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Incorporating explicit loss-of-load probability constraints in mathematical programming models for power system capacity planning
Incorporating explicit loss.of.load probability constraints in mathematical programming models for power system capacity planning A P Sanghvi M e t a Systems, 1911 N o r t h F o r t M y e r D r i v e , Suite 9 0 0 , Arlington, VA 22209, USA
I H Shavel ICF I n c o r p o r a t e d , 1 8 5 0 K S t r e e t , N.W., Suite 9 5 0 , W a s h i n g t o n , DC 2 0 0 0 6 , U S A
Current methodologies for incorporating reliability constraints in optimization models for capacity expansion planning are reviewed. In addition, a chance reliability constraint that can be incorporated into a mathematical programming model as a linear constraint is developed. Reliability is measured by loss-of-load probability. The constraint is based on a first-order approximation to the Gram-Charlier series representing the density function o f available capacity minus load. Numerical results showing good agreement between the linear approximation and actual loss-of-load probability are presented. Keywords: rel&bility, generation expansion, planning reliability calculations
I.
Introduction
This paper develops a linear constraint that can be incorporated explicitly into a linear programming (LP) formulation of an electric utility's capacity expansion planning problem. The constraint ensures that the optimal generation capacity expansion plan determined by the LP model provides a level of peak load reliability, measured by system loss-of-load probability (LOLP), that is equal to or better than a user-specified value. This methodology was used successfully in a capacity planning model that was developed to analyse certain strategic planning issues faced by the New England Power Pool (NEPool). II.
Literature review
An electric utility's generation capacity planning problem can be modelled as a mathematical optimization problem. Anderson 1 reviews a variety of such models, including Received: 14 December 1982
Vol 6 No 4 October 1984
0142-0615/84/040239-09
dynamic programming, linear programming (LP) and nonlinear programming formulations*. A typical formulation can be stated generally as: min [cl x + c z f ( x ) + csg(x)]
(i)
Ax = b
x~>O where x is a vector of decision variables representing an expansion strategy. The components of x represent the amounts of new generating capacity of each type to be added in each year of the horizon. In a formulation of broader scope, x can, for example, also include the incremental amounts of inertia capacity built in each year of the horizon, coal conversion activities, the amount of a certain type of load management installed and fuel stockpile decisions. The components of c 1 are unit capacity costs associated with the components of the vector x, and e2 denotes a * Such optimizations of models are particularly well-suited for strategic resource planning. They are able to solve large problems with relative ease, taking into account key factors such as uncertainty in load growth, uncertainty in water conditions (for hydroelectric generation) and the availability of a large number of resource options including demand-side alternatives. In particular, they are able to evaluate implicitly all feasible resource plans (mix, level and staging). They serve a complementary role with detailed production costing, reliability evaluation and financial analysis models, which by their very nature are more precise but cumbersome to solve and, therefore, usually restricted to the analysis of a few resource plans. In particular, a strategic planning model helps to characterize the general nature of the optimal resource plan. This information can then be used to structure a few resource plans with such characteristics. These plans are then subject to a 'finer tuned' analysis using detailed simulation and other models to arrive at the final selection.
$ 0 3 . 0 0 © 1 9 8 4 B u t t e r w o r t h & Co (Publishers) L t d
239
vector of unit fuel cost for each type of fuel used by tile power system. A vector of consumptions of each fuel type given the expansion plan x is denoted by f ( x ) and g(x) denotes a vector the components of which are the total generation by each plant type for the expansion strategy x. A vector of units' operating and maintenance (O&M) costs for each type of plant is denoted by ca. In a linear progranmring formulation, f(x) and g(x) are linear functions of x, A, and b. The constraints Ax = b ensure that in each time period the capacity expansion plan should meet projected load (both power and energy) and that no generating unit is loaded beyond its maximum availability. In addition, the model is likely to include a reserve margin constraint which requires that the "total available capacity at system peak in each period exceeds the system-peak load by a user-specified percentage, typically 20-25%. Such a constraint is of the type:
}~A ViCi >~PL(1 + RM)
(2)
i
where PL = peak load, A Vi = expected availability of unit i at peak, Ci = capacity of unit i, RM = reserve margin. The provision of a certain reserve margin, however, does not indicate anything specific about the peak reliability of a power system. Specifically, two capacity plans that provide the same reserve margin can have reliability implications, as measured by LOLP or other measures, that are markedly different because system reliability is a nonlinear function of key unit characteristics such as generating unit sizes, forced outage rates, maintenance requirements, etc. Generally, larger units and units with higher forced outage rates contribute less per megawatt to power system reliability than smaller units or units with lower forced outage rates:. Three other methodological approaches have been suggested for a more realistic and explicit incorporation of LOLP constraints in capacity planning models. Scherer and Joe 3 use binary integer variables to express system reliability explicitly as a constraint. A problem with their approach is that even their seven-plant problem example leads to 135 integer variables. For any realistic system, this would imply a problem that is too large, if not extremely cumbersome, for available mixed integer programming codes. To alleviate this situation, they use a heuristic procedure to prune the number of branches that need to be searched by a branchand-bound algorithm. Nevertheless, the use of such a procedure is likely to be computationally burdensome, especially if such a constraint were to be used in a more realistic formulation of the capacity expansion planning and dispatching problem than is contained in their paper. A recent model funded by EPRP contains a linear relation between the reserve margin and system LOLP, as a function of generating unit characteristics. First, potential candidates for unit additions are identified. These candidates are characterized by their unit size (MW), forced outage rate (%) and annual maintenance requirement (h). Given next year's peak load, this year's base system and the desired LOLP for next year, an iterative search procedure is used to estimate the total megawatt additions required if each candidate unit is used solely to augment the existing
240
generating mix. Tile effective load carrying capability-' ~I each potential candidate is determined by using a NewtonRaphson search procedure on the tail segment of the equivalent load curve obtained by applying the BoothBaleriaux probabilistic simulation procedure to all the units of the existing system. Each such simulation yields the reserve margin that would be required to meet next year's load with a desired LOLP if the given candidate unit were to be used solely. Based upon a large number of such experiments, a multivariate regression is carried out to estimate the following equation: RM i = c~+ ~(unit size/peak load) + 3,(forced outage rate) + 8 (maintenance downtime)
(3)
where RM i is the reserve margin that would be required if the desired LOLP is to be achieved by augmenting the existing system solely by units of type i; and a,/5, 7, 6, are the regression coefficients. Cote and Laughton s use a similar approach. They retain the linear reserve margin constraint of the type found in equation (2); however, they define the variable Ci as the effective capability of a generating unit of type i, which is based upon estimating the relation: G = oq + (3i/PL
(4)
The coefficients c~i and ]~i are derived from detailed calculations of the effective load carrying capability 2 of a unit under a number of different expansion schemes. Noonan and Giglio 6 developed an explicit chance constraint for capacity expansion models using offline simulations similar to those used in the EPRI study. Their work is discussed in the following section.
III. Chance-constrained formulation of peak load reliability To capture the stochastic nature of outages, it is necessary to formulate the capacity expansion problem with a chance constraint. The general form of such a constraint is: probability (available capacity ~> peak load) >~ 1 -- LOLP
(5)
This constraint guarantees that the probability that available capacity (a random variable) exceeds peak load (another random variable) is at least (1 -- LOLP). This formulation recognizes that there is always a positive probability of peak load exceeding available capacity. The constraint restricts the consideration of expansion strategies to those for which loss-of-load events occur with a small probability. Constraint (5) is generally referred to in the mathematical programming literature as a chance constraint v'8. Such constraints are probability statements and, therefore, cannot be incorporated directly within the capacity planning model. The most common procedure to incorporate such a constraint is to represent it by its 'deterministic equivalent '7. The latter represents a linear or nonlinear inequality such that its feasible solution set coincides with
Electrical Power & Energy Systems
the set of expansion plans that satisfy the probabilistic chance constraint (5).
Density function of load and available capacity
c(.)
Convolution
More specifically, let x be an expansion strategy*. Then the chance constraint (5) can be stated as:
Density function of the margin
Prob(C x - L
~> 0)~> 1
-
-
(6)
LOLP D
Capacity, MW
=
where C x denotes available capacity corresponding to the expansion strategy x, L denotes peak load, and LOLPD is the desired LOLP. Constraint (6) can be rewritten as: (7)
Prob (M x -K
0
where the capacity margin Mx under expansion strategy x is defined as the random variable ( C x - L). The chance constraint (7) can now be written as: FMx(0) ~< LOLPD
Capacity, MW
g
FMx(V)=f
(8)
fMx(t)d t
-Go
where FMx denotes the cumulative distribution function for the random variable Mx, i.e. FMx(t) = P(M x <. t). Assuming that the cumulative distribution function is strictly increasing, constraint (8) is satisfied if and only if:
oLP\ FMIx(LOLPD) ~> 0
(9)
That is, the chance constraint (5) is satisfied by the expansion alternative x if and only if the inverse of the cumulative distribution function of the corresponding margin random variable evaluated at LOLPD is nonnegative. Constraint (9) is referred to as the deterministic equivalent and replaces the chance constraint (5) in a mathematical programming model. Figure 1 depicts the reasoning used in translating the chance constraint to its deterministic equivalent. The convolution of the load and available capacity density functions, h (.) and g(-), yield the density function of the margin f ( . ) . The area in the left tail of this density function is the actual system LOLP. The interpretation of the term LOLP is identical to that used in the Booth-Baleriaux context. In fact, the Booth-Baleriaux 9'1°, procedure can be considered as an easily programmable recursive solution technique to compute the system LOLP as the area fo f ( m ) dm in the left tail• Finally, it is noteworthy that in Figure 1, the chance constraint is violated as FM~x(LOLPD) IS less than zero. •
.
,
-1
.
