AN INVESTIGATION OF THE USE OF OPTIMIZATION TECHNIQUES IN THE OVERALL EFFICIENCY ANALYSIS OF HYDROPOWER PLANTS

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1 AN INVESTIGATION OF THE USE OF OPTIMIZATION TECHNIQUES IN THE OVERALL EFFICIENCY ANALYSIS OF HYDROPOWER PLANTS E.W.Hirano Federal University of Santa Catarina Department of Mechanical Engineering E. Bazzo Federal University of Santa Catarina Department of Mechanical Engineering Abstract. This work proposes the optimization of unit commitment centered on the overall efficiency curve of the hydropower plants. The analysis of the overall efficiency of hydropower plants, as undertaken in this work, consists of studying the shape of the efficiency curve of a hydropower plant for all the range of power the plant can generate. The analysis tries to identify the possible ways of energy waste with the aid of the maximum overall efficiency curves, the main purpose is to reduce the water consumption for a given power demand by finding the best combination of operation points of the turbines installed. The Cyclic Coordinate search procedure is employed to optimize the operation of the power plants and simulations were performed to study cases where the plants have turbines with the same efficiency curves and where the plants have turbines with different efficiency curves. A consideration of few cases shows some values of possible energy waste. It is shown that, for even relatively small increases of about 1% or 2% in efficiency, the energy conservation in absolute terms is significant since the hydropower plants in Brazil have often a large power generating capacity. Keywords. Hydropower plants, Hydraulic turbines, Efficiency, Optimization 1. Introduction Hydraulic turbines are among the most efficient technologies for energy conversion, the efficiencies achieved by hydraulic turbines surpassed the level of 90% since the 1920 s (Creager and Justin, 1958). Although the efficiency of single turbines is a typical subject matter in most of the literature on the subject of turbomachinery (for instance, Pfleiderer and Peterman, 1972; Japikse and Baines, 1997), the analysis of the overall efficiency of an individual hydropower plant has not been as much a major concern, perhaps mainly because the efficiencies achieved nowadays are already high. The promotion of efficiency increases may not contribute to substantial improvements in terms of efficiency numbers, nevertheless, the energy conservation of small efficiency improvements can be significant when large amounts of energy are generated and some critical situations may require an optimized way of generating power from water. Techniques with low cost of implementation that can provide awareness about more rational processes of power generation should be investigated to indicate the extent of energy conservation possibilities. This work investigates the use of optimization techniques in the analysis of the overall efficiency of hydropower plants. The analysis proposed here can be related to the literature focusing on the problem of hydro unit scheduling with turbine considerations. Philpott et al. (2000), for instance, treat the problem of scheduling the turbines in a chain of stations down a river valley, considering the cost of a unit commitment at some instant in time (i.e., which of the generating units are on or off in every interval of the scheduling horizon), this cost is measured in terms of the water the unit is wasting and takes into account the nonlinear efficiencies of the turbines. Hreinsson (1988) presents a model to optimize the hourly power production of a purely hydroelectric system by minimizing losses in turbines and waterways. Piekutowski et al. (1994) describe a short term hydro optimization which incorporates turbine efficiencies and flow limits, penstock head losses, tailrace elevation and generator losses. Other examples can be found in the literature of the area. This work proposes the optimization of unit commitment centered on the overall efficiency curve of the hydropower plants, considering the core problem of runner water flow and turbine shaft power. The analysis of the overall efficiency of hydropower plants, as undertaken in this work, consists of studying the shape of the efficiency curve of a hydropower plant for all the range of power the plant can generate, considering the total number of turbines installed. An optimization technique allows the estimation of the maximum overall efficiency of a hydropower plant below which the operation of the turbines represents a waste of the water flow energy. The analysis also tries to identify the possible ways of energy waste with the aid of the maximum overall efficiency curves, the main purpose is to reduce the water consumption for a given power demand by finding the best combination of operation points of the turbines installed. 2. Problem Definition The Brazilian electric system is strongly based on hydropower generation. The total electrical power installed in Brazil is about 83 GW. Hydropower plants generate more than 79% of that amount (Aneel, 2003). The country also has a number of the largest hydropower plants in the world; most of the power available is generated by plants with a

