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A generation expansion model for electric utilities with stochastic stranded cost
Electrical Power and Energy Systems 24 (2002) 875±885
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A generation expansion model for electric utilities with stochastic stranded cost K. Jo Min*, P.S. Subramaniam 1 Department of Industrial and Manufacturing Systems Engineering (IMSE), 2019 Black Engineering, Iowa State University, Ames, IA 50011, USA Received 27 September 1999; received in revised form 15 February 2001; accepted 2 July 2001
Abstract We design and analyze a generation expansion planning model considering the stochastic nature of stranded cost. The stranded cost is the investment balance outstanding when a generation unit is forced to shut down because it is not economically competitive. This is an important issue nowadays due to the on-going transition of the electric power industry from regulated monopolies toward competition with substantial ®nancial uncertainties. In this paper, speci®cally, we mathematically formulate a generation expansion planning model with stochastic stranded cost based on the mean±variance method. Next, based on experts' estimation of elementary relations among projects over periods, we derive mathematical formulae for covariances among projects and periods. This is shown to enhance the implementability of the generation expansion planning model. Also, managerial insights are provided (e.g. a set of negatively correlated projects will reduce the ®nancial risk associated with stranded cost). Finally, by way of a numerical example, the features of our model are illustrated. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stranded cost; Mean±variance method; Expansion planning model
1. Introduction Currently, the electric power industry is undergoing a substantial transition from regulated monopolies toward competition. Hence, utilities are faced with various new concerns and problems in their operation, planning, and management [17]. One of the primary concerns under the new environment is that of stranded cost [8]. Stranded cost represents the investment balance outstanding when an asset of the utility (e.g. a generation unit) can no longer be operated due to economic reasons [11]. For example, if customers leave a utility for another provider to receive cheaper electricity, then the utility's generation unit can possibly be stranded [7,12]. From the electric utility's point of view, the conventional generation expansion planning [1,23] needs to be reconsidered. The reason is that, when the utilities face intense competition, the possibility of being stranded heightens the new ®nancial uncertainties in generation expansion planning. In this paper, we consider the generation expansion planning of electric utilities with stochastic stranded cost. * Corresponding author. Tel.: 11-515-294-8095; fax: 11-515-294-3524. E-mail address: [email protected] (K. Jo Min). 1 Logistics.com.
Speci®cally, we ®rst characterize the stochastic probability of a project (i.e. a generation unit) being stranded, and show how the corresponding risk factor can be incorporated into the decision making process. Next, by examining the correlations among periods and projects, we derive a general mathematical model as well as an approximate approach (especially for large size problems) for generation expansion planning. Finally, we provide managerial insights and an illustrative numerical example. In formulating the generation expansion planning models with the risk factor, we employ a method called mean± variance (M±V) analysis, which is widely utilized in the ®nance literature [9,18,20]. The M±V analysis assumes that expected return is desirable and that variance (a measure of risk) is undesirable. Martin [10] provides a mathematical programming model for this analysis, and demonstrates its usefulness with an example. Brown [4] uses the M±V analysis to make replacement decisions. The M±V analysis has been extended [9] to incorporate a 0, 1 binary decision variable for optimal project selection. For the generation expansion planning model, we will also utilize a 0, 1 binary decision variable for project selection. In this paper, we make an assumption that once a project is stranded, it remains stranded for all subsequent periods. This is reasonable given the prediction that competition will lead to cheaper electricity. That is, if the production cost for
0142-0615/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(02)00011-X
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K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
a project is higher than that of its competitor in a given period, it will likely remain higher in each of its subsequent periods. This assumption shows that the cash ¯ows over periods are correlated. In addition, there may be correlation among projects. For example, given that only a single pilot program with cheaper price will be implemented by a competitor, the demise of a project in one region of a utility may imply the survival of another project in another region. Given this observation, it is necessary that the correlations among periods and projects be quanti®ed in the generation expansion models. This is accomplished by deriving covariances among periods and projects via conditional probability estimation based on experts' opinion. Such model development is important as there are few capital budgeting models in the literature which quantify covariances between different periods and projects [6,19]. The rest of this paper is organized as follows. First, we introduce the basic assumptions, economic factors (e.g. asset) and the characterization of risk in our model. Next, we formulate the mathematical model for generation expansion with stranded cost and derive the covariance between periods for a project and between projects in all periods. We then obtain useful managerial insights and a numerical example is used to illustrate the features of our model. Finally, concluding remarks and comments on future research are presented.
2. Assumptions, economic factors, and risk in the model 2.1. Assumptions First, we denote any generation expansion activity, such as building a new generation unit or upgrading/refurbishing an existing unit, as a project. We assume for simplicity that the marginal cost of power generation from a project is constant with respect to the amount of power generated. Therefore, no partial shutdowns will be considered. If a project is economically infeasible to operate (or equivalently, stranded), a complete shutdown will be necessary. We also assume that, if a project is stranded, there would be no obligation on the part of the utility to serve the customers of that project (to be consistent with the trend to relax the obligation to serve in USA). Moreover, as in the case of numerous papers on capital budgeting [7], we will not consider in¯ation and tax effects in this model. In addition, we assume for clarity that the present value calculation will be computed with reference to the beginning of year 0. We also assume that all economic activities occur at the beginning of each year. Finally, in this paper, we assume that there is zero stranded cost recovery. The intent of this assumption is not to mitigate the effects of stranded cost by considering some partial stranded cost recovery. It may, however, be noted that our model can be extended to include the concept
of partial stranded cost recovery. This is achieved with an addition of a parameter d 0 # d # 1: This would mean that if a project is stranded it will generate d fraction of the revenue expected when it is not stranded. 2.2. Economic factors in the model We will now present all economic factors needed to formulate the generation expansion model. The factors described are similar to those assumed by numerous other papers [22]. We assume that initially (at the beginning of year 0), a ®nancial obligation (e.g. a loan) already exists and this obligation will be paid back by annual instalments at the beginning of year 1 on. The generation project has been built from the loan resulting in this ®nancial obligation. It is implicitly assumed that all projects under consideration (irrespective of their construction duration) will be operational at the beginning of year 1. Asset. Under the circumstances described thus far, a generation project k will have the asset (or worth) of Lk ; which is equal to the amount of the ®nancial obligation. Repayment of debt. The ®nancial obligation is serviced annually at a ®xed rate of interest. Let dkj represents the annual loan repayment amount for any year j for each project k. Depreciation. Often, an asset is viewed to depreciate annually, and it is viewed to have no economic value at the end of its life. Hence, in this paper, we will assume zero salvage value. We note that there are depreciation methods that can accommodate this assumption [7,21]. Let dekj represents the depreciation expense calculated for project k in period j. Fixed cost of operation. This expresses the cost of operating a plant, where the cost is independent of the level of production, for example, insurance and maintenance. This cost is represented by fkj for project k in period j. Variable cost of operation. This expresses the cost incurred for the operation of a plant, and is proportional to the level of production. This cost is represented by vkj for project k in period j. Revenue. This expresses the revenue received from a project for each period. Speci®cally, let qkj represents the amount of power generated by project k in period j. The price per unit of electricity may be the same for a group of projects located in the same region. However, they may be quite different for projects located in different regions. Let ukj represents the per unit selling price of electric power for project k in period j. Thus, the revenue from project k in period j will be given by rkj qkj ukj : Given these de®nitions and assumptions, let us characterize the stranded cost as follows. The total cost for project k in period j, denoted as ckj ; is given by ckj dkj 1 dekj 1 fkj 1 vkj
1
To accommodate the stochastic behavior with the occurrence of stranded cost, we split up ckj into two types. Let
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