From a practical standpoint, to specify the deterministic equivalent, one must first find the distribution of the margin M x and then invert it. The latter task is not straightforward and the former task can be more formidable, if not practically impossible, in most situations• One relatively simple case is when Mx has a normal distribution. In this case, the constraint (9) reduces to*:
EMx +
SMx [FMlx(1 -- LOLPD) ~< 01
(10)
where EMx = expected value of the margin ( C x - - L ) random * The components of the vector x specify the number of different types of generating units * More generally, this is true if the distribution of the random variable (Cx -- L) belongs to the class of all stable distributions 8
Vol 6 No 4 October 1984
~
F~xl(LC)LPD)
LOLPo Capacity{Mx), MW
Figure 1. Cumulative distribution function of the margin
variable, SMx = standard deviation of (Cx -- Z), and FMlx(.) denotes the inverse of the cumulative distribution function of the density function of ( C x - L), i.e. it is the fractile of order (1 -- LOLP) of the standardized variable of (Cx --L). There are several drawbacks to using a constraint based on a normality assumption. First, use of constraint (10) destroys the linearity of the optimization model. |t is relatively straightforward to show that the quadratic form on the left-hand side of (10) is concave, and, therefore, the feasible set for this constraint is convex. This property would permit the use of specialized nonlinear programming algorithms• Nevertheless, the ability to use widely available commercial LP packages to solve large problems is compromised because of the nonlinearity• A more important consideration is that although it is not unreasonable to assume that peak load L is normally distributed, the distribution of C x is generally difficult to assess in closed form and is not very close to being normal. C x is the convolution of the available capacity of each unit being considered in the expansion strategy x. In particular, under the expansion strategy x, if Yi denotes the available capacity of unit of type i, and n i denotes the number of units of type i, then C x = ~iniYi . Let Pi denote the probability that unit i is available; then the law of large numbers, the Lindberg-Feller theorem, suggests that, at least for large power systems, the normal distribution with mean Y~iniPiYi might provide an adequate representation for the random variable C x. However, such approximations are not sufficiently precise for reliability calculations in power system planning TM 12.
241
It is unlikely that the distribution of C x can be obtained in closed form for realistic cases. Thus, the task of determining tile distribution o f M x = Cx -- L becomes even more formidable. If the distribution for M x can be determined in closed form, it is still necessary to invert this distribution. Finally, even if the distribution o f M x can be inverted by a numerical procedure, this procedure must be invoked for all feasible expansion strategies. Thus, such a numerical procedure does not lend itself to a mathematical programming fornmlation where an explicit constraint is desired that is a well defined function of x and LOLP*.
where N(z)=lt/2rr)e
::/2
G1 = K s / K ~ ~2
G2 =: K4/I< 2 K1 = ,~, n i P i M W i -- IlL K2 = "~" n i P i M W 2 ( l -- Pi) + 0~, K3 = Z n i P i M W i 3 ( 1 -- 3pi + 2p 2)
Noonan and Giglio 6 recognized some of these problems. Nevertheless. they assumed that the random variable Cx is normally distributed. To overcome the problem of obtain. . . . -I mg a closed form for the reverted function b e y r ( ' ) , they developed a polynomial function based upon o~fline simulations. Whereas it is not immediately clear from their papers ]low this function was estimated, it seems that a number of calculations were carried out in which different combinations of unit additions, i.e. strategies x, are made to the existing system and the inverse function Fclx ];(LOLP) is calculated explicitly. Next, a regression analysis was carried out to estimate Fclx_7.(LOLP) as a polynomial function. This off line simulation and regression procedure appears to be very similar to that used in tile EPRI model 4 and, therefore, shares with it the characteristic that a considerable amount of preprocessing must be undertaken to set up tile optimization model for solution. Finally, it should be noted that the earliest use of the probabilistic reliability constraint (5) seems to be by Masse TM 14 and Morlat ~s. They refer to such constraints as 'guarantee conditions" and formulated them in the context of ensuring peak reliability as well as the ability of a hydrosystem to provide sufficient energy. However, most of their original papers are in French and it is not clear from the few papers in English (e.g. see References 14 and 15), and from the review paper by Anderson 1, if and how such constraints were ultimately incorporated in a numerical solution procedure.
IV. A deterministic equivalent chance LOLP constraint This section describes tile theoretical basis for developing an effective linearly approximate deterministic equivalent to the peak load chance constraint formulated in Section Ill. The approach here is based on the Gram-Charlier expansion of an arbitrary density function in terms of normal density expansion. Specifically, let f ( z ) denote the density function of the margin random variable M. Then the first four terms of the infinite Gram-Charlier series representation o f f ( z ) is:
K4 = ~ , / l i P i M W i ( l -- 7pi + 12 2 -- 6p 3) M W i = operating capability of a unit type i It i
=
number of generating units of type i
Pi = probability that a unit of type i is 'down', i.e.
the forced outage rate of unit i /aL = expected peak load o~ = variance of peak load Here K i is the ith cunmlant of tile density function ~t tile margin, f l z ) , i = 1. . . . . 4 and N O ) ( z ) is the ith derivative of N(z).
The Gram-Charlier expansion is an asymptotic expansion. that is, in the limit, as the number of terms approaches infinity the value of the series expansion approaches the value of the density function. However, accuracy is not necessarily improved by the addition of a finite number of terms. In practice, a few terms normally provide a good approximation to a density function. (See Cramer ~7 or Kendall and Stuart 18 for a detailed discussion.) Tile Gram-Charlier expansion has been used by Stremel and Rau ~1, among others, for calculating LOLP when peak load is not a random variable. Their results show the first four terms of the asymptotic Gram-Charlier expansion provide a reasonably accurate basis for computing LOLP, except for systems with few generating units. Furthermore the advantage of this cunmlant approach for calculating LOLP is that it is computationally faster than classical approaches, such as the Calabrese method 19, without a significant loss in accuracy. The Gram-Charlier expansion in equation ( 11 ) is a nonlinear function of unit size and, therefore, cannot be directly incorporated into an LP model of capacity planning. However, an effective linear approximation to this expansion can be developed. To derive this constraint, define X1, X > . . . , X N as the total amounts of capacity of type i, i.e. X i = n i M W i. The LOLP of the system can be written by integrating equation (11). In particular, it is evident that :
f ( z ) = N (z ) -- G,/3! N (3) (z) + G z/ 4 ! N (4) (z )
+ 10/6! G ~ N ( 6 ) ( z )
LOLP(X1, X2 . . . . . XN, L) = * Problems of this nature often severely limit the practical usefulness of mathematical optimization problems w i t h chance constraints. A good discussion of these and other problems in chance constrained programming is contained in Reference 16
242
Zl
Z l
(11)
j
f ( z ) dz =
f
N ( z ) dz
- G 1 / 3 ! N (z) ( z l ) + G 2 / 4 ! N (3) (z a) + 1 0/6! G 2N(s)(z1)
(12)
Electrical Power & Energy Systems
where
Z , = (PL -- ~ p i X i ) [(o~ + ~,piMWi(1 --Pi) Xi] -'/z N(2)(ZO = (Z] -- 1) N ( Z , ) N(a)(Z,) = (--Z 3 +
3Z,)N(Z,)
and N(S)(ZI) = ( - - Z s + 1 0 Z ~ -
15Z,)N(Z1)
Let the (N + 1)-tuple (X~ . . . . . XN, L) denote the total of capacity of type i, i = 1, 2 . . . . . N in year t, and L t the expected peak load in year t. By Taylor's theorem, the function LOLPt (X~, X~ . . . . . X } , L t) can be approximated by:
Experience has shown that this solution procedure produces good results and is computationally efficient, it requires less computer memory and much less execution time as each period's problem is an Nth the size of the full problem; solving the N smaller problems is much more efficient than solving the entire problem simultaneously. After solving the first time period problem, each successive single time period's problem is solved using the optimal basis from the previous period's solution. This 'hot basis start' greatly reduces the computational time for the N - - 1 single period problems for periods 2 through N. Such an algorithm, that also incorporates the linear constraint (14), was recently developed and used to solve a two-stage linear programming model under uncertainty for strategic capacity expansion planning using data front the New England Power Pool (N EPool)20' 21
L O L p t ( x { , . . . , X~r, L t) LOLP t
I ( X { - 1 . . . . . X~V-I,L t - l )
+ E 0 LOLP t - 1/aXi(Xt -- X I -1) i
+ 0 LOLpt-1/OL(L t - L
t-i)
(13)
Equation (13) provides the theoretical basis for the deterministic equivalent constraint in the capacity planning model. Specifically, the deterministic equivalent constraint can be written as the following linear constraint: L O L p t - l ( x l t - 3. . . . . X~v-3, Lt
+ ~ 0 LoLpt-1/OXi(X [-X
1)
V. Numerical accuracy of the linear approximation
t-l)
+ 0 LOLP/OL(L t - - L t - l ) <~ LOLPD
(14)
where LOLP D is the user-specified loss-of-load probability, and the partial derivatives 0 LOLP t - 1laX i and 0 LOLP t - 1/OL are evaluated at the point (X t - ]. . . . . X~v- 3, L t - 1). Expressions for these derivatives are given in the Appendix. This methodology can be easily extended to generate quadratic constraints or, more generally, polynomial constraints of higher order. Such constraints could be incorporated in nonlinear programming models for capacity expansion. Based on the good results obtained with a linear constraint, such extensions would be expected to produce excellent results, particularly in dynamic implementations. Constraint (14) is a linear constraint for ensuring a prespecified LOLP and can be incorporated, therefore, in a mathematical programming formulation of the capacity planning problem. Its use is well suited for the 'forward bootstrapping'-type solution procedure that is often used to solve capacity planning models. Sometimes, for computational reasons, the time periods in large capacity planning models are not solved simultaneously but are solved a single year at a time and in a 'forward pass' mode. That is, for a problem with an N-period planning horizon, N single period problems are solved sequentially in the natural order. The solution from period t -- 1 is 'wired' as an input for solving the next period problem. Next, the derivatives 0 LOLpt-1/ OX i and 0 LOLpt-x/OL are calculated using equation (12) and the equations in the Appendix. These calculations completely specify the coefficients of constraint (14) for the period t problem. This is what is meant by solving the optimization problem in a 'forward bootstrapping' mode.