2 capacity within the range of 0.4 GW to 4 GW. In addition, the construction in the country of the largest hydropower plant currently operating in the world with a capacity of 12.6 GW can be considered as a mile-stone in the development of hydropower in the world (Oud, 2002). As a result of this configuration, the Brazilian electric system is strongly dependent on the water levels of the storage dams, and affected by the uncertainty of future inflows. Critical low levels of reservoirs may lead to shortages in the supply of electrical energy, causing economic and social costs. The possibility of maximizing the power generation to increase the power availability for a given water flow is the motivation of this work. In this work it is studied a technique for improving the operation efficiency of impoundment hydropower plants, thus saving the stored water in an energy conservation effort. The technique consists of an optimization algorithm applied to the efficiency curves of hydraulic turbines. Finding the best operation points of each turbine has the objective of maximizing the overall efficiency of the power plant for every electrical power demand required from the plant by the external grid. The best configuration of operation points is searched by the technique, and this configuration can guide the operators to set the operation points in a different way than the procedures that may be usually employed. Simulations were performed to study cases where the plants have turbines with the same efficiency curves and where the plants have turbines with different efficiency curves. Although the most common case is the existence of plants with equal turbines, in a large hydropower system cases of plants having different turbines exist. A consideration of few cases shows some values of possible energy waste. It is shown that, for even relatively small increases of about 1% or 2% in efficiency, the energy conservation in absolute terms is significant since the hydropower plants in Brazil have often a large power generating capacity. An important contribution of this investigation is to provide awareness about the best operation procedures rather than to provide an on-line optimization technique. 3. Water Power Water used for power generation, such as in rivers, derives its capacity to perform work from the energy of the sun which is responsible by the operation of the hydrological cycle. The water evaporated by solar energy is condensed by various meteorological processes and precipitates on highlands to further flow as runoff to rivers, where it can be used to power generation. The fact that the hydrological cycle is dependent on meteorological factors a priori not controllable by man, imposes the need of proper planning of the hydropower generation system. This planning involves, among other issues, the planning of the generation reserve installed in the system and the hydro-scheduling problem with the long-range forecasting of water availability and the scheduling of reservoir water releases that may be constrained by factors such as hydraulically coupled plants or river navigation (Wood and Wollenberg, 1996). In computations for power and energy available from a hydroelectric plant during a given period, storage requirements are estimated based on the water flow demanded by the power output. Converting the number of MW of power needed to meet the requirements of output to its equivalent in cubic meter per second is a way of estimating this water flow. Choosing the points of turbine operation has an effect on this calculated water flow, and these effects are studied next. The problem investigated here can be considered as a minor issue of power development from water energy if compared to the general questions of hydropower operation, nevertheless, it is an opportunity of energy conservation that should be explored. 4. Efficiency of Hydraulic Turbines Power may be developed from water by three fundamental processes: by action of its weight, of its pressure, or its velocity. The power developed by turbines generally is a combination of these processes. The impulse type of turbines utilizes the kinetic energy of a high-velocity jet, and the reaction type of turbines utilizes the combined action of velocity and pressure. The study presented next is concerned with the reaction type and more specifically with the Francis turbine. If a steady discharge, Q, of water is available with a net head, H, the power that can be developed from this flow passing through the runner is (Pfleiderer & Peterman, 1972): P = η.ρ. H. Q. g (1) Where g is the local value of gravity acceleration, ρ is the value of water density, and η is the efficiency of the turbine. The efficiency depends mainly on the specific turbine and power being generated. For constant head and specific speed, the efficiency of a Francis turbine has a typical behavior as depicted in Fig. (1). A change in the value of the head also modifies the efficiency curve, reducing or increasing the total possible power output. Figure (1) illustrates the real data of efficiency as a function of power of two different Francis turbines (different manufacturers and different capacities) under the same conditions of head = 100 m and speed = 100 rpm. The different constitution of each turbine combined with these conditions results a maximum power output of 210 MW and 330 MW for the first and second turbines respectively. From Eq. (1), it can be promptly observed that, for a demanded power, the water flow required is a function of the efficiency, the head, the gravity acceleration, and the density. If the last three variables are considered constant, the