cSkj be the sunk cost that must be borne by the utility irrespective of whether project k is stranded or not, in period j. This is the sum of the repayment of debt and depreciation cost of project k in period j. Hence cSkj dkj 1 dekj
2
cN kj
Let be the cost that will reduce to zero if the plant becomes stranded. This is the sum of the operational and maintenance cost cN kj fkj 1 vkj
3
Hence ckj cSkj 1 cN kj
4
Let us now de®ne wkj to be a random variable, where wkj 0 if project k is stranded in period j, while wkj 1 if project k is not stranded in period j. Equivalently, for convenience of mathematical development, let us de®ne the following conditional probabilities. The probability of project k not being stranded in period j, given that it was not stranded in period j 2 1; is given by akj : We note that in period 1, this probability is de®ned unconditionally (as the project is operational only at the beginning of year 1). We further note that these conditional probabilities can be used to easily obtain the probabilities associated with wkj 0 or 1. This type of probability estimation appears in numerous papers [8]. This probability estimation is appropriate when past data are unavailable or inappropriate to be used in future projection [15]. This is particularly true in the current electric utility industry because there has never been such substantial transition from monopoly to competition in the past, and hence past data on stranded cost under a regulated monopoly may be irrelevant in the current competitive market. Furthermore, Sarin [15] claims that in such circumstances, the knowledge and experience of `experts' is often the only data source. The author then uses probability estimations for each possible outcome with expert opinions to make future projections. In this paper, by experts, we mean utility executives, consultants, and regulatory commissioners, etc. Further details on the experts' opinion approach can be found by Sarin [15]. Given the random variable of wkj ; the pro®t (revenue 2 total cost) for project k in period j will be qkj ukj 2 ckj if wkj 1: On the other hand, if wkj 0; the pro®t for project k in period j will be 2cSkj : Therefore, given wkj ; the pro®t for project k in period j is qkj ukj 2 ckj wkj 2 cSkj 1 2 wkj : This is simpli®ed to be S qkj ukj 2 cN 5 kj wkj 2 ckj 2.3. Characterization of risk With the random variable wkj introduced in Section 2.2, the resulting pro®t for a project is also a random variable. Hence, we can derive the expected value and the variance of
877
the pro®t for each project. The variance of the pro®t represents the degree of risk associated with each project. It is of importance to strike a balance between the total expected value of the pro®t and the corresponding total variance of the pro®t from all projects chosen. The reason is that, numerous decision makers demonstrate some degree of risk aversion, i.e. given the same expected value, lower variance (and hence risk) is preferred [3,13,14]. To address this issue of balance between the expected value versus the variance, we employ a frequently utilized method called the M±V method [9]. In this method, one attempts to minimize the total variance of all projects selected (including the covariance between projects, if any) subject to an expected value constraint. That is, for any given expected return on a portfolio, the portfolio with the smallest variance is preferred to all others; for any given portfolio variance, the portfolio with maximum expected return is preferred to all others. This assumption is called the M±V criterion [20]. We note that modi®cation of this assumption (e.g. expected value has priority over variance) can be studied via various sensitivity analyses, employing the M±V criterion as the benchmark. 3. The mathematical model 3.1. Basic mathematical formulation Given the set of assumptions and de®nitions under the M±V method, the mathematical formulation is described as follows. We recall from Eq. (5) that the pro®t from project k in S period j is expressed as qkj ukj 2 cN kj wkj 2 ckj : If I is the minimum attractive rate of return, then the present value of this pro®t for project k in period j would be expressed S j as qkj ukj 2 cN kj wkj 2 ckj = 1 1 I : Let us now de®ne Rk to be the present value of the pro®t from project k over the entire planning horizon. If T is the life of project k, Rk is simply the sum of the present value of the pro®t for project k, over each of the T periods. For simplicity, we assume that all projects have the same T periods of service starting from the beginning of year 1. We note that, mathematically, the relaxation of this assumption is straightforward (such relaxation does not seem to yield additional analytic results). Hence, Rk is calculated as S T qkj ukj 2 cN X kj wkj 2 ckj Rk 6 1 1 Ij j1 We note that the random variables wkj s are dependent between each other for all ks and all js, and hence have covariance. The expected value of Rk for project k is then S T qkj ukj 2 cN X kj E wkj 2 ckj mk E Rk 7 1 1 Ij j1
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K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
and the variance of Rk for project k is then:
s k2
Var Rk
2 T qkj ukj 2 cN X kj Var wkj 1 1 I2j
j1
12
N X X qki uki 2 cN ki qkj ukj 2 ckj 1 1 Ii1j
j,i
Cov wki ; wkj
(8)
Let xk ; k 1; ¼; N; be an indicator of prospective projects to be chosen at the beginning of year 0. Then let xk 1 if project k is chosen, otherwise xk 0: PNLet A be the total pro®t from all the projects. Then A k1 xk Rk : The mean of A is M
N X k1
xk E Rk
N X k1
xk m k
The variance of A is " # " #2 N N N X X X xk R k E x k Rk 2 xk m k V Var k1
k1
k1
Let k and h be two different candidate projects (k, h 1; ¼; N). Now let the covariance between any Rk and Rh ; i.e. Cov Rk ; Rh ; be represented by s kh : It can be veri®ed that the RHS is equal to V
N X k1
x2k s k2 1 2
XX k,h
xk xh s kh
Let E denotes the minimum pro®t requirement from all the projects chosen. According to the M±V method, the decision maker selects projects so as to minimize the total variance from all projects selected, as long as the expected pro®t is greater than or equal to E. The corresponding formulation is as follows: min V
x1 ;x2 ;¼;xn
s:t:
N X k1
N X k1
x2k s k2 1 2
XX k,h
xk xh s kh 9
xk m k $ E
xk is 1 if project k is selected and 0 otherwise for k 1; ¼; N: In order to optimize the objective function in Eq. (9), it is essential to estimate the variance of Rk ; s k2 for all k k 1; ¼; N: Moreover, in order to estimate s k2 ; it is necessary to estimate the covariance between wki and wkj i ± j (i, j 1; ¼; T), for project k. Likewise, for any two different projects h and k, we need to estimate the covariance between Rk and Rh ; s kh ; for all k and h (k, h 1; ¼; N). These covariances may not be zero as wki and wkj (and likewise Rk and Rh ) may be related. For example, if a
project is stranded in any period, it would remain so for each subsequent period thereafter. Also, if the decision maker's utility is quite inef®cient and ineffective, in general, both projects k and h are likely to be stranded at the same time. Therefore, a method to estimate these covariances is necessary. One way to estimate these covariances is suggested by Stevens and Bussey [19]. They assume that the correlation coef®cients between different periods of a single project, as well as the correlation coef®cients between different projects are determinable based on various assumptions on the behavior of cash ¯ows. For example, they assume that the correlation coef®cient relating two different projects is constant over all time periods. Their method's dependence on correlation coef®cients, however, is problematic because of the numerous restrictive assumptions and the fact that correlation coef®cients are generally derived from covariances (not the other way around). In contrast to this method, we will ®rst estimate some elementary relations among conditional probabilities with respect to wkj s. From these relations, we will derive the covariances between different periods for a project, and between different projects mathematically. A key advantage of this method over the previously mentioned method is that, the information required is relatively simple. That is, rather than ®rst estimating the correlation coef®cients downstream, we ®rst estimate elementary relations on conditional probabilities upstream. This is consistent with Sarin's approach where the problem is decomposed so that each expert is dealing with a relatively simple situation, with the information required from the experts being minimum [15]. 3.2. Modeling the covariance between periods for a project Here, we derive a mathematical method to estimate the covariance between two different periods i and j, of project k, i.e. Cov wki ; wkj : We ®rst need to model the probability distributions of the random variable wkj s for each project k in each period j. Let us recall that if the project k is not stranded in period j, then wkj 1: Also if the project k is stranded in that period, then wkj 0: Let us now de®ne the following distributions for any project k, having a T period planning horizon. Period I. Probability that project k is not stranded in this period is ak1 ; and the probability that it is stranded is 1 2 ak1 : Here, as explained earlier, ak1 is derived from experts' opinion. Hence, P wk1 1 ak1 and P wk1 0 1 2 ak1 : Now, if a project is not stranded in the ®rst year, it may or may not be stranded in the second year. However, if the project were to be stranded in the ®rst year itself, it is de®nite that it will remain stranded in the second year and beyond. Period II. Given this, we de®ne the following conditional
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
probability:
But
P wk2 1uwk1 1 ak2 and P wk2 0uwk1 1
Rh
1 2 ak2 ;
S T qht1 uht1 2 cN X ht1 2 cht1 1 1 It1
t11
and E Rh mh : Similarly, we can get Rk and E Rk : Hence
P wk2 1uwk1 0 0 and P wk2 0uwk1 0 1 Now P wk2 1 P wk1 1 P wk2 1uwk1 1 1 P wk1 0P wk2 1uwk1 0 (see Ref. [5] for details). Hence P wk2 1 ak1 ak2 and P wk2 0 1 2 ak1 ak2
10
Cov Rh ; Rk " E
! qht1 uht1 2
T X
Proposition 1. Given the following conditional probabilities, P wkj 1uwkj21 1 akj 1 , j # T and P wk1 1 ak1 ; for k 1; ¼; N; we have E wkj
z1
11
wht1 2 cSht1
S qkt2 ukt2 2 cN kt2 wkt2 2 ckt2 #
T X t21
X
akz
Var wkj 1 2
cN ht1
1 1 It1 !
t11
We now have the following proposition.
j Y
879
t11 to T t21 to T
2 mh mk 1 1 It2 " ! 1 N qht1 uht1 2 cht1 1 1 It11t2 !
j Y z1
!
akz
Cov wki wkj 1 2
j Y z1
i Y z1
qkt2 ukt2 2
akz
akz
E wht1 wkt2
12
! 2
!
