J. Ringlee, R. I 9 6 3 Econoniic system operation considering valve throttling losses, Part -Distribution of system loads by the method of dynamic pro- gramming.

Part III, Sasson, A. Some applications of optimization techniques to power systems problems, Proc. IEEE 62, No. Part , Steinberg, M. Thc theory of incremental rates and their practical application to load division, Elcc. Review of load flow calculations methods. Proc, IEEE 62 7. The system may be viewed as made of subsystems each of which involves a number of components. The degree of detail in component and subsystem models varies with the desired accuracy and relevance to the problem considered.

The electric power system also responds to and is affected by phenomena which can best be described as time series. Such a phenomena include power demand varia- tions and water resource availability. Our starting point in this chapter is the subject of modeling the energy source. In our case we deal with thermal plant as well as hydro plant models. This is followed by a treatment of the electric network components and load models. The modeling of the overall network subsystem is discussed next.

This is followed by a treatment of hydro network modeling including river flow dynamics and reservoir inflow forecast methodology. The chapter concludes with a summary of the objectives of optimal economic operation studies.

The most widely used renewable resource for electric power generation is hydro power. The future promises exciting developments with other renew- able sources such as wind power, solar energy, tidal power, etc. The purpose of this section is to briefly outline models for thermal and hydroelectric generation in general use for economic operational purposes. We start with a discussion of modeling the fuel cost variations with the active power generation for thermal generating plants.

This is followed by a treatment of hydro plant performance modeling. The medium of heat energy transfer to the turbines is steam produced in the boiler or nuclear reactor. Combustion turbines burn liquid or gaseous fuels, mostly light distillate oil or natural gas.

No intermediate steps are needed. In hydrocarbon fossil fuel steam units, fuel is burnt and energy is released in the form of heat in the boiler. Steam at high temperature and pressure is produced as a result. The steam is led via the drum to the turbines, where part of the thermal energy is transformed into mechanical form. The steam turbine drives the electric generator alternator.

The exhaust of the turbine is cooled in the condenser and the resulting water is pumped back to the boiler. A detailed study of the plant dynamic modeling is beyond the scope of our treatment. From an economic operational point of view, our interest is in an input-output model. The input in this case is the fuel cost and the output is the active power generation of the unit.

This is essentially an efficiency-type model and we discuss first various factors affecting our modeling efforts. Boiler efficiency depends on losses that may be classified as 1 stack heat losses, including those in the flue gas, moisture in the gas, and incomplete combustion of CO; 2 heat loss in the ash unburnt carbon ; 3 heat in steam for sootblowing; 4 heat in mill rejects for coal fired boilers : and 5 auxiliary power for mill groups, fans, and ash disposal. These losses are determined by variables related to the fuel, air, and feed water inputs to the boiler.

Certain variables are uncontrollable such as air and water temperature, moisture in the air and fuel for coal-fired as well as chemical analysis and calorific value of fuel. The condition of the boiler is an impor- tant factor which depends on the maintenance procedures. Thermal power plant performance and generating costs are also affected by turbo-alternator efficiency, which is not subject to such random changes as boiler efficiency.

This depends on thermal variables related to the Rankine cycle efficiency including temperature and pressure of steam at the turbine stop valve, high pressure, low pressure, cylinder exhaust, condenser, and low pressure and high pressure heaters. The internal condition of the turbine is a major factor for consideration. Prime causes for reduction in turbine internal efficiency include among other things deposition of solids onto blades and blade erosion.

It is thus evident that periodic updating of the model is necessary. Although initially prepared on the basis of input versus main unit output, the input-output relationship must be converted to input versus net plant sendout.

Auxiliary power requirements that depend on unit loading must be accounted for. The total cost of operation includes the fuel cost, cost of labor, supplies, and maintenance. However, no methods are presently available for expressing the latter as a function of the output. Arbitrary methods for determining these costs are used. The most common one is to assume the cost of labor, supplies, and maintenance to be a fixed percentage of the incoming fuel costs.

Let us introduce some terms used in connection with input-output models for thermal plants. For this purpose we refer to Fig. The average heat rate character- istic is obtained by dividing the input by the corresponding output. This is shown in Fig. The discontinuities in the cost curves eminently shown in the incremental heat rate curve of Fig.

