András István Fazekas a b & Éva V. Nagy c a Hungarian Power Companies Ltd., Budapest, Hungary. Available online: 29 Jun 2011

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1 This article was downloaded by: [András István Fazekas] On: 25 July 2011, At: 23:49 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Electric Power Components and Systems Publication details, including instructions for authors and subscription information: Modeling of Combined Heat and Power Generating Power Plant Units Using Markov Processes with Continuous Time Parameter and Discrete State Space András István Fazekas a b & Éva V. Nagy c a Hungarian Power Companies Ltd., Budapest, Hungary b Department For Energy Engineering, Budapest University of Technology and Economics, Budapest, Hungary c Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary Available online: 29 Jun 2011 To cite this article: András István Fazekas & Éva V. Nagy (2011): Modeling of Combined Heat and Power Generating Power Plant Units Using Markov Processes with Continuous Time Parameter and Discrete State Space, Electric Power Components and Systems, 39:10, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan, sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Electric Power Components and Systems, 39: , 2011 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Modeling of Combined Heat and Power Generating Power Plant Units Using Markov Processes with Continuous Time Parameter and Discrete State Space ANDRÁS ISTVÁN FAZEKAS 1;2 and ÉVA V. NAGY 3 1 Hungarian Power Companies Ltd., Budapest, Hungary 2 Department For Energy Engineering, Budapest University of Technology and Economics, Budapest, Hungary 3 Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary Abstract Power plant units with extraction and back-pressure turbines cogenerating heat and power play a decisive role in the heat and power generation of the former centrally planned Eastern European countries. Cogeneration requires that for a certain portion of the operating period, extraction-condensing and back-pressure steam turbine power plant units make less than their nominal capacity available to the power system. Consequently, the two-state reliability description is not suitable for differentiated reliability modeling of cogeneration power plants. This article briefly sets out a three-or-more-state reliability description of these power plant units using Markov processes with continuous time parameter and discrete state space. Keywords 1. Introduction power system reliability, cogeneration, power system planning Power plant units with back-pressure and extraction-condensing turbines are usually represented in system-level reliability calculations of power systems by a two-state reliability model (state representation) [1, 2], with the result that the external ambient temperature dependency of available capacity is not taken into account, at least not with satisfactory accuracy. Two-state reliability modeling for such power plant units thus leads to unrealistic values of system-level reliability indicators (e.g., loss-of-load probability [LOLP]) and diminishes the accuracy of reliability calculations [1, 2]. The two-state reliability description of power plant units assumes that a power plant unit is either available with full capacity or has completely failed, i.e., has lost its entire capacity. The newly developed procedure improves the accuracy of system-level reliability (LOLP) calculations for power generation systems by giving a more differentiated (and, thus, more accurate) description of the available capacity from extraction-condensing and back-pressure turbine power plants. Received 15 July 2010; accepted 28 November Address correspondence to Dr. András István Fazekas, Szentendrei ut , Budapest, H-1031, Hungary. afazekas@mvm.hu 1031

3 1032 A. I. Fazekas and É. V. Nagy It should be stressed that the novelty content of this conclusion lies not in the fact that the instantaneous maximum available power capacity of these power plant units is a function of instantaneous heat output and, therefore, of average daily ambient temperature, but in the recognition that this relationship permits their three-and-more-state reliability description by virtue of the desired random variable ( LPP max) being a transform of the random variable (& Tk ). This permits the calculation of the probability of occupation of each defined operating state (in the test period). Although multi-state reliability descriptions of power plant units have been developed previously [3 9], they have not yet been applied to the reliability modeling of extraction-condensing and back-pressure steam-turbine power plant units. 2. Resolution of Problem by Multi-state Reliability Modeling of Extraction-condensing and Back-pressure Power Plant Units It requires no further explanation to understand that the two-state model gives a very coarse approach to how these power plants actually run, especially since the current level of heat output very often means that derated power capacity is available to the power system [10]. This can be readily appreciated when it is known that these power plant units obey the relation L PP D f. PQ.T k //: (1) Equation (1) states that the maximum power output of the power plant (i.e., available capacity: L PP [MW] depends on the current heat output of the plant PQ [MW]). The latter depends on the external ambient temperature (T k [ ı C]), particularly in the case of heat for space heating purposes. An example of such a relationship is shown in Figure 1. The upper parallel line applies to the maximum loading of the power plant unit in question; the lines below it show the relation in the case of 80%, 60% and minimum loading (40%). Figure 2 shows the relationship between mean daily ambient temperature-dependent heat Figure 1. Relation between heat and power output for an extraction-condensing power plant unit.