cN kt2
j Y z1
akz i , j # T
13
The proof is in Appendix A. This proposition provides a simple derivation of the covariances between wkj for a project k, for each period j, of its T period planning horizon. This is used in the computation of the variance s k2 for project k, as shown in Eq. (8). 3.3. Modeling the covariance between projects To calculate the total variance V, in Eq. (9), we need to determine the covariance between projects in all periods. That is, for any two different projects h and k, we need to estimate the covariance between Rk and Rh ; s kh ; for all k and h (k, h 1; ¼; N). To model this, we need some additional information as to the relationships between the random variables wkj s, in each period j, for the k candidate projects. Let us recall that in our general model, we have N projects with T periods each. Let us select any two projects h and k out of the possible N projects. These have random variables de®ned by whj s and wkj s, for any period j, as speci®ed earlier. Now by de®nition
s kh Cov Rh Rk E Rh Rk 2 E Rh E Rk
14
qht1 uht1 2
cN ht1
cSkt2 E wht1 #
! 2 qkt2 ukt2 2
cN kt2
cSht1 E wkt2
1
cSht1 cSkt2
2 mh mk
We note that, for Cov Rh ; Rk ; we need to estimate E wht1 wkt2 : We also note that E wht1 wkt2 P wht1 1P wkt2 1uwht1 1: Now, there may be several different ways to characterize the relationship among these random variables. For example, the most general method may be to estimate P wkt2 1uwht1 1; as a parameter, for each projects k and h, and for all periods t1 and t2 : With this method, for N projects, each with T periods, we would need to estimate at least N N 2 1T 2 =2 parameters. Clearly, as the problem gets larger (with large T or N), the experts are asked to estimate a substantial number of parameters. This will not be effective as the estimation of this magnitude is quite demanding on the experts, and the reliability of such estimates may be questionable. The other problem which exists with this method is as follows. When we estimate conditional probabilities among random variables, we need to ensure that the probability estimations are consistent. By consistent, we mean that a certain probability relation exists between two random variables, there is only one unique of probability of occurrence of a certain event (e.g. if the sum of probabilities is not one, then consistency is not maintained; for example, see Ref. [5], p. 71). Because of these problems, we suggest the following
880
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
tractable, consistent approximation method. We recall that the probability of a project k not being stranded in period t2 t2 . 1 given that it was not stranded in period t2 2 1 was given by akt2 : Also the probability of project k not being stranded in period 1 is ak1 : We also recall that we need to determine P wkt2 1uwht1 1 (and hence E wht1 wkt2 ) to calculate Cov Rh ; Rk : Let us now specify the relationship between any two projects h and k in the ®rst period. Let the conditional probability that project k is not stranded in period 1, given that project h is not stranded in period 1 be de®ned as lk1 h1 : Also let the conditional probability that project k is not stranded in period 1, given that project h is stranded in period 1 be de®ned as uk1 h1 : Hence P wk1 1uwh1 1 lk1 h1 ; and P wk1 1uwh1 0 uk1 h1 : Let the parameter lk1 h1 be estimated by experts (similar to akt2 ). Now P wk1 1 P wh1 1P wk1 1uwh1 1 1 P wh1 0P wk1 1uwh1 0 ah1 lk1 h1 1 1 2 ah1 uk1h1. Hence
ak1 ah1 lk1 h1 1 1 2 ah1 uk1 h1
15
We note that we can easily compute Q uk1 h1 from Eq. (15). Also E wkt2 1 # t2 # T t2 in z1 akz as explained Q Section 3.3. Similarly E wht1 1 # t1 # T t1 a : z1 hz Should the projects h and k be independent, we would have P wk1 1uwh1 1 lk1 h1 ak1 : Hence, as the projects h and k approach independence, lk1 h1 approaches ak1 : Now for any period t2 t2 . 1 of project k, and period t1 of project h: P wkt2 1uwkt221 1; wht1 1
P wkt2 1; wkt221 1; wht1 1 P wkt221 1; wht1 1
Let us now assume that the projects h and k have little correlation. Given this, we may formulate the following approximation: P wkt2 1uwkt221 1; wht1 1
P wkt2 1; wkt221 1P wht1 1 P wkt221 1P wht1 1
P wkt2 1uwkt221 1 Given this approximation, we would now be able to determine E wht1 wkt2 (and consequently Cov Rh ; Rk ). Proposition 2. (a) For any period t1 1 # t1 # T of project h; and period 1 of project k 0 1 t1 Y 16 E wht1 wk1 @ ahy Alk1 h1 y1
(b) and for any period t1 1 # t1 # T; of project h, and t2
2 # t2 # T of project k, we have: 0 1 ! t1 t2 Y Y @ A E wht1 wkt2 ahy akz lk1 h1 y1
17
z2
Proof is shown in Appendix B. This proposition facilitates the derivation of the covariances between projects k and h (out of the possible N projects) in all their T periods. This is used in the computation of the total variance of all the projects selected as shown in Eq. (9). This proposition is based on aforementioned assumption that the projects h and k have little correlation. Hence, if the projects h and k have strong correlation, this proposition would be less relevant. On the other hand, if the projects h and k does have strong correlation, then one may aggregate these strongly correlated projects into one larger project for generation expansion planning purposes. Let us now consider three projects h, k and m, where project m is chosen out of the remaining N 2 2 projects. Similar to our treatment of projects h and k, let us assume that the probability that project m is not stranded in period 1, given that project k is not stranded in period 1 (i.e. P wm1 1uwk1 1 lm1 k1 ) is known. Since am1 is known, the probability that project m is not stranded in period 1, given that project k is stranded in period 1 (i.e. P wm1 1uwk1 0 um1 k1 ) may be easily computed am1 ak1 lm1 k1 1 1 2 ak1 um1 k1 : Given this information, all covariances between projects k and m may be computed as shown in Proposition 2. Similarly to characterize the relation between projects h and m, we would need to estimate lm1 h1 : The parameter um1 h1 may then be obtained using the relation am1 ak1 lm1 k1 1 1 2 ak1 um1 k1 ah1 lm1 h1 1 1 2 ah1 um1 h1 : Now, given lm1 h1 and um1 h1 ; we can compute all covariances between projects m and h in all periods. We note that, other than the de®nition of aht1 and akt2 to de®ne the covariance between two projects h and k (for any period t1 and t2, respectively), we need one additional parameter to be de®ned, namely lk1 h1 ; relating them in the ®rst period. Similarly for three projects, i.e. one additional project m, we need two additional parameters (i.e. lm1 k1 and lm1 h1 ). That is for three projects, we need three additional parameters relating the projects in the ®rst period. It may be easily veri®ed that for N projects, we would need N N 2 1=2 additional parameters relating the different projects in the ®rst period. We also note that our method is consistent. 3.4. Managerial insights Given the mathematical model and the characterization of covariances in the previous sections.
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885 Table 1 Market price for each period Period
Market price
1 2 3 4 5
50 40 30 20 15
Proposition 3. At any period t1 and t2, of two projects h and k respectively, let aht1 ; akt2 and lk1 h1 be estimated as speci®ed in the previous sections. If lk1 h1 , ak1 ; then s kh is negative. If lk1 h1 . ak1 ; then s kh is positive. If lk1 h1 ak1 ; then s kh is zero. The proof is shown in Appendix C. This proposition explains the importance of the parameter lk1 h1 in the selection of projects. If this parameter is small in magnitude (i.e. less than ak1 ), the covariance between
881
projects k and h in Eq. (9) will be negative, hence indicating a decrease in risk when both projects k and h are selected together. Conversely, if lk1 h1 . ak1 ; the covariance between projects k and h are positive, indicating an increase in risk when both projects k and h are selected together. So a decision maker should not only look for a set of negatively correlated projects to reduce the risk associated with the investment, but also try and avoid a set of positively correlated projects. Furthermore, an interesting observation is as follows. Recall that we have used the M±V objective of minimizing variance V given an expected pro®t level E. An alternative way to solve the problem is to have the objective of maximizing expected pro®t E, given a variance V. The solution to this problem could be different from the one solved here. The main difference between the objectives is the extent of the risk aversion of the decision maker.
4. Numerical example Let us assume that we have three candidate projects for
Table 2 Cash ¯ow for three candidate projects
Asset dkj
fkj
Project 1
Project 2
Project 3
3000.00 731.67 731.67 731.67 731.67 731.67
5000.00 1219.45 1219.45 1219.45 1219.45 1219.45
4000.00 975.56 975.56 975.56 975.56 975.56
300.00 300.00 100.00 100.00 100.00
400.00 400.00 400.00 400.00 400.00
350.00 350.00 150.00 150.00 150.00
X1j
Cap. 100 units
X2j
vkj
60.00 95.00 100.00 100.00 100.00
300.00 475.00 300.00 300.00 300.00
160.00 190.00 200.00 200.00 200.00
dekj
1000.00 800.00 600.00 400.00 200.00
1666.67 1333.33 1000.00 666.67 333.33
1333.33 1066.67 800.00 533.33 266.67
Per unit cost price Total cost per period
38.86 24.28 17.32 15.32 13.32
Cap. 200 units
X3j
800.00 950.00 1000.00 1000.00 1000.00
130.00 130.00 150.00 150.00 150.00
Per unit cost price 2331.67 2306.67 1731.67 1531.67 1331.67
25.54 20.54 18.10 16.43 14.76
Cap. 150 units 500.00 650.00 450.00 450.00 450.00
Per unit cost price 4086.12 3902.78 3619.45 3286.12 2952.78
31.59 23.40 15.84 14.06 12.28
3158.89 3042.23 2375.56 2108.89 1842.23
882
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
Table 3 Summary of projects 1, 2 and 3 Project 1 Units sold 65 95 100 100 100 m s 12 Project 2 Units sold 160 190 200 200 200 m s 22 s 12 Project 3 Units sold 130 130 150 150 150 m s 32 s 13 s 23
Exp unit pr. 50 40 30 20 15
Revenue 3250 3800 3000 2000 1500
Cost cf1j 1731.67 1531.67 1331.67 1131.67 931.67
Cost cv1j 600 775 400 400 400
a 1j
Exp unit pr.
Revenue
Cost cf2j
Cost cv2j
a 2j
50 40 30 20 15
8000 7600 6000 4000 3000
2886.12 2552.78 2219.45 1886.12 1552.78
1200 1350 1400 1400 1400
Exp unit pr.
Revenue
Cost cf3j
Cost cv3j
a 3j
50 40 30 20 15
6500 5200 3000 2400 2250
2308.89 2042.23 1775.56 1508.89 1242.23
850 1000 600 600 600
1.000 0.800 0.500 0.250 0.000 1708.29 4 918 300 23 519 749 211 516 770
generation expansion, each with ®ve periods (N 3; T 5). Let us assume that the market price remains the same for all the projects in a period. Let it have the variation across periods as shown in Table 1. The projects 1, 2 and 3 have costs as shown in Table 2. Table 3 shows the total revenue generated per period, costs per period and conditional probabilities in each period. For example, the conditional probabilities in each period of project 1 is a11 0:9; a12 1:0; a13 0:88; a14 0:875; a15 1:0: Further, let l21 11 0:8; l31 11 0:95; l31 21 0:9: Table 4 The possible choices of project selection x1
x2
x3
E
V
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
0 2134.29 3321.03 1708.29 5455.32 5029.32 3842.58 7163.61
8 53 4 62 35 6 37
293 896 918 189 781 171 034
0 182 324 300 507 084 983 768
0.900 1.000 0.888 0.875 1.000 2134.293 8 293 182.0
0.800 0.875 0.857 0.833 0.800 3321.03 53 896 324 0.0
Now from the M±V method (9), the problem is as follows: min V x21 s 12 1 x22 s 22 1 x23 s 32 1 2x1 x2 s 12 1 2x1 x3 s 13
x1 ;x2 ;x3
1 2x2 x3 s 23 s:t: m1 x1 1 m2 x2 1 m3 x3 $ E x1 ; x2 ; x3 Binary 0 or 1 where x1, x2, x3 are the choices of projects to be selected. Table 4 shows the table of possible choices of project selection. From this, we ®rst point out that, since l21 11 0:8 a21 ; s 12 0:0: Also l31 11 0:95 ,a31 1:0; s 13 is negative. Similarly, s 23 is also negative as l 31 21 , a31 : As for the optimal project selection, suppose the minimum pro®t from all projects selected should be E 3500; it is optimal to select projects 2 and 3. If E 5000; it is optimal to select projects 1 and 3. 5. Conclusion In this paper, we have mathematically formulated and
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
analyzed a generation expansion planning model for electric utilities facing highly stochastic stranded cost. The stochastic nature of the stranded cost is due to deregulation of the electric power industry, from regulated monopolies toward market competition. In formulating the generation expansion plan, we showed how the stochastic nature can be captured by a random variable based on experts' opinion and how the corresponding risk factor can be incorporated into the decisions making process via the M±V method. Next, by examining the covariances among periods and projects, we have derived conditions (i.e. smaller size problems) under which the formulation may be implemented easily. When the problem size is larger, an approximation technique is developed. In addition, we have derived managerial insights stating that, with a set of negatively correlated projects, the risk will be decreased. However with a set of positively correlated projects, the risk will be increased. Finally, we provide a numerical example illustrating the data requirements, the computation of covariances, and the optimal project selection based on the M±V method. Research in the area of generation expansion planning with stochastic stranded cost is only recently emerging. Hence, numerous future research problems exist. For example, instead of conventional probability estimation of experts' opinions, an alternative approach such as fuzzy sets [2] can be considered. Also, instead of the simple indicator random variable in this paper, one can consider factors that contribute to determine the indicator as random variables (e.g. electric power price, natural gas and coal prices for generation fuel). In fact, one can characterize the entire cash ¯ows as random variables. Such an alternative may require more sophisticated methodologies (e.g. chance constrained programming [16]). We believe it would be interesting to examine the bene®ts and costs of these alternative extensions. Acknowledgements We would like to thank the two anonymous referees, Dr Yuhong Yang, and Dr H.T. David for helpful comments. We are also grateful for ®nancial support from the Electric Power Research Center at Iowa State University and MidAmerican Energy Co.