These are loading output levels at which a new steam admission valve is being opened. As the valve is gradually lifted, the losses decrease until the valve is completely open.

The shape of the input-output curve in the neighborhood of the valve points is difficult to determine by actual testing. Most utility systems find it satis- factory to represent the input-output characteristic by a smooth curve which can be defined by a polynomial. The cost increases with the unit size.

Such tests may be carried out on an annual basis for each unit and pre- and postmajor overhaul tests on selected units. Heat rate evaluation on the basis of operating records does not involve unit outages and is therefore preferable from an economic point of view.

A third means is the manufacturer guaranteed performance data adjusted to actual oper- ating conditions. Updating performance characteristics through the per- formance correction factors is done at regular intervals.

It is important to note here that ASME's standard testing procedures call for boiler-turbine-generator to be in steady state operation for a number of hours before recording measurements for heat rate evalua- tion. Under day-to-day operating conditions this is never the case and thus operational heat rates deviate somewhat from the measured ones.

It is therefore deemed necessary to perform the regular update mentioned earlier. TABLE 2. Typical heat rate data for sample unit sizes for steam units using coal, oil, and gas as primary sources of energy are given in Table 2. We remark here that comparison of heat rates for units of the same size shows that coal-fired is the least expensive heatwise.

This is followed by oil-fired. Gas-fired plants are the most expensive. Moreover, the cost decreases as unit size increases. For economy operation problems treated in this book the fuel cost curve is modeled as a quadratic in the active power generation.

Hydroelectric installations are classified into two types-conventional and pumped storage. The conventional type is further classified into two classes, storage and run-of-river. During light load periods, water is pumped from the lower to the upper reservoirs using the most economic energy available as surplus from other sources in the system.

During peak load periods, water stored in the upper reservoir is released to generate power, displacing high cost fossil generation. Economic dispatch of systems with pumped-storage hydroelectric power presents a very challenging problem. Water not utilized is spilled over. The power station can be located in the stream or alongside. The latter is commonly referred to as a canal-type power station.

It is clear that in view of the unregulated flow, the generated electric power of a run-of-river plant is not a controllable variable. In periods with low power requirements water can be stored and utilized when the demand is high.

Economy dispatch of systems with storage- type hydro plants is the subject of further discussion in later chapters of this book. Figure 2.

The intake cafries the water directly to low-head turbines or to the pressure conduit in the case of medium- and high-head turbines. The intake is a canal or a concrete passageway.

The pressure conduit carries the water under pressure to the turbines. A closed pressure conduit is termed a penstock. To provide for pressure regulation a surge tank is installed along the penstock. This will prevent excessive pressure rises and drops during sudden load changes. Trash racks are provided at the inlet to the intake or pressure conduit to protect the turbine against floating and other foreign objects.

Water flowing through the hydroturbine runners is taken through draft tubes to the tailrace and tailrace reservoir. The tailrace is used to conduct the water from the draft tube to the tailwater reservoir, which is usually part of the original river at an elevation lower than the upper reservoir. In the reaction-type, water under pressure is only partly converted into velocity before it enters the turbine runner.

The Francis wheel, the Kaplan wheel, and the propeller wheel are the most commonly used turbines of the reaction-type. In the impulse-type turbine, water under pressure is entirely converted into velocity before entering the turbine. The Kaplan turbine is characterized by adjustable rotor and stator blades whereas the Francis turbine has adjustable stator blades.

The output power of hydro-turbo generation is a function of both the net hydraulic head and the rate of water discharge. An alternative form of writing Eq.

Typical performance curves are shown in Fig. The Kaplan turbine shows a superior performance characteristic as compared to the Francis-type. In the former there is a fairly flat efficiency curve over a range of discharge values above and below the discharge for maximum efficiency.

In the Francis turbine, efficiency falls off rather sharply both above and below the point of maximum efficiency. In addition to the model given by Eq. These are based in one way or the other on Eq. The reason for the numerous models is simply the diversity of installation characteristics. Probably the most used model is Glimn-Kirchmayer's, giving the variation of rate of discharge as a bi- quadratic function of effective head and active power generation: 2.

A more generalized form of a model is Hildebrand's, which takes the form 2. All of these various models can be interpreted as a consequence of Taylor expansion for a function of several variables.