4 Modeling of Power Plant Units with Markov Processes 1033 Figure 2. Variation of heat and power output of an extraction-condensing power plant unit with daily mean ambient temperature over a one-month period. output and the available (maximum) power. For extraction-condensing turbine plants, the maximum available power capacity decreases as heat output increases (for a backpressure plant, power output increases with heat output). Reliability modeling of these power plant units involving three or more states enables a considerable improvement in the accuracy of system-level reliability calculations [3, 7 9]. 3. Reliability Modeling of Extraction-condensing and Back-pressure Power Plant Units Using State Space Representation The operation of power plants, including extraction-condensing and back-pressure power plants, may be regarded as a Markov process with a continuous time parameter and a discrete state space. This means that the power plant unit has various defined operating states among which it passes consecutively in a random manner. The period for which it remains in each operating state also varies randomly. The state definitions together comprise the discrete state space. The non-zero capacity operating states may, for example, be defined as follows: P min L PP P 1! U 1 ; P 1 < L PP P 2! U 2 ; (2) P 2 < L PP P max! U 3 : In the equations, P min (MW) and P max (MW) are the minimum and maximum output power capacity, and P 1 (MW) and P 2 (MW) are two output values within the operating output range of the power plant unit, defined according to some consideration and subject to the relation P min < P 1 < P 2 < P max : (3)

5 1034 A. I. Fazekas and É. V. Nagy The current output (load) of the power plant unit is denoted by L PP (MW). The power plant is in one of the non-zero capacity operating states U 1, U 2, and U 3, regardless of output range L PP (MW). It follows from the definition that the power plant unit is always exclusively in one or other of states U 1, U 2, U 3, or D. State D is the completely failed operating state. The state space in this case thus comprises these discrete operating states: U D fu 1 ; U 2 ; U 3 ; Dg: (4) The operation of the power plant units may be conceived as a system that passes into various states as the continuous time parameter t runs through the specified period. For example, in the operation of an extraction-condensing power plant unit, changes of state as shown in Figure 3 are possible if the state space obeys Eq. (4). The operation of the power plant units may be conceived as a system that passes into various states as the continuous time parameter t runs through the specified period. For example, in the operation of an extraction-condensing power plant unit, changes of state as shown in Figure 3 are possible. Figure 4 shows an example of a specific process realization for an extractioncondensing power plant unit. In the case shown in the figure, the power plant unit changed state at times t 1 < t 2 < < t i <, i.e., passed from some defined operating state to another defined operating state, in which it remained for a random time t ic1 t i D i. From Figure 4, it may be seen that the change in state of the power plant unit is random, first, in respect of the consecutive operating states.u i / it passes into, and second, in respect of the time. i / for which it remains in each defined operating state. This means that as shown on the diagram operating states and their associated duration of persistence follow randomly upon one another. The operation of the power plant unit for a given interval can, therefore, uniquely be described by value pairs ŒU i ; i D t ic1 t i. In the case of the specific process realization of the diagram, the value pairs are as given in Table 1. There is, therefore, an unlimited number of possible Figure 3. Diagram of possible state changes in the case that the state space is defined by operating states U 1, U 2, U 3, and D.