Cov wk1 wk2 Ewk1 2 E wk1 wk2 2 E wk2 E wk1 wk2 2 E wk1 E wk2 Now E wk1 wk2 1 £ 1 £ P wk1 1; wk2 1 1 1 £ 0 £ P wk1 1; wk2 0 1 0 £ 1 £ P wk1 0; wk2 1 1 0 £ 0 £ P wk1 0; wk2 0 Therefore, E wk1 wk2 ak1 ak2 and Cov wk1 wk2 ak1 ak2 2 ak1 ak1 ak2 ak1 ak2 1 2 ak1 : Hence, the result is true for j 1: Let us now assumeQthat it is true for j n; 1 , Qj , T: Therefore, E wkn nz1 akz also P Wkn 1 nz1 akz : For any j n 1 1; n 1 1 # T P wkn11 1 P wkn 1P wkn 1 1 1uwkn 1 1 P wkn 0P wkn11 1uwkn 0
For any period j, at j 1; P wk1 1 ak1 ;
nY 11 z1
akz
Hence Qn 1 1 E wkn11 1 £ P wkn11 1 1 0 £ P wkn11 0 z1 akz Var wkn11 E wkn11 2 E wkn11 2 ! nY 11 nY 11 12 akz akz z1
z1
Also Cov wki wkn11 Ewki 2 E wki wkn11 2 E wkn11 E wki wkn11 2 E wki E wkn11 for any i , n 1 1: Now, E wki wkn11 1 £ 1 £ P wki 1; wkn11 1
nY 11 z1
akz
Therefore, Cov wki wkn11
nY 11 z1
akz 2
12
i Y z1
Appendix A. Proof of Proposition 1
883
i Y z1
akz
akz
nY 11 z1
! n11 Y z1
akz
akz :
By induction, Eqs. (11)±(13) follow for any j 1 # j # T: A
E wk1 1P wk1 1 1 0P wk1 0 ak1
Appendix B. Proof of Proposition 2
Var wk1 E wk1 2 E wk1 2
(a) At t1 1; E wh1 wk1 1 £ 1 £ P wh1 1 £ P wk1 1uwh1 1; as all other cases are zero, since wk1 and wh1 are binary. Hence E wh1 wk1 ah1 lk1 h1 : As in Appendix A, via mathematical induction, the result in Eq. (16) can be shown.
1 2 ak1 2 P wk1 1 1 0 2 ak1 2 P wk1 0 1 2 ak1 ak1
884
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
(b) At t1 1; t2 2; E wh1 wk2 P wh1 1P wk2 1uwh1 1 P wk2 1uwh1 1 P wk2 1; wk1 1uwh1 11P(wk2 1,wk1 0uwh1 1) ak2lk1 h1 (by assumption P wk2 1uwk1 1; wh1 1 P wk2 1uwk1 1: Hence E wh1 wk2 ah1 ak2 lk1 h1 : As in Appendix A, via mathematical induction, the result in Eq. (17) can be shown. A
(3a) At t2 1 and any t1, by Proposition 2a, we have 0 1 t1 Y E wht1 wkt2 @ ahy Alk1 h1 y1
Now E wht1 wk1 2 E wht1 E wk1 Hence
Qt1
y1
ahy lk1 h1 2 ak1 :
E wht1 wk1 , E wht1 ; E wk1 , 0; if lk1 h1 , ak1
(3b) At t2 t2 . 1 and any t1, by Proposition 2b, we have 0 1 ! t1 t2 Y Y E wht1 wkt2 @ ahy A akz lk1 h1
Appendix C. Proof of Proposition 3 From Eq. (14), we have Cov Rh ; Rk
X t11 to T t21 to T
1 1 1 It11t2
"
! qht1 uht1 2
y1
cN ht1
E wht1 wkt2 2 E wht1 E wkt2
qkt2 ukt2 2 cN kt2 E wht1 wkt2
2 qkt2 ukt2 2
cN kt2
cSkt2 E wht1 # cSht1 E wkt2
1
cSht1 cSkt2
Expanding the expressions for mh E Rh and mk E Rk ; we get " ! X 1 N qht1 uht1 2 cht1 Cov Rh ; Rk t11t2 t11 to T 1 1 I !
qkt2 ukt2 2 cN kt2 E wht1 wkt2 ! 2 qht1 uht1 2
cN ht1
2 qkt2 ukt2 2
cN kt2
cSkt2 E wht1 #
!
2
X t11 to T t21 to T
cSht1 E wkt2
1 1 1 It11t2
qkt2 ukt2 2
1
cSht1 cSkt2
"
! qht1 uht1 2
cN ht1
! cN kt2
E wht1 E wkt2 !
2 qht1 uht1 2
cN ht1
2 qkt2 ukt2 2
cN kt2
y1
ahy
t2 Y z1
akz
lk1 h1 21 ak1
E wht1 wkt2 , E wht1 ; E wkt2 , 0; if lk1 h1 , ak1 C2
!
t21 to T
t1 Y
Hence
! 2 qht1 uht1 2
z2
Now
!
cN ht1
C1
cSkt2 E wht1 #
! cSht1 E wkt2
1
cSht1 cSkt2
It now suf®ces to show that Cov Rh ; Rk will be negative if E wht1 wkt2 , E wht1 E wkt2 for all 1 # t1 # T and 1 # t2 # T:
From Eqs. (C1) and (C2), the result in Proposition 3 follows. Conversely, it follows that Cov(Rh, Rk) is positive if lk1 h1 . ak1 : Finally, if lk1 h1 ak1 ; then by de®nition Cov Rh ; Rk 0: A References [1] Ammons JC, McGinnis LF. A generation expansion planning model for electric utilities. Engng Econ 1984;30:205±25. [2] Bernardini A. What are the random and fuzzy sets and how to use them for uncertainty modeling in engineering systems? In: Elishakoff I, editor. Whys and hows in uncertainty modeling. New York: Springer, 1999. Chapter 2. [3] Bierman H, Smidt S. The capital budgeting decision. 3rd ed. New York: Macmillan, 1984. [4] Brown MJ. A mean±variance serial replacement decision model: the correlated case. Engng Econ 1993;38(3):237±47. [5] Freund JE. Mathematical statistics. 5th ed. New Jersey: Prentice-Hall, 1992. p. 71. [6] Giaccotto C. A simpli®ed approach to risk analysis in capital budgeting with serially correlated cash ¯ows. Engng Econ 1984;29(4):273± 86. [7] Grif®th M, Robinson T. Economic value and capital recoveryÐa regulatory model. Publ Util Fortnight 1985;11:30. [8] Kolbe AL, Tye WB. Compensation for the risk of stranded costs. Energy Policy 1996;24(12):1025±50. [9] Markowitz H. Mean variance analysis in portfolio choice and capital markets. New York: Basil Blackwell, 1987. [10] Martin AD. Mathematical programming of portfolio selections. Mgmt Sci 1955;1(2):152±66. [11] Of®ce of Consumer Advocate, Prepared Direct Testimony of D. Scott Norwood, RE: MidAmerican Energy Company, Docket no. APP-961, September 1996. [12] Pospisil R. Nuclear plants under deregulation. Electl World 1995:56±7. [13] Resnick S. Adventures in stochastic processes. Boston: Birkhauser, 1992. p. 73. [14] Robichek AA, Myers SC. Optimal ®nancing decisions. Englewood Cliffs, NJ: Prentice-Hall, 1965. p. 79±93.
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885 [15] Sarin RK. An approach for long term forecasting with an application to solar electric energy. Mgmt Sci 1979;25(6):543±54. [16] Sarper H. Capital rationing under risk: a chance constrained approach using uniformly distributed cash ¯ows and available budgets. Engng Econ 1993;39(1):49±76. [17] Schriefer J. More ¯exible power rates, more choice in suppliers. New Steel 1996:22±9. [18] Simaan Y. Estimation risk in portfolio selection: the mean variance versus the mean absolute deviation model. Mgmt Sci 1997;43(10):1437±46. [19] Stevens GT, Bussey LE. Formulating correlated cash ¯ow streams. Engng Econ 1973;18(1): 1±30.