3 quantity of water is determined since the efficiency is also a known value for being a function of the demanded power, a function defined by the efficiency curve (Fig. (1)). Figure 1. Efficiency curves of Francis turbines. 5. Overall Efficiency of a Hydropower Plant When considering more than one turbine, an overall efficiency has to be calculated to find the total water quantity required. This overall efficiency can be derived from Eq. (1). The total power output is the sum of the power of each unit, and the total water flow is the respective sum of flows calculated with the respective efficiencies: P Pn = η overall. H. ρ. g.( Q Qn ) (2) The variable η overall is the overall efficiency of the plant. It should be noted that the water flow in a single unit is a direct function of the power of this unit (Eq. (1)) and therefore the overall efficiency is a composition of the single efficiency curves of the turbines which results in a surface for two units or a hypersurface for more than two units. Figure (2) illustrates the efficiency surfaces and respective contour plots for the case of plants with two units. The left side of the figure illustrates the case where the plant has two equal units, and the right side illustrates the case where the plant has two different units. What can be seen is that, for a given power demand, there are many possible ways of combining the operation of the units, and a combination results an efficiency value. For the case of two units illustrated, a power demand is a linear constraint and the plane originated by the line of possible combinations that intersects the surface generates a curve of feasible efficiencies. The point where this curve has its maximum should be the point of operation. Since the contour plot of the left side of Fig. (2) has a symmetric shape, it is suggested that the maximum efficiency is obtained by simply dividing the demanded power in two and then setting the operation of the units based on this value. But taking into account the right side of Fig. (2), the result is not as obvious and a search for the best point must be done. Large power plants generally have more than two units and thus a specialized search, that is, optimization techniques, may be required. 6. Efficiency Optimization 6.1. Objective Function The problem of optimization is better stated if it is done so in terms of maximizing the overall efficiency. From Eq. (2), the objective function to be optimized is: n Pi i= 1 η overall = (3) n ρ. g. H. Qi i= 1

4 Figure 2. Efficiency surfaces and respective contour plots. Where n is the number of units in the plant. The constraint of the problem is the power demand: n P i = Power Demand (4) i= 1 Inputs to the optimization are the density of water, gravity acceleration, head, and water flows. The flows are functions of the power and efficiency curves of the turbines installed. During the computation the equations representing these relations must be considered: P1 Pn Q1... Qn = ρ. g. H. η1 ρ. g. H. ηn = (5) The optimization has to result the best combination of outputs of each unit that obey the total demanded power and at the same time maximizes the overall efficiency. The stated problem can be solved for different cases of head and for plants having equal units with the same efficiency curves, or for plants having different units with different efficiency curves Optimization Method This is a multivariate optimization with linear equality constraints (Gill and Wright, 1981). It can be considered a simple optimization problem since the efficiency curves are smooth functions with easily identified global maximum, not having local maximum points. There are many options available today for solving this type of optimization problem and reference is made to the literature related to the subject. The results presented next were obtained by a simple multidimensional search without the use of derivatives called the Cyclic Coordinate Method (Bazaraa and Shetty, 1979). This method is suited to the problem and performs well. The method of Cyclic Coordinates consists of maximizing a function f(x) (in this problem function f is Eq. (3)) of several variables proceeding in the following manner: given a vector of a possible solution x, a suitable direction d is