Vol 6 No 4 October 1984
In Section V some computational experience is described that examines the approximation associated with using the linear constraint (1 4) for evaluating the LOLP implications of different generation expansion strategies. It is noteworthy that the Gram-Charlier-based linear approximate LOLP constraint can also be utilized in a dynamic LP formulation of the capacity expansion planning problem; i.e. the approximation developed in this paper can also be used if the capacity planning LP for an N-period problem is solved simultaneously for all N-periods. This process is explained in Section VI.
This section presents the results of numerical experiments that were carried out to test the accuracy of the approximation provided by constraint (14). Specifically, a series of calculations was carried out to compare the LOLP calculated using the left-hand side of constraint (14), the LOLP approximation and the 'actual' LOLP calculated using equation (12). These experiments were undertaken for a generic utility. Situations corresponding to low-, medium- and high-load growth were examined. The experiments were designed to determine the capacity additions needed to bring tile system to a target LOLP after a specified increase in peak load. The first four terms of the Gram-Charlier approximation shown in equation (12) were used to calculate the actual system LOLP. The algorithm used to calculate additions based on actual LOLP added the minimum number of units of a given size and forced outage rate to the base system until the new system LOLP was less than the target LOLP. The size of the last unit added was then adjusted to achieve the target LOLP. To illustrate, suppose 250 MW units are being added and I 105 MW of this type of capacity are needed to achieve the target LOLP. This means that when four 250 MW units and one unit of 105 MW, with the same forced outage rate as the 250 MW unit, are added to the system, the LOLP of the augmented system wilt be the target LOLP. These experiments were carried out using a database representing a 6 GW power system. The system consists of 80 units ranging in size from 25 to 500 MW (Table 1). The system peak load is 5 GW in the base year and the base LOLP is 0.019 28. The peak load increments considered are from 250 MW to 3 000 MW in 250 MW steps. The base system was augmented with three unit sizes, 20,250, and
243
Table 1. Base system unit characteristics Unit size, MW
Number of units
Forced outage rates
25 50 i00 250 500
24 32 18 4 2
0.02 0.02 0.05 0.05 0.10
Table 2. Additions based on approximate and actual LOLP calculations for 50 MW unit additions
Load increment, MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2 500
255.32 510.64 765.98 1 021.32 1 276.67 1 532.03 1 787.40 2 042.77 2 298.16 2 553.54
255.47 510.93 766.40 1 021.86 1 277.33 1 532.79 1 788.26 2 043.72 2 299.19 2 554.65
0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999
4 4 5 5 5 5 5 5 6 6
500 MW. In interpreting the results presented in this paper, it should be noted that it is not the absolute size of a unit under consideration that is important; rather, the size of the unit relative to other units and relative to system peak load determines the accuracy of the approximation. For example, if all unit sizes were doubled and the system peak load were 12 GW, the 100 MW unit would look like the 50 MW unit in the present system. Tables 2 - 4 show the LOLP calculated using the four-term Gram-Charlier approximation, the LOLP calculated using the linear approximation and the ratio o f these two values for the 50 MW unit, the 250 MW unit, and the 500 MW unit, respectively. For simplicity, the four-term GramCharlier approximation is referred to as the actual LOLP to distinguish it from the linear approximation to this nonlinear function. A tolerance of 10 -7 was used for these calculations. This data illustrates the effects of unit size and forced outage rate. Reserve margin constraints ignore forced outage rates. This results in a relatively small error for the 50 MW unit. However, for larger units the effect of unit size is quite pronounced; larger units provide considerably less reliability per MW than smaller units of the same forced outage rate. Comparing typical units in such a system, a 50 MW unit with a forced outage rate of 0.02 provides about 28% more reliability per MW than a 500 MW unit with a forced outage rate of 0.10. A comparison of Tables 2 - 4 shows that the linear approximation is quite good for small unit sizes with low forced
244
outage rates. Tile approximation is less accurate li)r large units of higher forced outage rates. However, there is a consistent bias, i.e. for a given unit type, the ratio of actual to approximate LOLP is nearly constant as load is increased. As a consequence, the approximate LOLP can be replaced by a constant (the actual/approximate ratio) times the approximate LOLP and have a linear approximation that is accurate across a large range of values. In practice, it is a simple matter to determine appropriate correction factors for each type of unit by carrying out a few offline calculations. It is important to note that this consistency of bias depends upon using the initial system LOLP as the target LOLP. Extensive experiments with various power systems differing markedly in size and mix have indicated that when the target LOLP is the initial LOLP, the ratio of actual to approximate capacity additions is almost constant over a wide range of load increments. This property does not, however, hold if target LOLP and initial LOLP differ significantly. The incremental capacity needed to return the system to the target LOLP after an incremental increase in load is
Table 3. Additions based on approximate and actual LOLP calculations for 250 MW unit additions
Load increment, MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2 500
270.59 541.50 812.74 1 084.31 1 356.19 1 628.37 1 900.83 2 173.57 2 446.58 2 719.85
273.05 546.11 819.16 1 092.22 1 365.77 1 638.32 1 911.38 2 184.43 2 457.58 2 730.54
0.991 0.991 0.992 0.992 0.993 0.993 0.994 0.995 0.995 0.996
0 1 2 8 3 9 5 0 6 1
Table 4. Additions based on approximate and actual LOLP calculations for 500 MW unit additions
Load increment° MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
500 750 1 000 t 250 1 500 1 750 2 000 2 250 2 500
653.27 1 008.32 1 307.50 1 640.59 I 982.66 2 328.06 2 603.38 2 939.63 3 280.25
727.81 1 091.72 1 455.62 1 819.53 2 183.48 2 547.34 2 911.24 3 275.15 3 639.05
0.897 0.923 0.898 0.901 0.908 0.913 0.894 0.897 0.901
6 5 2 7 0 9 3 6 4
Electrical Power & Energy Systems
almost constant. Thus, the curve describing capacity additions per load increment needed to maintain the target LOLP is almost linear. Tests made using larger unit sizes do not show the linearity described previously. For units in the 800-1000 MW range, the first couple of units added provide less reliability per MW than subsequent units. This is consistent with the results presented by Garver 2. However, the nonlinearity is not so serious as to preclude the use of this methodology. It should be noted that care must be taken when using a Gram-Charlier expansion when large units are present. This is discussed further in Section VII.
VI. Applications to dynamic mathematical programming models In this section, a method is described to incorporate the linear peak load guarantee constraint in a dynamic mathematical programming model. Numerical results are also presented that indicate that reasonably accurate results can be expected from such an implementation. In Section IV, the 'forward bootstrapping' solution method was described in which time periods are solved sequentially and the solution of the t -- 1st period's problem is used to initialize t TM period's problem. This allows the calculation of the partial derivatives 8 LOLP t - l/OX i at the solution ( X t - 1 . . . . . Xfv-1, L t - 1). In a dynamic framework, one peak load reliability constraint is required for each period and the coefficients must be calculated before the solution process unfolds. The methodology proposed is to calculate coefficients for each period offline. As the target LOLP is known for each period, if the system mix at the beginning of each period were known, then the coefficients could be calculated. This suggests that a system mix may be extrapolated for each period and coefficients based on this mix may be calculated. Experiments were carried out to assess the error that might be introduced by incorrectly extrapolating the system mix. Even a gross error in extrapolating the mix did not result in an unacceptably large error. Thus, the advantage of this type of constraint in making tradeoffs between unit sizes and forced outage rates makes this methodology attractive. To simulate the use of this methodology in a dynamic linear programming framework, it was assumed that the system described in Section V was the 1982 system, a 3% load growth was assumed, and three systems were constructed for the year ending in 1996. Thus, the 1997 system faces a peak load of 7.8 GW. For the best guess at a year-end 1996 system, a mix was chosen similar to the 1982 mix and sufficient capacity was added to achieve a 0.019 28 LOLP, as calculated by the four-term Gram-Charlier approximation. In addition, two other potential systems were created for year-end 1996: one by adding base load units only and the other by adding peaking units only. This was designed to test the extreme expansion plans that might be identified by a dynamic LP model. These different expansion plans resulted in different partial derivatives L O L p t / a x i and, hence, different coefficients in the constraint (14). Thus, these experiments were designed
Vol 6 No 4 October 1984
to evaluate the error expected if a year-by-year optimization strategy were replaced by a dynamic optimization framework. Table 5 shows the types and numbers of units added in each case. Three experiments similar to those described in Section V were carried out to assess the relative error that may be found if coefficients, based on a base load/peaking unit mix of generation types similar to the 1982 mix, were used for the 1977 reliability constraint but only base load or only peaking units were added. Tables 6-8 show the capacity additions calculated using the linear constraint and the extrapolated unit mix and the values found using the actual LOLP and the extrapolated mix, base load additions only and peaking additions only for load increments of 2502 500 MW in 250 MW steps. The results show that the additions to the two alternative 1997 systems for each type of unit are not very different from the 1997 system with mixed additions. Thus, the methodology described in Section V will give good results with an extrapolated 1997 system, even if the actual 1997 system is quite different from the extrapolated system.
