Cogeneration planning under uncertainty. Part II: Decision theory-based assessment of planning alternatives

Applied Energy 88 (2011) 1075–1083

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Cogeneration planning under uncertainty. Part II: Decision theory-based assessment of planning alternatives Enrico Carpaneto a, Gianfranco Chicco a,⇑, Pierluigi Mancarella b, Angela Russo a a b

Dipartimento di Ingegneria Elettrica, Politecnico di Torino, corso Duca degli Abruzzi 24, I-10129 Torino, Italy Dept. of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, SW7 2AZ London, UK

a r t i c l e

i n f o

Article history: Received 16 March 2010 Received in revised form 6 August 2010 Accepted 16 August 2010 Available online 29 September 2010 Keywords: Cogeneration planning Decision theory Internal combustion engines Microturbines Uncertainty modeling Weighted regret criterion

a b s t r a c t This paper discusses specific models and analyses to select the best cogeneration planning solution in the presence of uncertainties on a long-term time scale, completing the approach formulated in the companion paper (Part I). The most convenient solutions are identified among a pre-defined set of planning alternatives according to decision theory-based criteria, upon definition of weighted scenarios and by using the exceeding probabilities of suitable economic indicators as decision variables. Application of the criteria to a real energy system with various technological alternatives operated under different control strategies is illustrated and discussed. The results obtained show that using the Net Present Cost indicator it is always possible to apply the decision theory concepts to select the best planning alternative. Other economic indicators like Discounted Payback Period and Internal Rate of Return exhibit possible application limits for cogeneration planning within the decision theory framework. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction This paper completes the formulation of the general framework introduced in the companion paper [1] by dealing with the details of taking into account large-scale uncertainties in the long-term time frame. In the multi-year time horizon for cogeneration planning, it is difficult to envision the expected trends of evolution of energy loads and, even more, of electricity and gas prices. Hence, proper alternative techniques of assessment of the foreseeable solutions have to be identified and applied. Literature studies have been based on sensitivity analyses of specific indicators (typically economic variables [2]) with respect to electricity and gas price variations. For instance, the simple payback period and the Internal Rate of Return have been used as indicators in [3] to carry out sensitivity analyses with respect to the variation of electricity price, fuel price and investment cost, for a Combined Heat and Power (CHP) application. Similarly, a deterministic approach has been used in [4] to find the most convenient technological alternative among a set of pre-defined candidate alternatives through minimization of the annualized total costs. Then, a sensitivity analysis has been performed to address the effects of upgraded performance of the equipment, reduction in the initial capital costs, and reduction in the electricity and gas prices. Sensitivity analyses have also been performed in [5] to test the robustness of the optimal solutions ⇑ Corresponding author. Tel.: +39 011 090 7141; fax: +39 011 090 7199. E-mail address: [email protected] (G. Chicco). 0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.08.019

found for cogeneration systems coupled to cooling generation devices in the presence of large variations of energy market prices. In general terms, sensitivity analyses are useful to get indications on the effects of pre-defined scenarios of variation of relevant variables. However, they give no information on how to combine the results obtained from different individual scenarios, and on the effects of actual occurrence of a scenario after the plant is installed. In order to get additional insights in this direction, different levels of involvement of the decision-maker can be considered [6]. In particular, the decision-maker can actively participate in the decision process, for instance choosing the scenarios to be analyzed and assigning to each of them a relative weight on the basis of specific expertise or personal preferences. In this way, the characteristics of alternative planning solutions available to the decision-maker can be explained by looking at the results obtained in each combined scenario considered. Approaches moving in this direction have been proposed for distributed generation siting and sizing in [7,8]. The nature of the results to be obtained (e.g., deterministic or probabilistic) is a further element driving the choice of the type of analysis. For instance, when taking into account uncertainty, the results can be conveniently expressed in probabilistic terms, providing the probability distributions of the planning outcomes. In this respect, it is possible to exploit the framework proposed in [1]. Instead of evaluating only best or worst cases, the hazard to which the decision-maker is exposed because of uncertainty is represented by the probability of occurrence of the outcomes. The relevant aspect is the evaluation of the probability of