5 determined and then f(x) is maximized from x in the direction d by a line search algorithm. By sequentially changing the vector d and maximizing the function in this direction, the global maximum is found. The Cyclic Coordinate method uses the coordinate axes (in this situation the power of each unit) as search directions. More specifically, the method searches along the directions d 1,, d n, where d j is a vector of zeros except for a one at the jth position. Thus, along the search direction d j, the power P j of unit j is changed, while the power levels of all other units are kept fixed. The direction is changed until the algorithm reaches the optimum point. Figure (3) illustrates this procedure. Figure 3. Cyclic Coordinates method. The demanded power imposes a linear constraint that can be implemented with the use of a penalty function, P(x), of the form: P ( x) = µ. h 2 ( x) (6) Where µ is an appropriately selected value above 0, and h(x) is a constraint function. The problem of optimization is then transformed into the optimization of the following objective function: g ( x) = f ( x) µ. h 2 ( x) (7) From Eq. (6), it is clear that if h 2 (x) is different from zero, the function g(x) will always be assigned a low value, thus not representing a optimization solution. Choosing a constraint function of the form: h ( x) = P Pn Power Demand (8) Makes h 2 (x) = 0 only for the possible solutions, penalizing the solutions that do not obey the constraint imposed by the value of demanded power. In Fig. (3), this penalty function could be represented as a straight line of feasible solutions. Additional care must be taken when computing the variable µ. For further understanding of the optimization technique, again is made reference to the optimization subject literature. 7. Overall Efficiency Analysis The main result of running an optimization algorithm applied to this problem of finding the best operation points of a group of units in a hydropower plant is the possibility of tracing the maximum overall efficiency curve of the plant. An overall efficiency curve of a power plant is the efficiency curve traced for the range of power that a hydroelectric plant can generate (Creager and Justin, 1958) and is a composition of the single efficiency curves of the units, depending on the shape of those curves. The maximum overall efficiency curve is composed of the maximum efficiencies provided by the best combination of unit operation points and is obtained by running the optimization algorithm for all the range of demanded power. Below the maximum overall efficiency a large number of different curves that do not represent the maximum efficiencies possible for a given power demand may exist.

6 Figure 4. Overall efficiency of a hydropower plant Equal Turbines For the most common case of plants having turbines of equal capacity and efficiency, the maximum overall efficiency curve has a shape similar to that of Fig. (4). The curve is plotted based on a Francis turbine of maximum capacity of 330 MW having the second efficiency curve depicted in Fig. (1). The number of units does not modify the shape of the curve except for a modification in an ending section of the curve as depicted in the figure. As already commented, the combinations of operation points that provide the maximum efficiency is obtained by simply dividing the power demand by the number of units installed. If this is not done the efficiency will have a smaller value than the value of the maximum overall efficiency curve. Although this conclusion could be obtained without the aid of an optimization algorithm, the optimization calls the attention to a possible way of energy waste. Figure (5) demonstrates the overall efficiency curve of a 4-unit plant as a function of the number of units in operation, it is seen that the points where the curves intersect are the points where the overall efficiency curve has an inflexion. Figure (6) shows the overall efficiency plotted against the percentage of the total power capacity of the plant in order to verify the points of inflexion in terms of this percentage. Energy waste may occur if the inflexion points are not observed and for a reduced power demand the plant continues to operate with a certain number of units when the operation of one unit should be stopped or started. For instance, the inflexion from the operation with 3 units to the operation with 4 units for the case in Fig. (5) and (6) occurs when the power demand has a value of 68.2% of the total power. Two cases can be analyzed: the operation of 3 units after the point, or the operation of 4 units before the point. In the first case, where 3 units are under operation, there is the possibility of not observing the inflexion point and maintaining 3 units operating from 68.2% to 75% of the total capacity (or the maximum capacity of the 3 units, 990 MW), this situation represents an overall efficiency of the 3 units (which is the same overall efficiency of the entire plant) of 91.43% for the plant under study. If 75% of the total capacity is generated with 4 units in operation, this overall efficiency would be 94.6%, a difference of 3.17% representing about 31.4 MW (3.17% of 990 MW) of energy waste due to the operation of 3 units after the inflexion point. In the second case, if the plant is taken with 4 units to a hypothetical point of 65% of the capacity (about 860 MW), a point before the inflexion point, this would represent a difference in efficiency of about 1.86% when compared to the operation with 3 units, and a energy waste of about 16 MW due to the operation of 4 units before the inflexion point. For this last case, since the 4-unit efficiency decreases abruptly, not observing the inflexion point could imply in a maximum difference between the operation with 4 or 3 units of 4.79% when the demanded power is 750 MW, a waste of MW. The worst case occurs when the demanded power is 260 MW (operation of 1 or 2 units), not observing the inflexion point could result in a difference of 12.97%, a waste of MW Different Turbines Plants having different turbines take more advantage of optimization because the solution to the question of choosing the best combination of operation points is not as obvious as it is in the case of equal turbines. Figures (7) and (8) illustrate two possible maximum efficiency curves for a plant having 2 units with a capacity of 210 MW and 2 units with a capacity of 330 MW, the respective efficiency curves are presented in Fig. (1). For a given demanded power, the combination of points is not the result of a simple division by the number of turbines and the algorithm has to provide the best combination. As an example, for a demanded power of 800 MW, the best combination would be 126 MW and 177 MW for the 210 MW units, 248 MW and 249 MW for the 330 MW units. This combination results an overall efficiency of 93.8%, and a simple division (all the units generating 200 MW) would result in an efficiency of 91.4%. This difference, 2.4%, accounts for a power waste of 19.2 MW.