Table 5. Additions to the base system between 1982 and 1996
Base/peaker mix Number of units 25 2O 2 2 1
Size, MW 25 50 250 500 0.54*
3 125.54 MW
Base
Peaker
Number of units
Size, MW
Number of units
1 5 4 1
40 100 37 250 500 0.95* 1 3 350.95 MW
Size, MW 25 50
0.93* 2 850.93 MW
* These units were added to achieve LOLPs of 0 . 0 1 9 2 8 . They were assumed to have a forced outage rate o f 0.02
Table 6. Capacity additions for 1997 - 50 MW units with forced outage rate 0.02
Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ peaker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2500
255.52 511.04 766.56 1 022.09 1 277.62 1 533.16 1 788.70 2 044.24 2 299.78 2555.31
255.53 511.07 766.60 1 022.15 1 277.69 1 533.23 1 788.78 2 044.33 2 299.88 2555.42
255.38 510.77 766.17 1 021.57 1 276.99 1 532.41 1 787.84 2 043.27 2 298.72 2554.16
255.92 511.84 767.76 1 023.68 1 279.60 1 535.51 1 791.43 2 047.35 2 303.37 2559.19
245
Table 7. Capacity additions for 1997 - 250 MW units with forced outage rate 0.05 Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ peaker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2500
270.34 540.80 811.38 1 082.08 t 352.80 1 623.85 1 894.91 2 166.09 2 437.41 2711.40
269.75 539.55 809.42 I 079.35 1 349.34 1 619.41 1 889.54 2 159.76 2 430.05 2700.42
271.05 532.38 814.02 1 085.90 1 358.06 1 630.48 1 903.15 2 176.07 2 449.22 2722.62
276.34 552.67 829.01 1 105.34 1 381.68 1 658.01 1 934.35 2 210.69 2 487.02 2763.26
Table 8. Capacity additions for 1997 - 500 MW unit with forced outage rate 0.10 Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ perker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
500 750 1 000 1 250 1 500 1 750 2000 2 250 2 500
631.70 973.72 1 265.58 1 593.53 1 929.77 2 270.29 2 541.00 2 874.24 3 153.66
620.11 938.37 1 242.51 1 565.52 1 869.12 2 178.56 2 501.65 2 794.39 3 107.57
654.53 1 009.68 1 309.73 ! 642.49 1 984.54 2 329.79 2 605.24 2 941.31 3 228.20
721.25 1 081.88 1 442.51 1 803.13 2 163.76 2 524.39 2 885.02 3 245.64 3 606.27
The results reported here and other experiments that have been carried out suggest the following general procedure to be effective in generating reliability (LOLP) constraints for each period in an N-period optimization model for capacity expansion planning that is solved simultaneously (dynamically) for all N-periods. Specifically, the linear constraint for period t is estimated using the following two-step procedure: (1) Project expected mix and capacity at the end of period t - I for t = 2 , . . . ,N*. Then calculate O L O L p t - I / 3 x i for t = 1. . . . . N - - 1 and b LOLP/OL using equation (12) (2) The LOLP constraint for year t is L O L p t - 1 + ~_ O L O L p t - 1 / O X i ( X it _ x i t - 1 )
i + / ) LOLP t - 1/OL (L t -- L t - 1) <~ LOLP/~ where LOLP t is the estimated actual loss-of-load probability in year t under the provided mix, and LOLP~ is the desired LOLP in year t.
246
VII.
Conclusions
This paper demonstrates the effectiveness of a methodolog? for developing explicit linear LOLP constraints to represent system reliability in optilnization models of capacity expansion planning. Such a procedure is more effective than using simple reserve margin constraints that do not help an optimization model sort out different units of different sizes, and forced LOLP constraints considerably enhance the ability (/t any resource mix optimization model, linear or nonlinear, to represent system peak reliability better than has been the case so far. In recent years, there have been a number of studies of the accuracy of expansions representing LOLP. The two types of commonly used expansions are Edgeworth and GramCharlier expansions. Levy and Kahn = have shown that tile Edgeworth expansion can lead to numerical inaccuracies when modelling small systems the forced outage rates of which sum to less than 1. A study by Schenk 2a has shown numerical inaccuracies with the Gram-Charlier expansion applied to an 1 800 MW system. However, TVA and several other organizations have successfully used Gram-Charlier 24 expansions for modelling reliability and production costing. Results are generally better for large systems with large numbers of units. This is because, in such cases, there will be only moderate skewness and consequently these expansions are more accurate. 18 The use of LOLP as the basis for developing a reliability constraint does not imply that it is the preferred indicator of system reliability. A discussion of the strengths and drawbacks of LOLP and other reliability indicators is beyond the scope of this paper. However, in spite of its drawbacks, LOLP is a better indicator of system reliability than system reserve margin.
VIII.
References
1 Anderson, D 'Models for determining least cost investments in electricity supply' Bell J. Econ. Manage. Sci. Vol 3 (Spring 1972) pp 267-301 Garver, L L 'Effective load carrying capability of generating units' IEEE Trans. Power Appar. & Syst. Vol 85 No 8 (August 1966) pp 910-919
3 Scherer, C R and Joe, L 'Electric power system planning with explicit stochastic reserves constraint' Manage. ScL Vol 23 No 9 (May 1977) pp 978-985 Electric Power Research Institute 'A generation planning system methodology and case study' Final report
EA-1807 submitted by Gordian Associates (August 1981) Cote, G and Laughton, M A 'Prediction of reserve requirements in generation planning' Int. J. E/ectr.
Power& Energ. Syst. Vol 2 (April 1980) pp 87-95
* Generally, this is easy for the first ten years of the planning horizon, as the base capacity expansion plan is known
Electrical Power & Energy Systems
Noonan, F and Giglio, R J 'Planning electric power generation: a non-linear mixed integer model employing Benders Decomposition' Manage. Sci. Vol 23 (1977) pp 946-956 Charnes, A and Cooper, W W 'Deterministic equivalents for optimizing and satisfying under chance constraints' Oper. Res. Vol 11 (1963) pp 18-39 Vajda, S 'Stochastic programming' in Abadie (Ed) Integer and nonlinear programming American Elsevier, USA (1970)
Baleriaux, H, Jamoulle, E and Fr Linard de Guertechnung 'Simulation de I'exploitation d'un parc de production d'electricite couple a des sations de pompage' ReviewE Vol 7 No 7 (1967) pp 3-24 10 Booth, R R 'Power system simulation model based upon probability analysis' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 62-69 11 Stremel, J P and Rau, N S 'The cumular~t method of calculating LOLP' IEEE Paper PES Summer Meeting, Portland, USA (July 1981 )
21 Sanghvi, A P, Shavel, I and Spann, R M 'Strategic planning for power system vulnerability: an optimization model for resource planning under uncertainty' IEEE Trans. Power Appar. & Syst. Vol PAS-101 (June 1982) pp 1420-1429 22
Levy, D J and Kahn, E P 'Accuracy of the Edgeworth approximation for LOLP calculations in small power systems' Proc. IEEE PES Summer Meeting Portland, USA (July 1981)
23 Schenk, K F 'Addendum to report analysis of reliability criteria for generation planning' National Energy Board, Canada (1979) 24 Stremel, J P 'Production costing for large-range generation expansion planning studies' Proc. IEEE PES Summer Meeting Portland, USA (July 1981)
Appendix This appendix presents the partial derivatives of LOLP that are needed to generate constraint (14). 0 L O L P / 3 X i = OZ1/3X i [N(Z0 -- G x / 3 ! N s ( Z , )
+ G2/4!N4(ZO + 10/6! G]N6(ZO]
12 Allan, R N and Takieddine, F N 'Generation modelling in power system reliability evaluation' Proc. IEEE Conf. Reliability of Power Supply Systems London, U K (February 1977) 13 Masse, P Les reserves et regulation de I'avenir dans la vie economique Hermourn, France (1946)
14 Masse, P 'Electrical investments' in Nelson (Ed) Marginal cost pricing in practice Prentice-Hall, USA (1964)
+ aG,/OX i [--N2(ZO/3! + I O / 6 ! N s ( Z , ) 2G,] + OG2/OX i [Ns(Z1)/4!]
(])
where OZ,/3X i = -- [p,o2L + ZnlpiMW~(1 --Pi)l -'j2
-- (l/2) (L -- Y.nipiMWi) (piMWi(l -- Pi)) 15 Morlat, G 'On instructions for the optimal management of seasonal reservoirs' in Nelson (Ed) Marginal cost pricing in practice Prentice-Hall, USA (1964)
x [02L + Y.niPiMW~(l --pi)] -3/2 OG1/OXi = piMW2(1 -- 3pi + 2p~)
16 Hogan, A, Morris, J and Thompson, H 'Decision
x [O2L+ T.niPiMW](1--pi)] -3/2
problems under risk and chance constrained programming: dilemmas in the transition' Manage. Sci. Vol 27 (1981) pp 698-716 17 Cramer, H Mathematical methods of statistics Princeton University Press, USA (1946)
-- 3/2 [Y, niMW~(1 -- 3pi + 2p~)] x [piMWi(1 --p/)] [o[ + 2niPiMW](1 _pi)]-s/2 OG2/OX i = PiMW3(l -- 7pi + 12p~-- 6p~) x [a2L + YmiPiMW{(1 -- Pi)l -z
18 Kendall, M G and Stuart, A The advanced theory of statistics Hafner, USA (1969)
-- 2 [~niPiMW~ (l -- 7Pi + 12Pi2 _ 6p i3)] X [PiMWi(l--Pi)] [02L+ P'niPiMW?(I --Pi)] -3
19 Billinton, R Power system reliability evaluation Gordon and Breach Science Publishers, USA (1970) 20 Sanghvi, A P, Shavel, I and Spann, R M 'Fuel diversification, stockpiling and system interconnection strategies for minimizing power system vulnerability to energy shortages' Institute of Management Science (TIMS) publication on Energy models and special studies, North-Holland, Netherlands (to be published)
Vol 6 No 4 October 1984
~)LOLPfi)L = N (Zi) OZx/OL -- G1/3 ! N(a)(Z ,) OZ ,/OL
+ G2/4!N(4)(Z,) aZ1/OL + I O I 6 ! G ~ N ( 6 ) (Z,) aZ,/OL
(2)
where OZ,/~L = [o2 + Z n i p i M W ~ ( 1 -- pi)]-1/2
247
I H Shavel ICF I n c o r p o r a t e d , 1 8 5 0 K S t r e e t , N.W., Suite 9 5 0 , W a s h i n g t o n , DC 2 0 0 0 6 , U S A
Current methodologies for incorporating reliability constraints in optimization models for capacity expansion planning are reviewed. In addition, a chance reliability constraint that can be incorporated into a mathematical programming model as a linear constraint is developed. Reliability is measured by loss-of-load probability. The constraint is based on a first-order approximation to the Gram-Charlier series representing the density function o f available capacity minus load. Numerical results showing good agreement between the linear approximation and actual loss-of-load probability are presented. Keywords: rel&bility, generation expansion, planning reliability calculations
I.