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[20] Stone BK. Risk, return and equilibrium: a general single period theory of asset selection and capital market equilibrium. Cambridge: MIT Press, 1970. [21] Thuesen GJ, Fabrycky WJ. Engineering economy. 6th ed. Englewood Cliffs, NJ: Prentice-Hall, 1984. p. 337±9. [22] Walter GB, Johnson MF, Johnson MS. Stranded cost and competitive forces in the electric utility industry. J Account, Audit Finance 1997;12(3):199±219. [23] Wang C, Min KJ. An integrated resource planning model for utilities with quanti®ed outage costs. Int J Electl Power Energy Syst 1998;20:517±24.
www.elsevier.com/locate/ijepes
A generation expansion model for electric utilities with stochastic stranded cost K. Jo Min*, P.S. Subramaniam 1 Department of Industrial and Manufacturing Systems Engineering (IMSE), 2019 Black Engineering, Iowa State University, Ames, IA 50011, USA Received 27 September 1999; received in revised form 15 February 2001; accepted 2 July 2001
Abstract We design and analyze a generation expansion planning model considering the stochastic nature of stranded cost. The stranded cost is the investment balance outstanding when a generation unit is forced to shut down because it is not economically competitive. This is an important issue nowadays due to the on-going transition of the electric power industry from regulated monopolies toward competition with substantial ®nancial uncertainties. In this paper, speci®cally, we mathematically formulate a generation expansion planning model with stochastic stranded cost based on the mean±variance method. Next, based on experts' estimation of elementary relations among projects over periods, we derive mathematical formulae for covariances among projects and periods. This is shown to enhance the implementability of the generation expansion planning model. Also, managerial insights are provided (e.g. a set of negatively correlated projects will reduce the ®nancial risk associated with stranded cost). Finally, by way of a numerical example, the features of our model are illustrated. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stranded cost; Mean±variance method; Expansion planning model
1. Introduction Currently, the electric power industry is undergoing a substantial transition from regulated monopolies toward competition. Hence, utilities are faced with various new concerns and problems in their operation, planning, and management [17]. One of the primary concerns under the new environment is that of stranded cost [8]. Stranded cost represents the investment balance outstanding when an asset of the utility (e.g. a generation unit) can no longer be operated due to economic reasons [11]. For example, if customers leave a utility for another provider to receive cheaper electricity, then the utility's generation unit can possibly be stranded [7,12]. From the electric utility's point of view, the conventional generation expansion planning [1,23] needs to be reconsidered. The reason is that, when the utilities face intense competition, the possibility of being stranded heightens the new ®nancial uncertainties in generation expansion planning. In this paper, we consider the generation expansion planning of electric utilities with stochastic stranded cost. * Corresponding author. Tel.: 11-515-294-8095; fax: 11-515-294-3524. E-mail address: [email protected] (K. Jo Min). 1 Logistics.com.
Speci®cally, we ®rst characterize the stochastic probability of a project (i.e. a generation unit) being stranded, and show how the corresponding risk factor can be incorporated into the decision making process. Next, by examining the correlations among periods and projects, we derive a general mathematical model as well as an approximate approach (especially for large size problems) for generation expansion planning. Finally, we provide managerial insights and an illustrative numerical example. In formulating the generation expansion planning models with the risk factor, we employ a method called mean± variance (M±V) analysis, which is widely utilized in the ®nance literature [9,18,20]. The M±V analysis assumes that expected return is desirable and that variance (a measure of risk) is undesirable. Martin [10] provides a mathematical programming model for this analysis, and demonstrates its usefulness with an example. Brown [4] uses the M±V analysis to make replacement decisions. The M±V analysis has been extended [9] to incorporate a 0, 1 binary decision variable for optimal project selection. For the generation expansion planning model, we will also utilize a 0, 1 binary decision variable for project selection. In this paper, we make an assumption that once a project is stranded, it remains stranded for all subsequent periods. This is reasonable given the prediction that competition will lead to cheaper electricity. That is, if the production cost for
0142-0615/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(02)00011-X
876
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
a project is higher than that of its competitor in a given period, it will likely remain higher in each of its subsequent periods. This assumption shows that the cash ¯ows over periods are correlated. In addition, there may be correlation among projects. For example, given that only a single pilot program with cheaper price will be implemented by a competitor, the demise of a project in one region of a utility may imply the survival of another project in another region. Given this observation, it is necessary that the correlations among periods and projects be quanti®ed in the generation expansion models. This is accomplished by deriving covariances among periods and projects via conditional probability estimation based on experts' opinion. Such model development is important as there are few capital budgeting models in the literature which quantify covariances between different periods and projects [6,19]. The rest of this paper is organized as follows. First, we introduce the basic assumptions, economic factors (e.g. asset) and the characterization of risk in our model. Next, we formulate the mathematical model for generation expansion with stranded cost and derive the covariance between periods for a project and between projects in all periods. We then obtain useful managerial insights and a numerical example is used to illustrate the features of our model. Finally, concluding remarks and comments on future research are presented.
2. Assumptions, economic factors, and risk in the model 2.1. Assumptions First, we denote any generation expansion activity, such as building a new generation unit or upgrading/refurbishing an existing unit, as a project. We assume for simplicity that the marginal cost of power generation from a project is constant with respect to the amount of power generated. Therefore, no partial shutdowns will be considered. If a project is economically infeasible to operate (or equivalently, stranded), a complete shutdown will be necessary. We also assume that, if a project is stranded, there would be no obligation on the part of the utility to serve the customers of that project (to be consistent with the trend to relax the obligation to serve in USA). Moreover, as in the case of numerous papers on capital budgeting [7], we will not consider in¯ation and tax effects in this model. In addition, we assume for clarity that the present value calculation will be computed with reference to the beginning of year 0. We also assume that all economic activities occur at the beginning of each year. Finally, in this paper, we assume that there is zero stranded cost recovery. The intent of this assumption is not to mitigate the effects of stranded cost by considering some partial stranded cost recovery. It may, however, be noted that our model can be extended to include the concept
of partial stranded cost recovery. This is achieved with an addition of a parameter d 0 # d # 1: This would mean that if a project is stranded it will generate d fraction of the revenue expected when it is not stranded. 2.2. Economic factors in the model We will now present all economic factors needed to formulate the generation expansion model. The factors described are similar to those assumed by numerous other papers [22]. We assume that initially (at the beginning of year 0), a ®nancial obligation (e.g. a loan) already exists and this obligation will be paid back by annual instalments at the beginning of year 1 on. The generation project has been built from the loan resulting in this ®nancial obligation. It is implicitly assumed that all projects under consideration (irrespective of their construction duration) will be operational at the beginning of year 1. Asset. Under the circumstances described thus far, a generation project k will have the asset (or worth) of Lk ; which is equal to the amount of the ®nancial obligation. Repayment of debt. The ®nancial obligation is serviced annually at a ®xed rate of interest. Let dkj represents the annual loan repayment amount for any year j for each project k. Depreciation. Often, an asset is viewed to depreciate annually, and it is viewed to have no economic value at the end of its life. Hence, in this paper, we will assume zero salvage value. We note that there are depreciation methods that can accommodate this assumption [7,21]. Let dekj represents the depreciation expense calculated for project k in period j. Fixed cost of operation. This expresses the cost of operating a plant, where the cost is independent of the level of production, for example, insurance and maintenance. This cost is represented by fkj for project k in period j. Variable cost of operation. This expresses the cost incurred for the operation of a plant, and is proportional to the level of production. This cost is represented by vkj for project k in period j. Revenue. This expresses the revenue received from a project for each period. Speci®cally, let qkj represents the amount of power generated by project k in period j. The price per unit of electricity may be the same for a group of projects located in the same region. However, they may be quite different for projects located in different regions. Let ukj represents the per unit selling price of electric power for project k in period j. Thus, the revenue from project k in period j will be given by rkj qkj ukj : Given these de®nitions and assumptions, let us characterize the stranded cost as follows. The total cost for project k in period j, denoted as ckj ; is given by ckj dkj 1 dekj 1 fkj 1 vkj
1
To accommodate the stochastic behavior with the occurrence of stranded cost, we split up ckj into two types. Let