This we discuss in the following: F. The tailrace elevation is a function of the discharge as well as spillage. The forebay elevation is a function of the reservoir geometry, natural water inflow, spillage, and water discharge. It is thus necessary to consider reservoir modeling in the case of variable head hydro plants.

These determine the active power generation available from the hydro plants. It should be noted that the evaporation and seepage losses are normally estimated by multiplying the storage by the appropriate coefficient. For simplicity these are assumed to be independent of the storage. Moreover, o t , the rate of water spillage, should be assumed zero unless certain overriding constraints are violated. The variation of storage or capacity of a reservoir of regular shape with the elevation can be computed with the formulas for the volumes of solids.

This for reservoirs on natural sites is determined from the elevation- storage curve. An area-elevation curve is constructed using the planimeter to find the area enclosed within each contour within the reservoir site. The integral of the area-elevation curve is the elevation-storage curve. A typical elevation-storage curve is shown in Fig.

It is important to note that natural factors will change the reservoir configuration over time. An example is sediment accumulation. It is thus important to update the reservoir model periodically. A mathematical model may be obtained by using Taylor's expansion. If we choose the first two terms, this will result in a linear relationship which corresponds to the mathematical artifice of a vertical-sided reservoir model.

A trapezoidal reservoir model results if we include the second-order term in the expansion. For a partially filled hemisphere model a third-order equation results. Of interest in our discussion are the requirements of each component as it affects the operation of the hydroelectric plant. Irrigation, flood control, water supply, stream flow augmentation, navigation, and recreation usages are among the possible purposes of a water resource development.

For these purposes the reservoir is regulated so that full requirements for each element are available under the design drought conditions.

For the former this is determined by the elevation of the spillway crest or the top of the spillway gates. The minimum level may be fixed by the elevation of the lowest outlet in the dam or by conditions of operating efficiency for the turbines. The second set is solely determined by the discharge capacity of the power plant as well as its efficiency.

The last inequality constraint would be imposed by irrigational, navigational, recreational, and flood prevention considerations. This is obtained by assuming constant efficiency, no spillage, invariant tailrace elevation, negligible pen- stock losses, and a vertical-sided reservoir.

Equations 2. In Eq. It is the objective of the present section to outline the fundamental models of the three basic components as well as the load to which the system must respond. The treatment here is for the case of balanced three-phase ac steady state operation, which is the mode of interest in optimal operation studies con- sidered in this work. The degree of detail of a synchronous machine model depends on the area of application in which one is interested. Greater detail is necessary for dynamic power systems studies involving transient analysis such as stability evaluation.

It is universal practice in synchronous machine modeling to apply the Park transformation to the three-phase armature coils. The result is a set of algebraic and differential equations relating to two equivalent orthogonal armature coils whose location is determined by the rotor position. The range of forms based on Park's trans- formation models is tremendous.

The primary source of these models has been power system transient stability analysis. The updating time interval involved in economic operation algorithms and controllers is longer than that for stability studies. Furthermore, the system is assumed to be in a sinusoidal steady state for modeling purposes. The control action in this case is to move the system from a certain steady state level to the next in an optimal way.

Less detailed models are thus considered sufficient for economic operation purposes. The order of magnitude of the problem becomes obvious if we note that the dynamical model of a synchronous machine with amortisseur windings, exciter, and turbine governor can involve fourteen differential equations.

Other controls such as the stabilizer and boiler offer added complexity. The synchronous machine model commonly used in optimal economy operation studies is that of the classical voltage source behind the equivalent characteristic reactance synchronous and armature reaction reactance. This model is a consequence of the equations based on the detailed Park's transformation.

The basic assumptions involved are those of balanced loading and sinusoidal steady state operation. We can thus avail ourselves of the conveniences offered by phasor diagram techniques. The armature resistance is neglected in this presentation since our object is to demonstrate the basis for limitations accounted for in our type of study. The machine terminal voltage V is taken as reference with the armature current I , lagging by an angle 4. A fictitious voltage source E represents the voltage required to be generated by the field current I ,.

The angle 6 is the torque angle. Here j denotes J-1 as usual. The first tells us that the generated power is limited to a certain maximum. This maximum is given by In practice the allowable maximum is about half the above value due to system stability considerations. The active generated power is contqolled mainly through the torque input from the turbine, which is regulated by the governor system.