6 Modeling of Power Plant Units with Markov Processes 1035 Figure 4. Diagram of possible state changes in the case that the state space is defined by operating states U 1, U 2, U 3, and D. specific process realizations, depending on the consecutive value pairs ŒU i ; i D t ic1 t i. It can be proved that the durations i, i.e., the durations of persistence in different defined operating states, follow an exponential distribution [1, 6, 7]. 4. Description of the Random Course of Operation of Power Plant Units as a Stochastic (Markov) Process with Continuous Time Parameter and a Discrete State Space Applying the three-or-more-state reliability model for power plant units with extractioncondensing and back-pressure turbines and the deduction of the reliability calculation s Table 1 Associated values of randomly varying durations and operating states X ti D U 1 U 2 U 3 U 1 U 2 D U 1 U 3 i D t ic1 t i

7 1036 A. I. Fazekas and É. V. Nagy input data form the novelty of the newly developed calculation method. In what follows, the basic principles of applying the three-or-more-state reliability model [7 9] for power plant units in question will be stated. For continuous time parameter, discrete state space Markov processes, introducing the notations t n 1 D t and t n D t C h, the conditional probability defining the Markov process takes the form P ŒX.t C h/ D j j X.t/ D i D p ij.t; h/: (5) This conditional probability is called the transition probability. For homogeneous Markov processes, the transition probability p ij depends solely on h, and not on t. Thus, the transition probability depends solely on the length of the interval h, and not on the time itself. The transition probabilities p ij. t/ may be arranged into a transition probability matrix (P. t//, where 2 3 p 11. t/ p 12. t/ p 1N. t/ p 21. t/ p 22. t/ p 2N. t/ P. t/ D 6 4 : : 7 : 5 D Œp ij. t/ : (6) p N1. t/ p N2. t/ p NN. t/ The matrix P. t/ giving the transition probabilities in Eq. (6) is a square matrix whose elements are non-negative numbers, and the sum of every row is one. Matrices with such properties are called stochastic matrices. The values of the transition intensities give the probabilities of state transitions for the unit of time in equations and q ij D lim t!0 q i D lim t!0 p ij. t/ ; i j; (7) t 1 p ii. t/ : (8) t For homogeneous Markov processes, the value of transition intensities is, by implication, constant; i.e., it does not vary with time. The transition intensity matrix A is defined as follows [3]: 2 3 q 1 q 12 q 1N 6 q 21 q 2 q 2N 7 A D 6 4 : : : 7 5 : (9) q N1 q N2 q N This matrix has the special property that every row adds up to zero. At sufficiently small time intervals h D t, the relations and P ŒX.t C t/ D j j X.t/ D i D p ij. t/ q ij t; (10) P ŒX.t C t/ D i j X.t/ D i D p ii. t/ 1 q i t: (11)

8 Modeling of Power Plant Units with Markov Processes 1037 apply for homogeneous Markov processes [3, 7]. In Eq. (11), p ii t is the probability that the process will have no change of state in interval t; i.e., the process will stay in state i until the end of interval t, providing that it was in state i at the beginning. The quantities q ij and q i in Eqs. (10) and (11) are called transition intensities. The stochastic matrix P. t/ giving the transition probabilities and the matrix A giving the transition intensities have the following relation: Here, P. t/ A D lim t!0 t I : (12) I D 6 7 4: : : 5 : (13) Equation (12) derives from Eqs. (7) and (8). Assuming that some power plant unit is in state i at a given moment, then in the following time interval t, it will pass into state j or it will remain in state i, i.e., not change state. Consequently, p ii. t/ C X i j p ij. t/ D 1: (14) This is shown in Figure 5. Using the foregoing in Eq. (13) gives 1 X q i D lim p ij. t/ D X q ij : (15) t!0 t i j i j Figure 5. Possible transitions from operating state i in the case that the state space is defined by operating states i, j, k, and l.

9 1038 A. I. Fazekas and É. V. Nagy The transition probability for successive time intervals h 1 and h 2 may be defined as the combination of the transition probabilities for each interval if h D h 1 C h 2 : (16) The transition probability (Figure 6) for the resultant interval h is p ij.h/ D X k p ik.h 1 /p kj.h 2 /: (17) The above is known as the Chapman-Kolmogorov equation. The unconditional probability distribution for a random variable X.t/ may be defined as follows, assuming that X.t/ D i and P ŒX.t/ D i D p i.t/: p i.t C t/ D p i.t/p ii. t/ C X p j.t/p j i. t/: (18) j i The matrix form of Eq. (18) is [3] p.t C t/ D p.t/p. t/; (19) where p.t C t/ D Œp i.t C t/ ; (20) and p.t/ D Œp i.t/ : (21) Figure 6. Transition probability for successive time intervals.