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
cSkj be the sunk cost that must be borne by the utility irrespective of whether project k is stranded or not, in period j. This is the sum of the repayment of debt and depreciation cost of project k in period j. Hence cSkj dkj 1 dekj
2
cN kj
Let be the cost that will reduce to zero if the plant becomes stranded. This is the sum of the operational and maintenance cost cN kj fkj 1 vkj
3
Hence ckj cSkj 1 cN kj
4
Let us now de®ne wkj to be a random variable, where wkj 0 if project k is stranded in period j, while wkj 1 if project k is not stranded in period j. Equivalently, for convenience of mathematical development, let us de®ne the following conditional probabilities. The probability of project k not being stranded in period j, given that it was not stranded in period j 2 1; is given by akj : We note that in period 1, this probability is de®ned unconditionally (as the project is operational only at the beginning of year 1). We further note that these conditional probabilities can be used to easily obtain the probabilities associated with wkj 0 or 1. This type of probability estimation appears in numerous papers [8]. This probability estimation is appropriate when past data are unavailable or inappropriate to be used in future projection [15]. This is particularly true in the current electric utility industry because there has never been such substantial transition from monopoly to competition in the past, and hence past data on stranded cost under a regulated monopoly may be irrelevant in the current competitive market. Furthermore, Sarin [15] claims that in such circumstances, the knowledge and experience of `experts' is often the only data source. The author then uses probability estimations for each possible outcome with expert opinions to make future projections. In this paper, by experts, we mean utility executives, consultants, and regulatory commissioners, etc. Further details on the experts' opinion approach can be found by Sarin [15]. Given the random variable of wkj ; the pro®t (revenue 2 total cost) for project k in period j will be qkj ukj 2 ckj if wkj 1: On the other hand, if wkj 0; the pro®t for project k in period j will be 2cSkj : Therefore, given wkj ; the pro®t for project k in period j is qkj ukj 2 ckj wkj 2 cSkj 1 2 wkj : This is simpli®ed to be S qkj ukj 2 cN 5 kj wkj 2 ckj 2.3. Characterization of risk With the random variable wkj introduced in Section 2.2, the resulting pro®t for a project is also a random variable. Hence, we can derive the expected value and the variance of
877
the pro®t for each project. The variance of the pro®t represents the degree of risk associated with each project. It is of importance to strike a balance between the total expected value of the pro®t and the corresponding total variance of the pro®t from all projects chosen. The reason is that, numerous decision makers demonstrate some degree of risk aversion, i.e. given the same expected value, lower variance (and hence risk) is preferred [3,13,14]. To address this issue of balance between the expected value versus the variance, we employ a frequently utilized method called the M±V method [9]. In this method, one attempts to minimize the total variance of all projects selected (including the covariance between projects, if any) subject to an expected value constraint. That is, for any given expected return on a portfolio, the portfolio with the smallest variance is preferred to all others; for any given portfolio variance, the portfolio with maximum expected return is preferred to all others. This assumption is called the M±V criterion [20]. We note that modi®cation of this assumption (e.g. expected value has priority over variance) can be studied via various sensitivity analyses, employing the M±V criterion as the benchmark. 3. The mathematical model 3.1. Basic mathematical formulation Given the set of assumptions and de®nitions under the M±V method, the mathematical formulation is described as follows. We recall from Eq. (5) that the pro®t from project k in S period j is expressed as qkj ukj 2 cN kj wkj 2 ckj : If I is the minimum attractive rate of return, then the present value of this pro®t for project k in period j would be expressed S j as qkj ukj 2 cN kj wkj 2 ckj = 1 1 I : Let us now de®ne Rk to be the present value of the pro®t from project k over the entire planning horizon. If T is the life of project k, Rk is simply the sum of the present value of the pro®t for project k, over each of the T periods. For simplicity, we assume that all projects have the same T periods of service starting from the beginning of year 1. We note that, mathematically, the relaxation of this assumption is straightforward (such relaxation does not seem to yield additional analytic results). Hence, Rk is calculated as S T qkj ukj 2 cN X kj wkj 2 ckj Rk 6 1 1 Ij j1 We note that the random variables wkj s are dependent between each other for all ks and all js, and hence have covariance. The expected value of Rk for project k is then S T qkj ukj 2 cN X kj E wkj 2 ckj mk E Rk 7 1 1 Ij j1
878
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
and the variance of Rk for project k is then:
s k2
Var Rk
2 T qkj ukj 2 cN X kj Var wkj 1 1 I2j
j1
12
N X X qki uki 2 cN ki qkj ukj 2 ckj 1 1 Ii1j
j,i
Cov wki ; wkj
(8)
Let xk ; k 1; ¼; N; be an indicator of prospective projects to be chosen at the beginning of year 0. Then let xk 1 if project k is chosen, otherwise xk 0: PNLet A be the total pro®t from all the projects. Then A k1 xk Rk : The mean of A is M
N X k1
xk E Rk
N X k1
xk m k
The variance of A is " # " #2 N N N X X X xk R k E x k Rk 2 xk m k V Var k1
k1
k1
Let k and h be two different candidate projects (k, h 1; ¼; N). Now let the covariance between any Rk and Rh ; i.e. Cov Rk ; Rh ; be represented by s kh : It can be veri®ed that the RHS is equal to V
N X k1
x2k s k2 1 2
XX k,h
xk xh s kh
Let E denotes the minimum pro®t requirement from all the projects chosen. According to the M±V method, the decision maker selects projects so as to minimize the total variance from all projects selected, as long as the expected pro®t is greater than or equal to E. The corresponding formulation is as follows: min V
x1 ;x2 ;¼;xn
s:t:
N X k1
N X k1
x2k s k2 1 2
XX k,h
xk xh s kh 9
xk m k $ E
xk is 1 if project k is selected and 0 otherwise for k 1; ¼; N: In order to optimize the objective function in Eq. (9), it is essential to estimate the variance of Rk ; s k2 for all k k 1; ¼; N: Moreover, in order to estimate s k2 ; it is necessary to estimate the covariance between wki and wkj i ± j (i, j 1; ¼; T), for project k. Likewise, for any two different projects h and k, we need to estimate the covariance between Rk and Rh ; s kh ; for all k and h (k, h 1; ¼; N). These covariances may not be zero as wki and wkj (and likewise Rk and Rh ) may be related. For example, if a
project is stranded in any period, it would remain so for each subsequent period thereafter. Also, if the decision maker's utility is quite inef®cient and ineffective, in general, both projects k and h are likely to be stranded at the same time. Therefore, a method to estimate these covariances is necessary. One way to estimate these covariances is suggested by Stevens and Bussey [19]. They assume that the correlation coef®cients between different periods of a single project, as well as the correlation coef®cients between different projects are determinable based on various assumptions on the behavior of cash ¯ows. For example, they assume that the correlation coef®cient relating two different projects is constant over all time periods. Their method's dependence on correlation coef®cients, however, is problematic because of the numerous restrictive assumptions and the fact that correlation coef®cients are generally derived from covariances (not the other way around). In contrast to this method, we will ®rst estimate some elementary relations among conditional probabilities with respect to wkj s. From these relations, we will derive the covariances between different periods for a project, and between different projects mathematically. A key advantage of this method over the previously mentioned method is that, the information required is relatively simple. That is, rather than ®rst estimating the correlation coef®cients downstream, we ®rst estimate elementary relations on conditional probabilities upstream. This is consistent with Sarin's approach where the problem is decomposed so that each expert is dealing with a relatively simple situation, with the information required from the experts being minimum [15]. 3.2. Modeling the covariance between periods for a project Here, we derive a mathematical method to estimate the covariance between two different periods i and j, of project k, i.e. Cov wki ; wkj : We ®rst need to model the probability distributions of the random variable wkj s for each project k in each period j. Let us recall that if the project k is not stranded in period j, then wkj 1: Also if the project k is stranded in that period, then wkj 0: Let us now de®ne the following distributions for any project k, having a T period planning horizon. Period I. Probability that project k is not stranded in this period is ak1 ; and the probability that it is stranded is 1 2 ak1 : Here, as explained earlier, ak1 is derived from experts' opinion. Hence, P wk1 1 ak1 and P wk1 0 1 2 ak1 : Now, if a project is not stranded in the ®rst year, it may or may not be stranded in the second year. However, if the project were to be stranded in the ®rst year itself, it is de®nite that it will remain stranded in the second year and beyond. Period II. Given this, we de®ne the following conditional
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
probability:
But
P wk2 1uwk1 1 ak2 and P wk2 0uwk1 1
Rh
1 2 ak2 ;
S T qht1 uht1 2 cN X ht1 2 cht1 1 1 It1
t11
and E Rh mh : Similarly, we can get Rk and E Rk : Hence
P wk2 1uwk1 0 0 and P wk2 0uwk1 0 1 Now P wk2 1 P wk1 1 P wk2 1uwk1 1 1 P wk1 0P wk2 1uwk1 0 (see Ref. [5] for details). Hence P wk2 1 ak1 ak2 and P wk2 0 1 2 ak1 ak2
10
Cov Rh ; Rk " E
! qht1 uht1 2
T X
Proposition 1. Given the following conditional probabilities, P wkj 1uwkj21 1 akj 1 , j # T and P wk1 1 ak1 ; for k 1; ¼; N; we have E wkj
z1
11
wht1 2 cSht1
S qkt2 ukt2 2 cN kt2 wkt2 2 ckt2 #
T X t21
X
akz
Var wkj 1 2
cN ht1
1 1 It1 !
t11
We now have the following proposition.
j Y
879
t11 to T t21 to T
2 mh mk 1 1 It2 " ! 1 N qht1 uht1 2 cht1 1 1 It11t2 !
j Y z1
!
akz
Cov wki wkj 1 2
j Y z1
i Y z1
qkt2 ukt2 2
akz
akz
E wht1 wkt2
12
! 2
!