7 Figure 5. Overall efficiency as a function of number of units. Figure 6. Overall efficiency as a function of percentage of total power installed. The issue of inflexion points discussed in the previous section is also important in the case of different turbines. In addition, there is also the need to consider what turbine should be kept under operation and what turbine should not as the demanded power is reduced. Figure (7) shows the overall efficiency curve when the turbines with capacity of 330 MW are maintained under operation and the turbines with minor capacity are firstly stopped. Figure (8) shows the opposite situation where the major units sequentially stop operating when the demanded power is reduced and the 210 MW units are maintained under operation until the minimum possible power. It is inferred by the two situations that if the major units are kept operating as the power is reduced, the overall efficiency curve is smoother than the other case. This signifies that the average efficiency is larger than the average efficiency of the case when the minor units are preferred and therefore it is recommended to maintain the major units operating when the demand is reduced. 8. Conclusion The previous exposition treated the core problem of optimizing the operation points of hydraulic turbines in hydropower plants, considering the turbines shaft power and the runners water flows. Two general types of hydropower plants were studied, plants with turbines of equal efficiency, and with different efficiency. The potential ways of energy waste were analyzed as well as the levels of conservation that can be achieved. As an important note in this conclusion remark, a discussion on the possible levels of energy waste must be done. If the ways of water energy waste eventually happen, even low levels of waste of 1% to 3% account for large power differences in large power plants. The levels found in the discussion above, for example, are the same levels of power of a SHP small hydropower plant 30 MW. In comparison with a thermal plant this power could represent 86 MW of fuel, if admitting a standard 35% of thermal plant efficiency. Greater energy wastes are expected for larger plants. Since the implementation of these techniques is practically costless, the awareness provided by running an optimization algorithm applied to the particular turbines of a hydropower plant could avoid the possible energy waste

8 situations, in a supplementary energy conservation effort. Further studies on the integration of the considerations about the overall efficiency curves of hydropower plants with other unit scheduling algorithms could also promote good results towards energy conservation in energy conversion processes of hydropower developments. Figure 7. Overall efficiency with different turbines. Figure 8. Overall efficiency with different turbines (maximum average efficiency). 9. References Aneel, 2003, Banco de Dados da Geração, Agência Nacional de Energia Elétrica Corporate Data, June 2003, url: Bazaraa, M.S., Shetty, C.M., 1979, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, Inc., New York, USA. Creager, W.P., Justin, J.D., 1958, Hydroelectric Handbook, John Wiley & Sons, Inc., New York, USA, 2nd edition. Gill, P.E., Murray, W., Wright, M.H., 1981, Practical Optimization Hartcourt Brace and Company, Publishers, Academic Press, London, UK. Hreinsson, E. B., 1988, Optimal short-term operation of a purely hydroelectric system, IEEE Transactions on Power Systems, Vol. 3, No. 3. pp Japikse, D., Baines, N. C., 1997, Introduction to Turbomachinery, Concepts ETI, Woburn, MA, USA. Oud, E., 2002, The evolving context for hydropower development, Energy Policy, Vol. 30, pp Pfleiderer, C., Peterman, H., 1972, Strömungsmachinen Springer-Verlag, Berlin/Heidelberg, Germany, 4 th edition. Philpott, A.B., Craddock, M., Waterer, H., 2000, Hydro-electric unit commitment subject to uncertain demand, European Journal of Operational Research. Vol. 125, pp Piekutowski, M.R., Litwinowicz, T., Frowd, R.J., 1994, Optimal short-term scheduling for a large-scale cascaded hydro system IEEE Transactions on Power Systems, Vol. 9, No. 2. pp

9 Wood, A., Wollenberg, B.F., 1996, Power Generation Operation and Control John Wiley & Sons, Inc., New York,USA, 2 nd edition. 10. Copyright Notice The authors, E.W.Hirano and E.Bazzo, are the only responsible for the printed material included in his paper.