Introduction
This paper develops a linear constraint that can be incorporated explicitly into a linear programming (LP) formulation of an electric utility's capacity expansion planning problem. The constraint ensures that the optimal generation capacity expansion plan determined by the LP model provides a level of peak load reliability, measured by system loss-of-load probability (LOLP), that is equal to or better than a user-specified value. This methodology was used successfully in a capacity planning model that was developed to analyse certain strategic planning issues faced by the New England Power Pool (NEPool). II.
Literature review
An electric utility's generation capacity planning problem can be modelled as a mathematical optimization problem. Anderson 1 reviews a variety of such models, including Received: 14 December 1982
Vol 6 No 4 October 1984
0142-0615/84/040239-09
dynamic programming, linear programming (LP) and nonlinear programming formulations*. A typical formulation can be stated generally as: min [cl x + c z f ( x ) + csg(x)]
(i)
Ax = b
x~>O where x is a vector of decision variables representing an expansion strategy. The components of x represent the amounts of new generating capacity of each type to be added in each year of the horizon. In a formulation of broader scope, x can, for example, also include the incremental amounts of inertia capacity built in each year of the horizon, coal conversion activities, the amount of a certain type of load management installed and fuel stockpile decisions. The components of c 1 are unit capacity costs associated with the components of the vector x, and e2 denotes a * Such optimizations of models are particularly well-suited for strategic resource planning. They are able to solve large problems with relative ease, taking into account key factors such as uncertainty in load growth, uncertainty in water conditions (for hydroelectric generation) and the availability of a large number of resource options including demand-side alternatives. In particular, they are able to evaluate implicitly all feasible resource plans (mix, level and staging). They serve a complementary role with detailed production costing, reliability evaluation and financial analysis models, which by their very nature are more precise but cumbersome to solve and, therefore, usually restricted to the analysis of a few resource plans. In particular, a strategic planning model helps to characterize the general nature of the optimal resource plan. This information can then be used to structure a few resource plans with such characteristics. These plans are then subject to a 'finer tuned' analysis using detailed simulation and other models to arrive at the final selection.
$ 0 3 . 0 0 © 1 9 8 4 B u t t e r w o r t h & Co (Publishers) L t d
239
vector of unit fuel cost for each type of fuel used by tile power system. A vector of consumptions of each fuel type given the expansion plan x is denoted by f ( x ) and g(x) denotes a vector the components of which are the total generation by each plant type for the expansion strategy x. A vector of units' operating and maintenance (O&M) costs for each type of plant is denoted by ca. In a linear progranmring formulation, f(x) and g(x) are linear functions of x, A, and b. The constraints Ax = b ensure that in each time period the capacity expansion plan should meet projected load (both power and energy) and that no generating unit is loaded beyond its maximum availability. In addition, the model is likely to include a reserve margin constraint which requires that the "total available capacity at system peak in each period exceeds the system-peak load by a user-specified percentage, typically 20-25%. Such a constraint is of the type:
}~A ViCi >~PL(1 + RM)
(2)
i
where PL = peak load, A Vi = expected availability of unit i at peak, Ci = capacity of unit i, RM = reserve margin. The provision of a certain reserve margin, however, does not indicate anything specific about the peak reliability of a power system. Specifically, two capacity plans that provide the same reserve margin can have reliability implications, as measured by LOLP or other measures, that are markedly different because system reliability is a nonlinear function of key unit characteristics such as generating unit sizes, forced outage rates, maintenance requirements, etc. Generally, larger units and units with higher forced outage rates contribute less per megawatt to power system reliability than smaller units or units with lower forced outage rates:. Three other methodological approaches have been suggested for a more realistic and explicit incorporation of LOLP constraints in capacity planning models. Scherer and Joe 3 use binary integer variables to express system reliability explicitly as a constraint. A problem with their approach is that even their seven-plant problem example leads to 135 integer variables. For any realistic system, this would imply a problem that is too large, if not extremely cumbersome, for available mixed integer programming codes. To alleviate this situation, they use a heuristic procedure to prune the number of branches that need to be searched by a branchand-bound algorithm. Nevertheless, the use of such a procedure is likely to be computationally burdensome, especially if such a constraint were to be used in a more realistic formulation of the capacity expansion planning and dispatching problem than is contained in their paper. A recent model funded by EPRP contains a linear relation between the reserve margin and system LOLP, as a function of generating unit characteristics. First, potential candidates for unit additions are identified. These candidates are characterized by their unit size (MW), forced outage rate (%) and annual maintenance requirement (h). Given next year's peak load, this year's base system and the desired LOLP for next year, an iterative search procedure is used to estimate the total megawatt additions required if each candidate unit is used solely to augment the existing
240
generating mix. Tile effective load carrying capability-' ~I each potential candidate is determined by using a NewtonRaphson search procedure on the tail segment of the equivalent load curve obtained by applying the BoothBaleriaux probabilistic simulation procedure to all the units of the existing system. Each such simulation yields the reserve margin that would be required to meet next year's load with a desired LOLP if the given candidate unit were to be used solely. Based upon a large number of such experiments, a multivariate regression is carried out to estimate the following equation: RM i = c~+ ~(unit size/peak load) + 3,(forced outage rate) + 8 (maintenance downtime)
(3)
where RM i is the reserve margin that would be required if the desired LOLP is to be achieved by augmenting the existing system solely by units of type i; and a,/5, 7, 6, are the regression coefficients. Cote and Laughton s use a similar approach. They retain the linear reserve margin constraint of the type found in equation (2); however, they define the variable Ci as the effective capability of a generating unit of type i, which is based upon estimating the relation: G = oq + (3i/PL
(4)
The coefficients c~i and ]~i are derived from detailed calculations of the effective load carrying capability 2 of a unit under a number of different expansion schemes. Noonan and Giglio 6 developed an explicit chance constraint for capacity expansion models using offline simulations similar to those used in the EPRI study. Their work is discussed in the following section.
III. Chance-constrained formulation of peak load reliability To capture the stochastic nature of outages, it is necessary to formulate the capacity expansion problem with a chance constraint. The general form of such a constraint is: probability (available capacity ~> peak load) >~ 1 -- LOLP
(5)
This constraint guarantees that the probability that available capacity (a random variable) exceeds peak load (another random variable) is at least (1 -- LOLP). This formulation recognizes that there is always a positive probability of peak load exceeding available capacity. The constraint restricts the consideration of expansion strategies to those for which loss-of-load events occur with a small probability. Constraint (5) is generally referred to in the mathematical programming literature as a chance constraint v'8. Such constraints are probability statements and, therefore, cannot be incorporated directly within the capacity planning model. The most common procedure to incorporate such a constraint is to represent it by its 'deterministic equivalent '7. The latter represents a linear or nonlinear inequality such that its feasible solution set coincides with
Electrical Power & Energy Systems
the set of expansion plans that satisfy the probabilistic chance constraint (5).
Density function of load and available capacity
c(.)
Convolution
More specifically, let x be an expansion strategy*. Then the chance constraint (5) can be stated as:
Density function of the margin
Prob(C x - L
~> 0)~> 1
-
-
(6)
LOLP D
Capacity, MW
=
where C x denotes available capacity corresponding to the expansion strategy x, L denotes peak load, and LOLPD is the desired LOLP. Constraint (6) can be rewritten as: (7)
Prob (M x -K
0
where the capacity margin Mx under expansion strategy x is defined as the random variable ( C x - L). The chance constraint (7) can now be written as: FMx(0) ~< LOLPD
Capacity, MW
g
FMx(V)=f
(8)
fMx(t)d t
-Go
where FMx denotes the cumulative distribution function for the random variable Mx, i.e. FMx(t) = P(M x <. t). Assuming that the cumulative distribution function is strictly increasing, constraint (8) is satisfied if and only if:
oLP\ FMIx(LOLPD) ~> 0
(9)
That is, the chance constraint (5) is satisfied by the expansion alternative x if and only if the inverse of the cumulative distribution function of the corresponding margin random variable evaluated at LOLPD is nonnegative. Constraint (9) is referred to as the deterministic equivalent and replaces the chance constraint (5) in a mathematical programming model. Figure 1 depicts the reasoning used in translating the chance constraint to its deterministic equivalent. The convolution of the load and available capacity density functions, h (.) and g(-), yield the density function of the margin f ( . ) . The area in the left tail of this density function is the actual system LOLP. The interpretation of the term LOLP is identical to that used in the Booth-Baleriaux context. In fact, the Booth-Baleriaux 9'1°, procedure can be considered as an easily programmable recursive solution technique to compute the system LOLP as the area fo f ( m ) dm in the left tail• Finally, it is noteworthy that in Figure 1, the chance constraint is violated as FM~x(LOLPD) IS less than zero. •
.
,
-1
.