10 Modeling of Power Plant Units with Markov Processes 1039 The vectors of Eqs. (20) and (21) are row vectors. Equations (10) and (11), with the appropriate substitutions, yield p i.t C t/ p i.t/.1 q i t/ C X j i p j.t/q j i t: (22) From Eq. (22), p i.t C t/ t p i.t/ p i.t/q i C X j i p j.t/q j i : (23) Taking the boundary transition t! 0 gives the differential equation [3] Pp i.t/ D p i.t/q i C X j i p j.t/q j i : (24) Here, Pp i.t/ is the time derivative of p i.t/,. where The solution of the matrix differential of Eq. (25) is dpi.t/ dt /. The matrix equation form is Pp.t/ D p.t/a; (25) Pp.t/ D Œ Pp i.t/ : (26) p.t/ D p.0/e At : (27) Here, p.0/ is a row vector containing the initial conditions, and e At D I C At C A 2 t2 2Š C : (28) For most practical applications, it is the long-term probabilities that are of interest [3]. This means the value of the required p i.t/ as t! 1. What has to be found, therefore, is the probability distribution. Assuming that there exists such a probability, the change of p i.t/ must steadily decrease with increasing t. Consequently, the matrix differential in Eq. (25) simplifies to the form 0 D pa; (29) In this case, the row vector p gives the probability of persisting in each defined operating state as t! 1. Furthermore, the relation idn X p i D 1 (30) id1 also applies. The solutions to the system of Eqs. (29) and (30) yield the long-term probabilities of persisting in each operating state, i.e., the probability distribution of remaining in each operating state after a long period of operation.

11 1040 A. I. Fazekas and É. V. Nagy 5. Deduction of Three-and-more-state Reliability Description s Input Data There is a clear requirement for a means of defining the data to be input to three-ormore-state reliability descriptions of such power plant units. The basic quantity to be determined is the probability that a power plant unit is in a given power output range (corresponding to a state of operation). This probability may be determined by virtue of the defining operational feature of extraction-condensing and back-pressure power plant units that their maximum power capacity is at all times a function of their instantaneous heat output. The instantaneous heat output, in turn, depends on the daily mean ambient temperature, whose cumulative distribution function (CDF) is known for any geographical region. Consequently, the probability distribution function of the instantaneous power capacity may also be determined by appropriate transformation of the daily mean ambient temperature CDF. It is thus possible to calculate the probability that the power output capacity of a specific power plant will be in a given range at any time. The calculation is as follows: 1. The space-heating heat power demand other conditions being equal is a function of external ambient temperature PQ f0 D PQ f0.t k / and Q f0 D Q f0.t k /. 2. The probability distribution of the external ambient temperature & Tk [ ı C] as a random variable is known for a particular geographical point from meteorological data (i.e., the probability of occurrence of each value of the ambient temperature) as a random variable (i.e., the function F.& Tk / is known). 3. For extraction-condensing and back-pressure power plant units, the maximum power output capacity is a function heat power output, i.e., the function of Eq. (1) exists and is known. 4. The output power L PP (MW) of an extraction-condensing or back-pressure power plant unit corresponds to the random variable L PP D P;max (MW). This random variable, as mentioned above, is a function of the power plant unit s current heat output power PQ (MW), and the latter is a function of the randomly varying ambient temperature T k ( ı C). The random variable P;max (MW) is, thus, ultimately a random variable of the ambient temperature & Tk ( ı C): P;max D f.& Tk /: (31) 5. F.& Tk / and Eq. (1) permit determination of F. P max /. 6. Knowing F. Pmax /, it is possible to calculate the probability that a power plant unit will be in a given power range; i.e., the long-term probability of being in each defined operating state may be determined. 6. Derivation of the Distribution Function of the Maximum Power Output Random Variable by Transformation of the Duration Diagram of the Daily Mean Ambient Temperature If L PP (MW) denotes an arbitrary value of the maximum power capacity of a power plant unit (power variable) and l (MW) a specific value of the load, then T.l/ (hr) is the duration within a given period when the relation L PP l holds. It thus represents the number of hours in which the maximum power capacity of an extraction-condensing or back-pressure cogeneration power plant unit is greater than or equal to a defined power