cN kt2
j Y z1
akz i , j # T
13
The proof is in Appendix A. This proposition provides a simple derivation of the covariances between wkj for a project k, for each period j, of its T period planning horizon. This is used in the computation of the variance s k2 for project k, as shown in Eq. (8). 3.3. Modeling the covariance between projects To calculate the total variance V, in Eq. (9), we need to determine the covariance between projects in all periods. That is, for any two different projects h and k, we need to estimate the covariance between Rk and Rh ; s kh ; for all k and h (k, h 1; ¼; N). To model this, we need some additional information as to the relationships between the random variables wkj s, in each period j, for the k candidate projects. Let us recall that in our general model, we have N projects with T periods each. Let us select any two projects h and k out of the possible N projects. These have random variables de®ned by whj s and wkj s, for any period j, as speci®ed earlier. Now by de®nition
s kh Cov Rh Rk E Rh Rk 2 E Rh E Rk
14
qht1 uht1 2
cN ht1
cSkt2 E wht1 #
! 2 qkt2 ukt2 2
cN kt2
cSht1 E wkt2
1
cSht1 cSkt2
2 mh mk
We note that, for Cov Rh ; Rk ; we need to estimate E wht1 wkt2 : We also note that E wht1 wkt2 P wht1 1P wkt2 1uwht1 1: Now, there may be several different ways to characterize the relationship among these random variables. For example, the most general method may be to estimate P wkt2 1uwht1 1; as a parameter, for each projects k and h, and for all periods t1 and t2 : With this method, for N projects, each with T periods, we would need to estimate at least N N 2 1T 2 =2 parameters. Clearly, as the problem gets larger (with large T or N), the experts are asked to estimate a substantial number of parameters. This will not be effective as the estimation of this magnitude is quite demanding on the experts, and the reliability of such estimates may be questionable. The other problem which exists with this method is as follows. When we estimate conditional probabilities among random variables, we need to ensure that the probability estimations are consistent. By consistent, we mean that a certain probability relation exists between two random variables, there is only one unique of probability of occurrence of a certain event (e.g. if the sum of probabilities is not one, then consistency is not maintained; for example, see Ref. [5], p. 71). Because of these problems, we suggest the following
880
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
tractable, consistent approximation method. We recall that the probability of a project k not being stranded in period t2 t2 . 1 given that it was not stranded in period t2 2 1 was given by akt2 : Also the probability of project k not being stranded in period 1 is ak1 : We also recall that we need to determine P wkt2 1uwht1 1 (and hence E wht1 wkt2 ) to calculate Cov Rh ; Rk : Let us now specify the relationship between any two projects h and k in the ®rst period. Let the conditional probability that project k is not stranded in period 1, given that project h is not stranded in period 1 be de®ned as lk1 h1 : Also let the conditional probability that project k is not stranded in period 1, given that project h is stranded in period 1 be de®ned as uk1 h1 : Hence P wk1 1uwh1 1 lk1 h1 ; and P wk1 1uwh1 0 uk1 h1 : Let the parameter lk1 h1 be estimated by experts (similar to akt2 ). Now P wk1 1 P wh1 1P wk1 1uwh1 1 1 P wh1 0P wk1 1uwh1 0 ah1 lk1 h1 1 1 2 ah1 uk1h1. Hence
ak1 ah1 lk1 h1 1 1 2 ah1 uk1 h1
15
We note that we can easily compute Q uk1 h1 from Eq. (15). Also E wkt2 1 # t2 # T t2 in z1 akz as explained Q Section 3.3. Similarly E wht1 1 # t1 # T t1 a : z1 hz Should the projects h and k be independent, we would have P wk1 1uwh1 1 lk1 h1 ak1 : Hence, as the projects h and k approach independence, lk1 h1 approaches ak1 : Now for any period t2 t2 . 1 of project k, and period t1 of project h: P wkt2 1uwkt221 1; wht1 1
P wkt2 1; wkt221 1; wht1 1 P wkt221 1; wht1 1
Let us now assume that the projects h and k have little correlation. Given this, we may formulate the following approximation: P wkt2 1uwkt221 1; wht1 1
P wkt2 1; wkt221 1P wht1 1 P wkt221 1P wht1 1
P wkt2 1uwkt221 1 Given this approximation, we would now be able to determine E wht1 wkt2 (and consequently Cov Rh ; Rk ). Proposition 2. (a) For any period t1 1 # t1 # T of project h; and period 1 of project k 0 1 t1 Y 16 E wht1 wk1 @ ahy Alk1 h1 y1
(b) and for any period t1 1 # t1 # T; of project h, and t2
2 # t2 # T of project k, we have: 0 1 ! t1 t2 Y Y @ A E wht1 wkt2 ahy akz lk1 h1 y1
17
z2
Proof is shown in Appendix B. This proposition facilitates the derivation of the covariances between projects k and h (out of the possible N projects) in all their T periods. This is used in the computation of the total variance of all the projects selected as shown in Eq. (9). This proposition is based on aforementioned assumption that the projects h and k have little correlation. Hence, if the projects h and k have strong correlation, this proposition would be less relevant. On the other hand, if the projects h and k does have strong correlation, then one may aggregate these strongly correlated projects into one larger project for generation expansion planning purposes. Let us now consider three projects h, k and m, where project m is chosen out of the remaining N 2 2 projects. Similar to our treatment of projects h and k, let us assume that the probability that project m is not stranded in period 1, given that project k is not stranded in period 1 (i.e. P wm1 1uwk1 1 lm1 k1 ) is known. Since am1 is known, the probability that project m is not stranded in period 1, given that project k is stranded in period 1 (i.e. P wm1 1uwk1 0 um1 k1 ) may be easily computed am1 ak1 lm1 k1 1 1 2 ak1 um1 k1 : Given this information, all covariances between projects k and m may be computed as shown in Proposition 2. Similarly to characterize the relation between projects h and m, we would need to estimate lm1 h1 : The parameter um1 h1 may then be obtained using the relation am1 ak1 lm1 k1 1 1 2 ak1 um1 k1 ah1 lm1 h1 1 1 2 ah1 um1 h1 : Now, given lm1 h1 and um1 h1 ; we can compute all covariances between projects m and h in all periods. We note that, other than the de®nition of aht1 and akt2 to de®ne the covariance between two projects h and k (for any period t1 and t2, respectively), we need one additional parameter to be de®ned, namely lk1 h1 ; relating them in the ®rst period. Similarly for three projects, i.e. one additional project m, we need two additional parameters (i.e. lm1 k1 and lm1 h1 ). That is for three projects, we need three additional parameters relating the projects in the ®rst period. It may be easily veri®ed that for N projects, we would need N N 2 1=2 additional parameters relating the different projects in the ®rst period. We also note that our method is consistent. 3.4. Managerial insights Given the mathematical model and the characterization of covariances in the previous sections.
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885 Table 1 Market price for each period Period
Market price
1 2 3 4 5
50 40 30 20 15
Proposition 3. At any period t1 and t2, of two projects h and k respectively, let aht1 ; akt2 and lk1 h1 be estimated as speci®ed in the previous sections. If lk1 h1 , ak1 ; then s kh is negative. If lk1 h1 . ak1 ; then s kh is positive. If lk1 h1 ak1 ; then s kh is zero. The proof is shown in Appendix C. This proposition explains the importance of the parameter lk1 h1 in the selection of projects. If this parameter is small in magnitude (i.e. less than ak1 ), the covariance between
881
projects k and h in Eq. (9) will be negative, hence indicating a decrease in risk when both projects k and h are selected together. Conversely, if lk1 h1 . ak1 ; the covariance between projects k and h are positive, indicating an increase in risk when both projects k and h are selected together. So a decision maker should not only look for a set of negatively correlated projects to reduce the risk associated with the investment, but also try and avoid a set of positively correlated projects. Furthermore, an interesting observation is as follows. Recall that we have used the M±V objective of minimizing variance V given an expected pro®t level E. An alternative way to solve the problem is to have the objective of maximizing expected pro®t E, given a variance V. The solution to this problem could be different from the one solved here. The main difference between the objectives is the extent of the risk aversion of the decision maker.
4. Numerical example Let us assume that we have three candidate projects for
Table 2 Cash ¯ow for three candidate projects
Asset dkj
fkj
Project 1
Project 2
Project 3
3000.00 731.67 731.67 731.67 731.67 731.67
5000.00 1219.45 1219.45 1219.45 1219.45 1219.45
4000.00 975.56 975.56 975.56 975.56 975.56
300.00 300.00 100.00 100.00 100.00
400.00 400.00 400.00 400.00 400.00
350.00 350.00 150.00 150.00 150.00
X1j
Cap. 100 units
X2j
vkj
60.00 95.00 100.00 100.00 100.00
300.00 475.00 300.00 300.00 300.00
160.00 190.00 200.00 200.00 200.00
dekj
1000.00 800.00 600.00 400.00 200.00
1666.67 1333.33 1000.00 666.67 333.33
1333.33 1066.67 800.00 533.33 266.67
Per unit cost price Total cost per period
38.86 24.28 17.32 15.32 13.32
Cap. 200 units
X3j
800.00 950.00 1000.00 1000.00 1000.00
130.00 130.00 150.00 150.00 150.00
Per unit cost price 2331.67 2306.67 1731.67 1531.67 1331.67
25.54 20.54 18.10 16.43 14.76
Cap. 150 units 500.00 650.00 450.00 450.00 450.00
Per unit cost price 4086.12 3902.78 3619.45 3286.12 2952.78
31.59 23.40 15.84 14.06 12.28
3158.89 3042.23 2375.56 2108.89 1842.23
882
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
Table 3 Summary of projects 1, 2 and 3 Project 1 Units sold 65 95 100 100 100 m s 12 Project 2 Units sold 160 190 200 200 200 m s 22 s 12 Project 3 Units sold 130 130 150 150 150 m s 32 s 13 s 23
Exp unit pr. 50 40 30 20 15
Revenue 3250 3800 3000 2000 1500
Cost cf1j 1731.67 1531.67 1331.67 1131.67 931.67
Cost cv1j 600 775 400 400 400
a 1j
Exp unit pr.
Revenue
Cost cf2j
Cost cv2j
a 2j
50 40 30 20 15
8000 7600 6000 4000 3000
2886.12 2552.78 2219.45 1886.12 1552.78
1200 1350 1400 1400 1400
Exp unit pr.