The reactive power generated as given by Eq. Loading of the generator is normally carried out according to its capabil- ity curve. The curve is frequently called the reactive capability curve since it shows the active power loading versus the reactive power loading and the limits of output for the rating and at various hydrogen gas pressures.

Constant power factor lines are also indicated on this curve. A typical capability curve is shown in Fig. In region I, field heating dictates the limits and hence the curve in this region is a constant field current locus. Armature heating is the limiting factor in region 11; the curve in this region is a constant apparent power MVA locus.

Region I11 of the curve is limited by end heating and stability. These parameters are the series resistance and inductance, shunt capacitance, and shunt conductance. The line resistance and inductive reactance are of importance in almost all problems. For some studies it is possible to omit the shunt capacitance and conductance and thus simplify the equivalent circuit considerably.

The determination of the parameters on the basis of line length, conductors used, and conductor spacing as they are mounted on the supporting structure has been the subject of extensive studies. For our purpose it is assumed that the parameters are available. It is further assumed that the system is operating in the sinusoidal steady state. The result is two second-order differential equations relating the voltage and current variations along the line.

The subscripts s and r stand for sending and receiving end variables, respectively. An equivalent circuit favored by power system engineers is the n circuit. It is on the basis of Eq. Usually no more than three terms are required. The result, obtained rigorously, can be obtained by the intuitive assumption that the lines series impedance is lumped and the shunt admittance Y is divided equally with each half placed at an end of the line.

A final model is the short line up to 80 km model and in this case the shunt admittance is neglected altogether. The line is thus represented only by its series impedance, as shown in Fig. The need for high power transmission capacity and reduced trans- mission losses requires high transmission voltages. This is made possible by the use of power transformers. As is the case with the synchronous machine and transmission line, the modeling of the power transformer has been the focus of attention of many investigators.

For our purposes the most basic form of a transformer model is the equivalent T circuit shown in Fig. The left and right series impedances are essentially those of the primary and secondary winding. I Req, t "I ye1 Fig. I admittance relates to the magnetic circuit path exciting current. If one is interested in obtaining a n representation, then the use of the Y-A trans- formation will facilitate that. However, this is not normally done since for most transformers operated at power frequencies 50 or 60 Hz , the exciting current will be small.

Consequently the shunt branch can be moved to either end of the circuit, and the L circuit shown in Fig. In the figures, subscript 1 denotes primary while 2 denotes sec- ondary quantities. The prime refers to a quantity viewed or referred to another side.

Thus V ; is the secondary voltage referred to primary. The resistance is denoted by R, and reactance is X. Conductance is symbolized by gc while susceptance is symbolized by b,. For most applications, the shunt branch may be removed altogether.

Find the optimal schedule neglecting losses, when the demand is MW. Determine optimal generation schedule neglecting losses. Determine optimal schedule neglecting losses. It also has a minimum limit set by stable boiler operation. The economic dispatch problem now is to schedule generation to minimize cost, subject to the equality constraint. As soon as a plant reaches the limit maximum or minimum its output is fixed at that point and is maintained a constant.

The other plants are operated at equal incremental costs. Obtain the optimal schedule if the load varies from 50 — MW. Solution: The incremental fuel costs of the two plants are evaluated at their lower limits and upper limits of generation.

Initially, additional load is taken up by unit 2, till such time its incremental fuel cost becomes equal to 2. Beyond this, the two units are operated with equal incremental fuel costs. Further loads are taken up by unit 1. Fig 8. For any particular load, the schedule for each unit for economic dispatch can be obtained. Example 4. In example 3, what is the saving in fuel cost for the economic schedule compared to the case where the load is shared equally.

The load is MW. There is an increase in cost of unit 1 since PG1 increases and decrease in cost of unit 2 since PG2 decreases. The assumption here is that the load varies linearly between maximum and minimum values. Experiences with large systems has shown that the loss of accuracy is not significant if this approximation is used. An average set of loss coefficients may be used over the complete daily cycle in the coordination of incremental production costs and incremental transmission losses.

An incremental change in load of 4 MW requires generation to be increased by 6 MW. Some features of the site may not work correctly.

DOI: El-Hawary and Gustav S. El-Hawary , G. Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Figures, Tables, and Topics from this paper. IBM Power Systems. Citation Type. Has PDF.