12 Modeling of Power Plant Units with Markov Processes 1041 value l (MW). The duration diagram of the maximum (available) power capacity may be derived from the duration diagram of the ambient temperature for the period via the relation L PP D f. PQ i.t k //. The duration diagram of the maximum power capacity may be transformed in several steps into a probability distribution function of maximum power capacity. The first step transforms time values on the abscissa into relative time values. The values on the abscissa of the resulting diagram may be understood as values of the probability that the maximum power capacity is greater than or equal to a given maximum power capacity, i.e., they give the probability P ŒL PP l. The second transformation step is the exchange of the abscissa and the ordinate. This puts the probability on the ordinate and the maximum power capacity on the abscissa. The curve obtained from these two transformations may be regarded as the complementary curve of the probability distribution curve of random variable L PP (i.e., of the probability distribution of L PP ), the curve t.l/ D 1 F LPP.l/: (32) In this equation, t.l/ is the relative duration ( ) when L PP l, and F LPP.l/ is the cumulative probability function of L PP. It follows that the cumulative probability function of L PP is defined by the relation F LPP.l/ D 1 t.l/ (33) and gives the relative duration ( ) in which L PP < l. The relative duration may also be understood as a probability measure. Then the relative duration is equal to the probability (gives the probability) that L PP < l, or F LPP.l/ D P ŒL PP < l : (34) F LPP.l/ D 1 t.l/; therefore, gives the probability that the maximum power capacity is less than a given value l. The relation thus defined can be proved to satisfy the following requirements of a probability distribution: 1: F LPP.l/ 0: (35) 2: The function is monotonically increasing, so that for 8l 1 and every 8l 2, if the relation l 1 < l 2 holds, then F LPP.l 1 / F LPP.l 2 /; i.e., 8l 1 and 8l 2,..l 1 < l 2 /.F LPP.l 1 / < F LPP.l 2 //. (36) 3: lim l! 1 F L PP.l/ D 0: (37) 4: lim l!c1 F L PP.l/ D 1: (38) 5: F LPP.l/ D P.L PP < l/ is continuous from the left at every point; i.e., for 8c 2 R, lim F L PP.l/ D F LPP.c/: (39) l!c 0 Since the function above defined can be proved to satisfy the requirements 1 5, it may be regarded as the distribution function of random variable L PP. It is known from probability theory that if the distribution function of a random variable F LPP.l/ is continuous, and

13 1042 A. I. Fazekas and É. V. Nagy F 0 L PP.l/ exists everywhere except at a finite number of points, then the derivative function may be regarded as the density function of L PP, i.e., F 0 L PP.l/ D f.l/: (40) In this equation, f LPP.l/ is the density function of F LPP.l/. It follows that F LPP.l/ D Z l 1 f LPP.z/dz: (41) The distribution function is therefore the integral of the density function. This means that the random variable L PP also has a density function. From probability theory, the density function f LPP.l/ is defined by the relation f LPP.l/ D df L PP.l/ dl This density function has the following properties: D t 0.l/: (42) 1: f LPP.l/ 0: (the density function cannot be negative). (43) 2: 3: Z C1 1 Z l2 f LPP.l/dl D 1: (44) l 1 f LPP.l/dl D P Œl 1 L PP l 2 : (45) This relation means that the integral of the density function over an arbitrary interval Œl 1 I l 2 is equal to the probability that the random variable falls in the interval Œl 1 I l Results and Conclusions The power plant system modeled in the calculations comprised 46 power plant units, and the total installed power capacity was 9710 MW. Table 2 gives the reliability and capacity figures for the power plant units making up the power plant system. Combined heat and power generating power plant units accounted for 4660 MW of the total installed power capacity; i.e., cogeneration power plant units contributed some 48% of installed power capacity. The power plant system included approximately 300 gas-engine cogeneration power plant units having installed power capacity in the MW range, which were, for the purposes of computation, modeled as a single aggregated power plant unit. As a further simplification, it was assumed that the characteristic curve of the 12 combined-cycle cogeneration power plant units (CCGT), having a total installed power capacity of 1800 MW, effectively coincides with the characteristic curve of extractioncondensing power plant units; i.e., it was assumed that the maximum power capacity of these power plant units is also a function of the current heat output. In reality, the relationship between the power capacity and heat output of CCGT power plant units is not so rigid. Comparative calculations were performed for two different cases using the above calculation principle. In the first (calculations 1V), power plant units cogenerating electricity and heat were modeled with a two-state reliability description. This case corresponds to the current practice. In the second case (calculations 2V), the reliability description of