Revenue
Cost cf3j
Cost cv3j
a 3j
50 40 30 20 15
6500 5200 3000 2400 2250
2308.89 2042.23 1775.56 1508.89 1242.23
850 1000 600 600 600
1.000 0.800 0.500 0.250 0.000 1708.29 4 918 300 23 519 749 211 516 770
generation expansion, each with ®ve periods (N 3; T 5). Let us assume that the market price remains the same for all the projects in a period. Let it have the variation across periods as shown in Table 1. The projects 1, 2 and 3 have costs as shown in Table 2. Table 3 shows the total revenue generated per period, costs per period and conditional probabilities in each period. For example, the conditional probabilities in each period of project 1 is a11 0:9; a12 1:0; a13 0:88; a14 0:875; a15 1:0: Further, let l21 11 0:8; l31 11 0:95; l31 21 0:9: Table 4 The possible choices of project selection x1
x2
x3
E
V
0 1 0 0 1 1 0 1
0 0 1 0 1 0 1 1
0 0 0 1 0 1 1 1
0 2134.29 3321.03 1708.29 5455.32 5029.32 3842.58 7163.61
8 53 4 62 35 6 37
293 896 918 189 781 171 034
0 182 324 300 507 084 983 768
0.900 1.000 0.888 0.875 1.000 2134.293 8 293 182.0
0.800 0.875 0.857 0.833 0.800 3321.03 53 896 324 0.0
Now from the M±V method (9), the problem is as follows: min V x21 s 12 1 x22 s 22 1 x23 s 32 1 2x1 x2 s 12 1 2x1 x3 s 13
x1 ;x2 ;x3
1 2x2 x3 s 23 s:t: m1 x1 1 m2 x2 1 m3 x3 $ E x1 ; x2 ; x3 Binary 0 or 1 where x1, x2, x3 are the choices of projects to be selected. Table 4 shows the table of possible choices of project selection. From this, we ®rst point out that, since l21 11 0:8 a21 ; s 12 0:0: Also l31 11 0:95 ,a31 1:0; s 13 is negative. Similarly, s 23 is also negative as l 31 21 , a31 : As for the optimal project selection, suppose the minimum pro®t from all projects selected should be E 3500; it is optimal to select projects 2 and 3. If E 5000; it is optimal to select projects 1 and 3. 5. Conclusion In this paper, we have mathematically formulated and
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
analyzed a generation expansion planning model for electric utilities facing highly stochastic stranded cost. The stochastic nature of the stranded cost is due to deregulation of the electric power industry, from regulated monopolies toward market competition. In formulating the generation expansion plan, we showed how the stochastic nature can be captured by a random variable based on experts' opinion and how the corresponding risk factor can be incorporated into the decisions making process via the M±V method. Next, by examining the covariances among periods and projects, we have derived conditions (i.e. smaller size problems) under which the formulation may be implemented easily. When the problem size is larger, an approximation technique is developed. In addition, we have derived managerial insights stating that, with a set of negatively correlated projects, the risk will be decreased. However with a set of positively correlated projects, the risk will be increased. Finally, we provide a numerical example illustrating the data requirements, the computation of covariances, and the optimal project selection based on the M±V method. Research in the area of generation expansion planning with stochastic stranded cost is only recently emerging. Hence, numerous future research problems exist. For example, instead of conventional probability estimation of experts' opinions, an alternative approach such as fuzzy sets [2] can be considered. Also, instead of the simple indicator random variable in this paper, one can consider factors that contribute to determine the indicator as random variables (e.g. electric power price, natural gas and coal prices for generation fuel). In fact, one can characterize the entire cash ¯ows as random variables. Such an alternative may require more sophisticated methodologies (e.g. chance constrained programming [16]). We believe it would be interesting to examine the bene®ts and costs of these alternative extensions. Acknowledgements We would like to thank the two anonymous referees, Dr Yuhong Yang, and Dr H.T. David for helpful comments. We are also grateful for ®nancial support from the Electric Power Research Center at Iowa State University and MidAmerican Energy Co.
Cov wk1 wk2 Ewk1 2 E wk1 wk2 2 E wk2 E wk1 wk2 2 E wk1 E wk2 Now E wk1 wk2 1 £ 1 £ P wk1 1; wk2 1 1 1 £ 0 £ P wk1 1; wk2 0 1 0 £ 1 £ P wk1 0; wk2 1 1 0 £ 0 £ P wk1 0; wk2 0 Therefore, E wk1 wk2 ak1 ak2 and Cov wk1 wk2 ak1 ak2 2 ak1 ak1 ak2 ak1 ak2 1 2 ak1 : Hence, the result is true for j 1: Let us now assumeQthat it is true for j n; 1 , Qj , T: Therefore, E wkn nz1 akz also P Wkn 1 nz1 akz : For any j n 1 1; n 1 1 # T P wkn11 1 P wkn 1P wkn 1 1 1uwkn 1 1 P wkn 0P wkn11 1uwkn 0
For any period j, at j 1; P wk1 1 ak1 ;
nY 11 z1
akz
Hence Qn 1 1 E wkn11 1 £ P wkn11 1 1 0 £ P wkn11 0 z1 akz Var wkn11 E wkn11 2 E wkn11 2 ! nY 11 nY 11 12 akz akz z1
z1
Also Cov wki wkn11 Ewki 2 E wki wkn11 2 E wkn11 E wki wkn11 2 E wki E wkn11 for any i , n 1 1: Now, E wki wkn11 1 £ 1 £ P wki 1; wkn11 1
nY 11 z1
akz
Therefore, Cov wki wkn11
nY 11 z1
akz 2
12
i Y z1
Appendix A. Proof of Proposition 1
883
i Y z1
akz
akz
nY 11 z1
! n11 Y z1
akz
akz :
By induction, Eqs. (11)±(13) follow for any j 1 # j # T: A
E wk1 1P wk1 1 1 0P wk1 0 ak1
Appendix B. Proof of Proposition 2
Var wk1 E wk1 2 E wk1 2
(a) At t1 1; E wh1 wk1 1 £ 1 £ P wh1 1 £ P wk1 1uwh1 1; as all other cases are zero, since wk1 and wh1 are binary. Hence E wh1 wk1 ah1 lk1 h1 : As in Appendix A, via mathematical induction, the result in Eq. (16) can be shown.
1 2 ak1 2 P wk1 1 1 0 2 ak1 2 P wk1 0 1 2 ak1 ak1
884
K. Jo Min, P.S. Subramaniam / Electrical Power and Energy Systems 24 (2002) 875±885
(b) At t1 1; t2 2; E wh1 wk2 P wh1 1P wk2 1uwh1 1 P wk2 1uwh1 1 P wk2 1; wk1 1uwh1 11P(wk2 1,wk1 0uwh1 1) ak2lk1 h1 (by assumption P wk2 1uwk1 1; wh1 1 P wk2 1uwk1 1: Hence E wh1 wk2 ah1 ak2 lk1 h1 : As in Appendix A, via mathematical induction, the result in Eq. (17) can be shown. A
(3a) At t2 1 and any t1, by Proposition 2a, we have 0 1 t1 Y E wht1 wkt2 @ ahy Alk1 h1 y1
Now E wht1 wk1 2 E wht1 E wk1 Hence
Qt1
y1
ahy lk1 h1 2 ak1 :
E wht1 wk1 , E wht1 ; E wk1 , 0; if lk1 h1 , ak1
(3b) At t2 t2 . 1 and any t1, by Proposition 2b, we have 0 1 ! t1 t2 Y Y E wht1 wkt2 @ ahy A akz lk1 h1
Appendix C. Proof of Proposition 3 From Eq. (14), we have Cov Rh ; Rk
X t11 to T t21 to T
1 1 1 It11t2
"
! qht1 uht1 2
y1
cN ht1
E wht1 wkt2 2 E wht1 E wkt2
qkt2 ukt2 2 cN kt2 E wht1 wkt2
2 qkt2 ukt2 2
cN kt2
cSkt2 E wht1 # cSht1 E wkt2
1
cSht1 cSkt2
Expanding the expressions for mh E Rh and mk E Rk ; we get " ! X 1 N qht1 uht1 2 cht1 Cov Rh ; Rk t11t2 t11 to T 1 1 I !
qkt2 ukt2 2 cN kt2 E wht1 wkt2 ! 2 qht1 uht1 2
cN ht1
2 qkt2 ukt2 2
cN kt2
cSkt2 E wht1 #
!
2
X t11 to T t21 to T
cSht1 E wkt2
1 1 1 It11t2
qkt2 ukt2 2
1
cSht1 cSkt2
"
! qht1 uht1 2
cN ht1
! cN kt2
E wht1 E wkt2 !
2 qht1 uht1 2
cN ht1
2 qkt2 ukt2 2
cN kt2
y1
ahy
t2 Y z1
akz
lk1 h1 21 ak1
E wht1 wkt2 , E wht1 ; E wkt2 , 0; if lk1 h1 , ak1 C2
!
t21 to T
t1 Y
Hence
! 2 qht1 uht1 2
z2
Now
!
cN ht1
C1
cSkt2 E wht1 #
! cSht1 E wkt2
1
cSht1 cSkt2
It now suf®ces to show that Cov Rh ; Rk will be negative if E wht1 wkt2 , E wht1 E wkt2 for all 1 # t1 # T and 1 # t2 # T:
From Eqs. (C1) and (C2), the result in Proposition 3 follows. Conversely, it follows that Cov(Rh, Rk) is positive if lk1 h1 . ak1 : Finally, if lk1 h1 ak1 ; then by de®nition Cov Rh ; Rk 0: A References [1] Ammons JC, McGinnis LF. A generation expansion planning model for electric utilities. Engng Econ 1984;30:205±25. [2] Bernardini A. What are the random and fuzzy sets and how to use them for uncertainty modeling in engineering systems? In: Elishakoff I, editor. Whys and hows in uncertainty modeling. New York: Springer, 1999. Chapter 2. [3] Bierman H, Smidt S. The capital budgeting decision. 3rd ed. New York: Macmillan, 1984. [4] Brown MJ. A mean±variance serial replacement decision model: the correlated case. Engng Econ 1993;38(3):237±47. [5] Freund JE. Mathematical statistics. 5th ed. New Jersey: Prentice-Hall, 1992. p. 71. [6] Giaccotto C. A simpli®ed approach to risk analysis in capital budgeting with serially correlated cash ¯ows. Engng Econ 1984;29(4):273± 86. [7] Grif®th M, Robinson T. Economic value and capital recoveryÐa regulatory model. Publ Util Fortnight 1985;11:30. [8] Kolbe AL, Tye WB. Compensation for the risk of stranded costs. Energy Policy 1996;24(12):1025±50. [9] Markowitz H. Mean variance analysis in portfolio choice and capital markets. New York: Basil Blackwell, 1987. [10] Martin AD. Mathematical programming of portfolio selections. Mgmt Sci 1955;1(2):152±66. [11] Of®ce of Consumer Advocate, Prepared Direct Testimony of D. Scott Norwood, RE: MidAmerican Energy Company, Docket no. APP-961, September 1996. [12] Pospisil R. Nuclear plants under deregulation. Electl World 1995:56±7. [13] Resnick S. Adventures in stochastic processes. Boston: Birkhauser, 1992. p. 73. [14] Robichek AA, Myers SC. Optimal ®nancing decisions. Englewood Cliffs, NJ: Prentice-Hall, 1965. p. 79±93.
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