From a practical standpoint, to specify the deterministic equivalent, one must first find the distribution of the margin M x and then invert it. The latter task is not straightforward and the former task can be more formidable, if not practically impossible, in most situations• One relatively simple case is when Mx has a normal distribution. In this case, the constraint (9) reduces to*:
EMx +
SMx [FMlx(1 -- LOLPD) ~< 01
(10)
where EMx = expected value of the margin ( C x - - L ) random * The components of the vector x specify the number of different types of generating units * More generally, this is true if the distribution of the random variable (Cx -- L) belongs to the class of all stable distributions 8
Vol 6 No 4 October 1984
~
F~xl(LC)LPD)
LOLPo Capacity{Mx), MW
Figure 1. Cumulative distribution function of the margin
variable, SMx = standard deviation of (Cx -- Z), and FMlx(.) denotes the inverse of the cumulative distribution function of the density function of ( C x - L), i.e. it is the fractile of order (1 -- LOLP) of the standardized variable of (Cx --L). There are several drawbacks to using a constraint based on a normality assumption. First, use of constraint (10) destroys the linearity of the optimization model. |t is relatively straightforward to show that the quadratic form on the left-hand side of (10) is concave, and, therefore, the feasible set for this constraint is convex. This property would permit the use of specialized nonlinear programming algorithms• Nevertheless, the ability to use widely available commercial LP packages to solve large problems is compromised because of the nonlinearity• A more important consideration is that although it is not unreasonable to assume that peak load L is normally distributed, the distribution of C x is generally difficult to assess in closed form and is not very close to being normal. C x is the convolution of the available capacity of each unit being considered in the expansion strategy x. In particular, under the expansion strategy x, if Yi denotes the available capacity of unit of type i, and n i denotes the number of units of type i, then C x = ~iniYi . Let Pi denote the probability that unit i is available; then the law of large numbers, the Lindberg-Feller theorem, suggests that, at least for large power systems, the normal distribution with mean Y~iniPiYi might provide an adequate representation for the random variable C x. However, such approximations are not sufficiently precise for reliability calculations in power system planning TM 12.
241
It is unlikely that the distribution of C x can be obtained in closed form for realistic cases. Thus, the task of determining tile distribution o f M x = Cx -- L becomes even more formidable. If the distribution for M x can be determined in closed form, it is still necessary to invert this distribution. Finally, even if the distribution o f M x can be inverted by a numerical procedure, this procedure must be invoked for all feasible expansion strategies. Thus, such a numerical procedure does not lend itself to a mathematical programming fornmlation where an explicit constraint is desired that is a well defined function of x and LOLP*.
where N(z)=lt/2rr)e
::/2
G1 = K s / K ~ ~2
G2 =: K4/I< 2 K1 = ,~, n i P i M W i -- IlL K2 = "~" n i P i M W 2 ( l -- Pi) + 0~, K3 = Z n i P i M W i 3 ( 1 -- 3pi + 2p 2)
Noonan and Giglio 6 recognized some of these problems. Nevertheless. they assumed that the random variable Cx is normally distributed. To overcome the problem of obtain. . . . -I mg a closed form for the reverted function b e y r ( ' ) , they developed a polynomial function based upon o~fline simulations. Whereas it is not immediately clear from their papers ]low this function was estimated, it seems that a number of calculations were carried out in which different combinations of unit additions, i.e. strategies x, are made to the existing system and the inverse function Fclx ];(LOLP) is calculated explicitly. Next, a regression analysis was carried out to estimate Fclx_7.(LOLP) as a polynomial function. This off line simulation and regression procedure appears to be very similar to that used in tile EPRI model 4 and, therefore, shares with it the characteristic that a considerable amount of preprocessing must be undertaken to set up tile optimization model for solution. Finally, it should be noted that the earliest use of the probabilistic reliability constraint (5) seems to be by Masse TM 14 and Morlat ~s. They refer to such constraints as 'guarantee conditions" and formulated them in the context of ensuring peak reliability as well as the ability of a hydrosystem to provide sufficient energy. However, most of their original papers are in French and it is not clear from the few papers in English (e.g. see References 14 and 15), and from the review paper by Anderson 1, if and how such constraints were ultimately incorporated in a numerical solution procedure.
IV. A deterministic equivalent chance LOLP constraint This section describes tile theoretical basis for developing an effective linearly approximate deterministic equivalent to the peak load chance constraint formulated in Section Ill. The approach here is based on the Gram-Charlier expansion of an arbitrary density function in terms of normal density expansion. Specifically, let f ( z ) denote the density function of the margin random variable M. Then the first four terms of the infinite Gram-Charlier series representation o f f ( z ) is:
K4 = ~ , / l i P i M W i ( l -- 7pi + 12 2 -- 6p 3) M W i = operating capability of a unit type i It i
=
number of generating units of type i
Pi = probability that a unit of type i is 'down', i.e.
the forced outage rate of unit i /aL = expected peak load o~ = variance of peak load Here K i is the ith cunmlant of tile density function ~t tile margin, f l z ) , i = 1. . . . . 4 and N O ) ( z ) is the ith derivative of N(z).
The Gram-Charlier expansion is an asymptotic expansion. that is, in the limit, as the number of terms approaches infinity the value of the series expansion approaches the value of the density function. However, accuracy is not necessarily improved by the addition of a finite number of terms. In practice, a few terms normally provide a good approximation to a density function. (See Cramer ~7 or Kendall and Stuart 18 for a detailed discussion.) Tile Gram-Charlier expansion has been used by Stremel and Rau ~1, among others, for calculating LOLP when peak load is not a random variable. Their results show the first four terms of the asymptotic Gram-Charlier expansion provide a reasonably accurate basis for computing LOLP, except for systems with few generating units. Furthermore the advantage of this cunmlant approach for calculating LOLP is that it is computationally faster than classical approaches, such as the Calabrese method 19, without a significant loss in accuracy. The Gram-Charlier expansion in equation ( 11 ) is a nonlinear function of unit size and, therefore, cannot be directly incorporated into an LP model of capacity planning. However, an effective linear approximation to this expansion can be developed. To derive this constraint, define X1, X > . . . , X N as the total amounts of capacity of type i, i.e. X i = n i M W i. The LOLP of the system can be written by integrating equation (11). In particular, it is evident that :
f ( z ) = N (z ) -- G,/3! N (3) (z) + G z/ 4 ! N (4) (z )
+ 10/6! G ~ N ( 6 ) ( z )
LOLP(X1, X2 . . . . . XN, L) = * Problems of this nature often severely limit the practical usefulness of mathematical optimization problems w i t h chance constraints. A good discussion of these and other problems in chance constrained programming is contained in Reference 16
242
Zl
Z l
(11)
j
f ( z ) dz =
f
N ( z ) dz
- G 1 / 3 ! N (z) ( z l ) + G 2 / 4 ! N (3) (z a) + 1 0/6! G 2N(s)(z1)
(12)
Electrical Power & Energy Systems
where
Z , = (PL -- ~ p i X i ) [(o~ + ~,piMWi(1 --Pi) Xi] -'/z N(2)(ZO = (Z] -- 1) N ( Z , ) N(a)(Z,) = (--Z 3 +
3Z,)N(Z,)
and N(S)(ZI) = ( - - Z s + 1 0 Z ~ -
15Z,)N(Z1)
Let the (N + 1)-tuple (X~ . . . . . XN, L) denote the total of capacity of type i, i = 1, 2 . . . . . N in year t, and L t the expected peak load in year t. By Taylor's theorem, the function LOLPt (X~, X~ . . . . . X } , L t) can be approximated by:
Experience has shown that this solution procedure produces good results and is computationally efficient, it requires less computer memory and much less execution time as each period's problem is an Nth the size of the full problem; solving the N smaller problems is much more efficient than solving the entire problem simultaneously. After solving the first time period problem, each successive single time period's problem is solved using the optimal basis from the previous period's solution. This 'hot basis start' greatly reduces the computational time for the N - - 1 single period problems for periods 2 through N. Such an algorithm, that also incorporates the linear constraint (14), was recently developed and used to solve a two-stage linear programming model under uncertainty for strategic capacity expansion planning using data front the New England Power Pool (N EPool)20' 21
L O L p t ( x { , . . . , X~r, L t) LOLP t
I ( X { - 1 . . . . . X~V-I,L t - l )
+ E 0 LOLP t - 1/aXi(Xt -- X I -1) i
+ 0 LOLpt-1/OL(L t - L
t-i)
(13)
Equation (13) provides the theoretical basis for the deterministic equivalent constraint in the capacity planning model. Specifically, the deterministic equivalent constraint can be written as the following linear constraint: L O L p t - l ( x l t - 3. . . . . X~v-3, Lt
+ ~ 0 LoLpt-1/OXi(X [-X
1)
V. Numerical accuracy of the linear approximation
t-l)
+ 0 LOLP/OL(L t - - L t - l ) <~ LOLPD
(14)
where LOLP D is the user-specified loss-of-load probability, and the partial derivatives 0 LOLP t - 1laX i and 0 LOLP t - 1/OL are evaluated at the point (X t - ]. . . . . X~v- 3, L t - 1). Expressions for these derivatives are given in the Appendix. This methodology can be easily extended to generate quadratic constraints or, more generally, polynomial constraints of higher order. Such constraints could be incorporated in nonlinear programming models for capacity expansion. Based on the good results obtained with a linear constraint, such extensions would be expected to produce excellent results, particularly in dynamic implementations. Constraint (14) is a linear constraint for ensuring a prespecified LOLP and can be incorporated, therefore, in a mathematical programming formulation of the capacity planning problem. Its use is well suited for the 'forward bootstrapping'-type solution procedure that is often used to solve capacity planning models. Sometimes, for computational reasons, the time periods in large capacity planning models are not solved simultaneously but are solved a single year at a time and in a 'forward pass' mode. That is, for a problem with an N-period planning horizon, N single period problems are solved sequentially in the natural order. The solution from period t -- 1 is 'wired' as an input for solving the next period problem. Next, the derivatives 0 LOLpt-1/ OX i and 0 LOLpt-x/OL are calculated using equation (12) and the equations in the Appendix. These calculations completely specify the coefficients of constraint (14) for the period t problem. This is what is meant by solving the optimization problem in a 'forward bootstrapping' mode.