14 Modeling of Power Plant Units with Markov Processes 1043 Table 2 Modeled power system Denomination Nominal capacity (MW) Number of units ( ) Availability factor (two-state reliability description) ( ) Nuclear power plant (condensing power plant) Conventional natural gas-fired power plant units Conventional lignite-fired power plant units CCGT power plant units (CHP) Coal-fired power plant units with back-pressure and extraction-condensing steam turbines (CHP) Diesel engines with combined a heat and power generation (CHP) Open-cycle gas turbine units Total (power plant system) a Modeled as an aggregated power plant unit. CHP: combined heat and power generation. power plant units cogenerating electricity and heat involved reliability modeling as in Eq. (4). The improvement in accuracy gained by using the new calculation method may be determined by comparing the LOLP values obtained from calculations 1V and 2V. For the sake of clarity, the calculation results are given together in Table 3. (The reliability characteristics of the power plant units incorporated into sample calculations were strongly distorted in the negative direction for illustrative clarity. Consequently, the Table 3 Results in summary Calculation version Principal characteristics of calculation version LOLP value ( ) Deviation from base value (%) 1V 2V Two-state reliability description of power plant units Reliability modeling of power plant units according to Eq. (4) ' D LOLP D 0: ' D LOLP D 0: C21.5

15 1044 A. I. Fazekas and É. V. Nagy resulting LOLP values are considerably higher than those that occur in reality.) The table shows the deviation from the base values. Calculation 2V, i.e., the three-state reliability modeling of power plant units cogenerating electricity and heat, resulted in an increase in the LOLP value of around 22%. This is the improvement in accuracy of system reliability analysis achieved by a more differentiated reliability calculation. The new procedure improves the accuracy of system-level reliability (LOLP) calculations for power generation systems by giving a more differentiated (and, thus, more accurate) description of the available capacity from extraction-condensing and backpressure turbine power plants, using a three-or more-state space representation. The reliability modeling of the power plant unit is based on Markov process with a continuous time parameter and discrete state space. Applying the three-or-more-state reliability model for power plant units with extraction-condensing and back-pressure turbine and the deduction of the reliability calculation s input data form the novelty of the newly developed calculation method. References 1. Roberts, N. H., Mathematical Methods in Reliability Engineering, New York: McGraw-Hill, Hall, J. D., Ringlee, R. J., and Wood, A. J., Frequency and duration method for power system reliability calculations: Part I generation system model, IEEE Trans., Vol. PAS-87, pp , Endrenyi, J., Reliability Modelling in Power Systems, Chichester, New York, London, and Toronto: John Wiley & Sons, pp , , Young, L., and Singh, C., Reliability evaluation of composite power systems using Markov cut-set method, IEEE Trans. Power Syst., Vol. 25, No. 2, pp , Dehghani, M., and Nikravesh, S., State-space model parameter identification in large-scale power systems, IEEE Trans. Power Syst., Vol. 23, No. 3, pp , Armstadter, B. L., Reliability Mathematics. Fundamentals, Practices, Procedures, New York: McGraw-Hill, Billinton, R., and Allan, R. N., Reliability Evaluation of Power Systems, New York and London: Plenum Press, Billinton, R., and Allan, R. N., Reliability Evaluation of Engineering Systems. Concepts and Techniques, New York and London: Plenum Press, Billinton, R., Power System Reliability Evaluation, New York, London, and Paris: Gordon and Beach Science Publishers, Galloway, C. D., Garver, L. L., Ringlee, R. J., and Wood, A. J., Frequency and duration method for power system reliability calculations: Part III generation system planning, IEEE Trans., Vol. PAS-88, pp , 1988.