Vol 6 No 4 October 1984
In Section V some computational experience is described that examines the approximation associated with using the linear constraint (1 4) for evaluating the LOLP implications of different generation expansion strategies. It is noteworthy that the Gram-Charlier-based linear approximate LOLP constraint can also be utilized in a dynamic LP formulation of the capacity expansion planning problem; i.e. the approximation developed in this paper can also be used if the capacity planning LP for an N-period problem is solved simultaneously for all N-periods. This process is explained in Section VI.
This section presents the results of numerical experiments that were carried out to test the accuracy of the approximation provided by constraint (14). Specifically, a series of calculations was carried out to compare the LOLP calculated using the left-hand side of constraint (14), the LOLP approximation and the 'actual' LOLP calculated using equation (12). These experiments were undertaken for a generic utility. Situations corresponding to low-, medium- and high-load growth were examined. The experiments were designed to determine the capacity additions needed to bring tile system to a target LOLP after a specified increase in peak load. The first four terms of the Gram-Charlier approximation shown in equation (12) were used to calculate the actual system LOLP. The algorithm used to calculate additions based on actual LOLP added the minimum number of units of a given size and forced outage rate to the base system until the new system LOLP was less than the target LOLP. The size of the last unit added was then adjusted to achieve the target LOLP. To illustrate, suppose 250 MW units are being added and I 105 MW of this type of capacity are needed to achieve the target LOLP. This means that when four 250 MW units and one unit of 105 MW, with the same forced outage rate as the 250 MW unit, are added to the system, the LOLP of the augmented system wilt be the target LOLP. These experiments were carried out using a database representing a 6 GW power system. The system consists of 80 units ranging in size from 25 to 500 MW (Table 1). The system peak load is 5 GW in the base year and the base LOLP is 0.019 28. The peak load increments considered are from 250 MW to 3 000 MW in 250 MW steps. The base system was augmented with three unit sizes, 20,250, and
243
Table 1. Base system unit characteristics Unit size, MW
Number of units
Forced outage rates
25 50 i00 250 500
24 32 18 4 2
0.02 0.02 0.05 0.05 0.10
Table 2. Additions based on approximate and actual LOLP calculations for 50 MW unit additions
Load increment, MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2 500
255.32 510.64 765.98 1 021.32 1 276.67 1 532.03 1 787.40 2 042.77 2 298.16 2 553.54
255.47 510.93 766.40 1 021.86 1 277.33 1 532.79 1 788.26 2 043.72 2 299.19 2 554.65
0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999
4 4 5 5 5 5 5 5 6 6
500 MW. In interpreting the results presented in this paper, it should be noted that it is not the absolute size of a unit under consideration that is important; rather, the size of the unit relative to other units and relative to system peak load determines the accuracy of the approximation. For example, if all unit sizes were doubled and the system peak load were 12 GW, the 100 MW unit would look like the 50 MW unit in the present system. Tables 2 - 4 show the LOLP calculated using the four-term Gram-Charlier approximation, the LOLP calculated using the linear approximation and the ratio o f these two values for the 50 MW unit, the 250 MW unit, and the 500 MW unit, respectively. For simplicity, the four-term GramCharlier approximation is referred to as the actual LOLP to distinguish it from the linear approximation to this nonlinear function. A tolerance of 10 -7 was used for these calculations. This data illustrates the effects of unit size and forced outage rate. Reserve margin constraints ignore forced outage rates. This results in a relatively small error for the 50 MW unit. However, for larger units the effect of unit size is quite pronounced; larger units provide considerably less reliability per MW than smaller units of the same forced outage rate. Comparing typical units in such a system, a 50 MW unit with a forced outage rate of 0.02 provides about 28% more reliability per MW than a 500 MW unit with a forced outage rate of 0.10. A comparison of Tables 2 - 4 shows that the linear approximation is quite good for small unit sizes with low forced
244
outage rates. Tile approximation is less accurate li)r large units of higher forced outage rates. However, there is a consistent bias, i.e. for a given unit type, the ratio of actual to approximate LOLP is nearly constant as load is increased. As a consequence, the approximate LOLP can be replaced by a constant (the actual/approximate ratio) times the approximate LOLP and have a linear approximation that is accurate across a large range of values. In practice, it is a simple matter to determine appropriate correction factors for each type of unit by carrying out a few offline calculations. It is important to note that this consistency of bias depends upon using the initial system LOLP as the target LOLP. Extensive experiments with various power systems differing markedly in size and mix have indicated that when the target LOLP is the initial LOLP, the ratio of actual to approximate capacity additions is almost constant over a wide range of load increments. This property does not, however, hold if target LOLP and initial LOLP differ significantly. The incremental capacity needed to return the system to the target LOLP after an incremental increase in load is
Table 3. Additions based on approximate and actual LOLP calculations for 250 MW unit additions
Load increment, MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2 500
270.59 541.50 812.74 1 084.31 1 356.19 1 628.37 1 900.83 2 173.57 2 446.58 2 719.85
273.05 546.11 819.16 1 092.22 1 365.77 1 638.32 1 911.38 2 184.43 2 457.58 2 730.54
0.991 0.991 0.992 0.992 0.993 0.993 0.994 0.995 0.995 0.996
0 1 2 8 3 9 5 0 6 1
Table 4. Additions based on approximate and actual LOLP calculations for 500 MW unit additions
Load increment° MW
Additions based on actual LOLP calculations
Additions based on approximate LOLP calculations
Ratio (actual/ approximate)
500 750 1 000 t 250 1 500 1 750 2 000 2 250 2 500
653.27 1 008.32 1 307.50 1 640.59 I 982.66 2 328.06 2 603.38 2 939.63 3 280.25
727.81 1 091.72 1 455.62 1 819.53 2 183.48 2 547.34 2 911.24 3 275.15 3 639.05
0.897 0.923 0.898 0.901 0.908 0.913 0.894 0.897 0.901
6 5 2 7 0 9 3 6 4
Electrical Power & Energy Systems
almost constant. Thus, the curve describing capacity additions per load increment needed to maintain the target LOLP is almost linear. Tests made using larger unit sizes do not show the linearity described previously. For units in the 800-1000 MW range, the first couple of units added provide less reliability per MW than subsequent units. This is consistent with the results presented by Garver 2. However, the nonlinearity is not so serious as to preclude the use of this methodology. It should be noted that care must be taken when using a Gram-Charlier expansion when large units are present. This is discussed further in Section VII.
VI. Applications to dynamic mathematical programming models In this section, a method is described to incorporate the linear peak load guarantee constraint in a dynamic mathematical programming model. Numerical results are also presented that indicate that reasonably accurate results can be expected from such an implementation. In Section IV, the 'forward bootstrapping' solution method was described in which time periods are solved sequentially and the solution of the t -- 1st period's problem is used to initialize t TM period's problem. This allows the calculation of the partial derivatives 8 LOLP t - l/OX i at the solution ( X t - 1 . . . . . Xfv-1, L t - 1). In a dynamic framework, one peak load reliability constraint is required for each period and the coefficients must be calculated before the solution process unfolds. The methodology proposed is to calculate coefficients for each period offline. As the target LOLP is known for each period, if the system mix at the beginning of each period were known, then the coefficients could be calculated. This suggests that a system mix may be extrapolated for each period and coefficients based on this mix may be calculated. Experiments were carried out to assess the error that might be introduced by incorrectly extrapolating the system mix. Even a gross error in extrapolating the mix did not result in an unacceptably large error. Thus, the advantage of this type of constraint in making tradeoffs between unit sizes and forced outage rates makes this methodology attractive. To simulate the use of this methodology in a dynamic linear programming framework, it was assumed that the system described in Section V was the 1982 system, a 3% load growth was assumed, and three systems were constructed for the year ending in 1996. Thus, the 1997 system faces a peak load of 7.8 GW. For the best guess at a year-end 1996 system, a mix was chosen similar to the 1982 mix and sufficient capacity was added to achieve a 0.019 28 LOLP, as calculated by the four-term Gram-Charlier approximation. In addition, two other potential systems were created for year-end 1996: one by adding base load units only and the other by adding peaking units only. This was designed to test the extreme expansion plans that might be identified by a dynamic LP model. These different expansion plans resulted in different partial derivatives L O L p t / a x i and, hence, different coefficients in the constraint (14). Thus, these experiments were designed
Vol 6 No 4 October 1984
to evaluate the error expected if a year-by-year optimization strategy were replaced by a dynamic optimization framework. Table 5 shows the types and numbers of units added in each case. Three experiments similar to those described in Section V were carried out to assess the relative error that may be found if coefficients, based on a base load/peaking unit mix of generation types similar to the 1982 mix, were used for the 1977 reliability constraint but only base load or only peaking units were added. Tables 6-8 show the capacity additions calculated using the linear constraint and the extrapolated unit mix and the values found using the actual LOLP and the extrapolated mix, base load additions only and peaking additions only for load increments of 2502 500 MW in 250 MW steps. The results show that the additions to the two alternative 1997 systems for each type of unit are not very different from the 1997 system with mixed additions. Thus, the methodology described in Section V will give good results with an extrapolated 1997 system, even if the actual 1997 system is quite different from the extrapolated system.
Table 5. Additions to the base system between 1982 and 1996
Base/peaker mix Number of units 25 2O 2 2 1
Size, MW 25 50 250 500 0.54*
3 125.54 MW
Base
Peaker
Number of units
Size, MW
Number of units
1 5 4 1
40 100 37 250 500 0.95* 1 3 350.95 MW
Size, MW 25 50
0.93* 2 850.93 MW
* These units were added to achieve LOLPs of 0 . 0 1 9 2 8 . They were assumed to have a forced outage rate o f 0.02
Table 6. Capacity additions for 1997 - 50 MW units with forced outage rate 0.02
Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ peaker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2500
255.52 511.04 766.56 1 022.09 1 277.62 1 533.16 1 788.70 2 044.24 2 299.78 2555.31
255.53 511.07 766.60 1 022.15 1 277.69 1 533.23 1 788.78 2 044.33 2 299.88 2555.42
255.38 510.77 766.17 1 021.57 1 276.99 1 532.41 1 787.84 2 043.27 2 298.72 2554.16
255.92 511.84 767.76 1 023.68 1 279.60 1 535.51 1 791.43 2 047.35 2 303.37 2559.19
245
Table 7. Capacity additions for 1997 - 250 MW units with forced outage rate 0.05 Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ peaker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
250 500 750 1 000 1 250 1 500 1 750 2 000 2 250 2500
270.34 540.80 811.38 1 082.08 t 352.80 1 623.85 1 894.91 2 166.09 2 437.41 2711.40
269.75 539.55 809.42 I 079.35 1 349.34 1 619.41 1 889.54 2 159.76 2 430.05 2700.42
271.05 532.38 814.02 1 085.90 1 358.06 1 630.48 1 903.15 2 176.07 2 449.22 2722.62
276.34 552.67 829.01 1 105.34 1 381.68 1 658.01 1 934.35 2 210.69 2 487.02 2763.26
Table 8. Capacity additions for 1997 - 500 MW unit with forced outage rate 0.10 Approximate Load (based on increment, approxiMW mations
Actual (based on base load/ perker system)
Actual (based on baseload additions)
Actual (based on peaker additions)
500 750 1 000 1 250 1 500 1 750 2000 2 250 2 500
631.70 973.72 1 265.58 1 593.53 1 929.77 2 270.29 2 541.00 2 874.24 3 153.66
620.11 938.37 1 242.51 1 565.52 1 869.12 2 178.56 2 501.65 2 794.39 3 107.57
654.53 1 009.68 1 309.73 ! 642.49 1 984.54 2 329.79 2 605.24 2 941.31 3 228.20
721.25 1 081.88 1 442.51 1 803.13 2 163.76 2 524.39 2 885.02 3 245.64 3 606.27
The results reported here and other experiments that have been carried out suggest the following general procedure to be effective in generating reliability (LOLP) constraints for each period in an N-period optimization model for capacity expansion planning that is solved simultaneously (dynamically) for all N-periods. Specifically, the linear constraint for period t is estimated using the following two-step procedure: (1) Project expected mix and capacity at the end of period t - I for t = 2 , . . . ,N*. Then calculate O L O L p t - I / 3 x i for t = 1. . . . . N - - 1 and b LOLP/OL using equation (12) (2) The LOLP constraint for year t is L O L p t - 1 + ~_ O L O L p t - 1 / O X i ( X it _ x i t - 1 )
i + / ) LOLP t - 1/OL (L t -- L t - 1) <~ LOLP/~ where LOLP t is the estimated actual loss-of-load probability in year t under the provided mix, and LOLP~ is the desired LOLP in year t.
246
VII.
Conclusions
This paper demonstrates the effectiveness of a methodolog? for developing explicit linear LOLP constraints to represent system reliability in optilnization models of capacity expansion planning. Such a procedure is more effective than using simple reserve margin constraints that do not help an optimization model sort out different units of different sizes, and forced LOLP constraints considerably enhance the ability (/t any resource mix optimization model, linear or nonlinear, to represent system peak reliability better than has been the case so far. In recent years, there have been a number of studies of the accuracy of expansions representing LOLP. The two types of commonly used expansions are Edgeworth and GramCharlier expansions. Levy and Kahn = have shown that tile Edgeworth expansion can lead to numerical inaccuracies when modelling small systems the forced outage rates of which sum to less than 1. A study by Schenk 2a has shown numerical inaccuracies with the Gram-Charlier expansion applied to an 1 800 MW system. However, TVA and several other organizations have successfully used Gram-Charlier 24 expansions for modelling reliability and production costing. Results are generally better for large systems with large numbers of units. This is because, in such cases, there will be only moderate skewness and consequently these expansions are more accurate. 18 The use of LOLP as the basis for developing a reliability constraint does not imply that it is the preferred indicator of system reliability. A discussion of the strengths and drawbacks of LOLP and other reliability indicators is beyond the scope of this paper. However, in spite of its drawbacks, LOLP is a better indicator of system reliability than system reserve margin.
VIII.
References
1 Anderson, D 'Models for determining least cost investments in electricity supply' Bell J. Econ. Manage. Sci. Vol 3 (Spring 1972) pp 267-301 Garver, L L 'Effective load carrying capability of generating units' IEEE Trans. Power Appar. & Syst. Vol 85 No 8 (August 1966) pp 910-919
3 Scherer, C R and Joe, L 'Electric power system planning with explicit stochastic reserves constraint' Manage. ScL Vol 23 No 9 (May 1977) pp 978-985 Electric Power Research Institute 'A generation planning system methodology and case study' Final report
EA-1807 submitted by Gordian Associates (August 1981) Cote, G and Laughton, M A 'Prediction of reserve requirements in generation planning' Int. J. E/ectr.
Power& Energ. Syst. Vol 2 (April 1980) pp 87-95
* Generally, this is easy for the first ten years of the planning horizon, as the base capacity expansion plan is known
Electrical Power & Energy Systems
Noonan, F and Giglio, R J 'Planning electric power generation: a non-linear mixed integer model employing Benders Decomposition' Manage. Sci. Vol 23 (1977) pp 946-956 Charnes, A and Cooper, W W 'Deterministic equivalents for optimizing and satisfying under chance constraints' Oper. Res. Vol 11 (1963) pp 18-39 Vajda, S 'Stochastic programming' in Abadie (Ed) Integer and nonlinear programming American Elsevier, USA (1970)
Baleriaux, H, Jamoulle, E and Fr Linard de Guertechnung 'Simulation de I'exploitation d'un parc de production d'electricite couple a des sations de pompage' ReviewE Vol 7 No 7 (1967) pp 3-24 10 Booth, R R 'Power system simulation model based upon probability analysis' IEEE Trans. Power Appar. & Syst. Vol PAS-91 (1972) pp 62-69 11 Stremel, J P and Rau, N S 'The cumular~t method of calculating LOLP' IEEE Paper PES Summer Meeting, Portland, USA (July 1981 )
21 Sanghvi, A P, Shavel, I and Spann, R M 'Strategic planning for power system vulnerability: an optimization model for resource planning under uncertainty' IEEE Trans. Power Appar. & Syst. Vol PAS-101 (June 1982) pp 1420-1429 22
Levy, D J and Kahn, E P 'Accuracy of the Edgeworth approximation for LOLP calculations in small power systems' Proc. IEEE PES Summer Meeting Portland, USA (July 1981)
23 Schenk, K F 'Addendum to report analysis of reliability criteria for generation planning' National Energy Board, Canada (1979) 24 Stremel, J P 'Production costing for large-range generation expansion planning studies' Proc. IEEE PES Summer Meeting Portland, USA (July 1981)
Appendix This appendix presents the partial derivatives of LOLP that are needed to generate constraint (14). 0 L O L P / 3 X i = OZ1/3X i [N(Z0 -- G x / 3 ! N s ( Z , )
+ G2/4!N4(ZO + 10/6! G]N6(ZO]
12 Allan, R N and Takieddine, F N 'Generation modelling in power system reliability evaluation' Proc. IEEE Conf. Reliability of Power Supply Systems London, U K (February 1977) 13 Masse, P Les reserves et regulation de I'avenir dans la vie economique Hermourn, France (1946)
14 Masse, P 'Electrical investments' in Nelson (Ed) Marginal cost pricing in practice Prentice-Hall, USA (1964)
+ aG,/OX i [--N2(ZO/3! + I O / 6 ! N s ( Z , ) 2G,] + OG2/OX i [Ns(Z1)/4!]
(])
where OZ,/3X i = -- [p,o2L + ZnlpiMW~(1 --Pi)l -'j2
-- (l/2) (L -- Y.nipiMWi) (piMWi(l -- Pi)) 15 Morlat, G 'On instructions for the optimal management of seasonal reservoirs' in Nelson (Ed) Marginal cost pricing in practice Prentice-Hall, USA (1964)
x [02L + Y.niPiMW~(l --pi)] -3/2 OG1/OXi = piMW2(1 -- 3pi + 2p~)
16 Hogan, A, Morris, J and Thompson, H 'Decision
x [O2L+ T.niPiMW](1--pi)] -3/2
problems under risk and chance constrained programming: dilemmas in the transition' Manage. Sci. Vol 27 (1981) pp 698-716 17 Cramer, H Mathematical methods of statistics Princeton University Press, USA (1946)
-- 3/2 [Y, niMW~(1 -- 3pi + 2p~)] x [piMWi(1 --p/)] [o[ + 2niPiMW](1 _pi)]-s/2 OG2/OX i = PiMW3(l -- 7pi + 12p~-- 6p~) x [a2L + YmiPiMW{(1 -- Pi)l -z
18 Kendall, M G and Stuart, A The advanced theory of statistics Hafner, USA (1969)
-- 2 [~niPiMW~ (l -- 7Pi + 12Pi2 _ 6p i3)] X [PiMWi(l--Pi)] [02L+ P'niPiMW?(I --Pi)] -3
19 Billinton, R Power system reliability evaluation Gordon and Breach Science Publishers, USA (1970) 20 Sanghvi, A P, Shavel, I and Spann, R M 'Fuel diversification, stockpiling and system interconnection strategies for minimizing power system vulnerability to energy shortages' Institute of Management Science (TIMS) publication on Energy models and special studies, North-Holland, Netherlands (to be published)
Vol 6 No 4 October 1984
~)LOLPfi)L = N (Zi) OZx/OL -- G1/3 ! N(a)(Z ,) OZ ,/OL
+ G2/4!N(4)(Z,) aZ1/OL + I O I 6 ! G ~ N ( 6 ) (Z,) aZ,/OL
(2)
where OZ,/~L = [o2 + Z n i p i M W ~ ( 1 -- pi)]-1/2
247