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List of symbols a b c e f j n% p r t x y A C CAB CCHP Fr J

(index) technological alternative generic real number Net Present Cost instance (€) (subscript) electrical objective function (index) Monte Carlo extraction percentage of exceeding probability probability generic random number (instance of the RV r) (subscript) thermal (index) scenario (index) long-term time frame (year) number of technological alternatives cost (€) investment cost of the Auxiliary Boiler (€) investment cost of the CHP system (€) Cumulative Distribution Function of the RV r number of Monte Carlo extractions

occurrence of a specific outcome, compared to a threshold level of exceeding probability defined by the decision-maker, as typically done within the domain of application of risk analysis techniques [3,9–12]. In this paper, an approach based on decision theory concepts [13] is used to enable the decision-maker to maintain a significant level of interaction in drawing the scenarios to be analyzed and in interpreting the results obtained. Decision theory has been applied to cogeneration planning by the authors in [14] by considering a single objective to be minimized (the expected value of the Net Present Cost). This paper extends the analysis in different directions to model and discuss the large-scale uncertainty issues identified in [1]. The comprehensive mathematical formulation provided is used with different objective functions (to be minimized or maximized) and different decision criteria (minimum expected value, minimax weighted regret, and a mixed optimist–pessimist criterion). The planning outcomes are then expressed with their probability distributions, enabling the decision-maker to choose the level of exceeding probability to be considered. The proposed framework is applied to select the most convenient CHP solution (type, size and control strategy) among a predefined set of alternatives, considering the multiple scenarios of long-term evolution of energy loads and prices identified by the decision-maker. A business-as-usual case is considered as the reference alternative, with separate production (SP) of electricity from the electricity distribution system (EDS) and heat generated in conventional boilers. The other technological alternatives include different cogeneration technologies such as microturbines (MTs) or internal combustion engines (ICEs), with specified electrical and thermal rated power. Each technological alternative is operated under one of the control strategies (or operating modes [15]) described in the companion paper [1], namely, on–off operation, electrical load-following and thermal load-following. This paper is organized as follows. Section 2 illustrates the formulation of the optimization problems in the long-term time frame. Section 3 describes the decision criteria adopted to cope with large-scale uncertainty. Section 4 shows and discusses the results obtained in the application case defined in [1] with specific technological alternatives, control strategies, economic indicators, and decision theory criteria. Section 5 summarizes the concluding remarks referred to the application of the decision theory concepts within the comprehensive multiple time frame approach.

R X Y

a c(y) d

jr lr m #

1 t n

x U N X r

regret felt by the decision-maker (€) number of scenarios number of long-term time frames (years) weighting factor for the optimist–‘‘pessimist” criterion fuel price at year y (€/kWh) discount rate annual rate of increase for the RV r expected value of the RV r dummy variable Discounted Payback Period instance (years) (subscript) price availability coefficient of the CHP unit control strategy of the CHP unit planning alternative cash flow (€) number of control strategies number of planning alternatives (RV) generic random variable

2. Formulation of the optimization problem in the long-term time frame The input information to formulate the optimization problem is based on the random variables (RVs) obtained through the approach illustrated in [1] for short- and medium-term time frames. These RVs represent typical economic indicators adopted for the analysis of investments in the cogeneration sector, such as the Net Present Cost (NPC), the Net Present Value (NPV), the Discounted Payback Period (DPP), and the Internal Rate of Return (IRR) [2,16]. This section illustrates how the various economic indicators can be adopted in the formulation of optimization problems in the decision theory-based planning framework to deal with the long-term time frame issues. Specific limitations in the use of the DPP and IRR indicators within this framework are further discussed in Section 4.5. A planning alternative x = 1, . . ., X is defined as the pair (a, n) given by the technological alternative a with the associated CHP control strategy n, in addition to separate production from the EDS and from the auxiliary boiler (AB). Considering an economic indicator (RV r with n% exceeding probability) to be minimized (as for NPC and DPP), the optimization problem is formulated as ^

xr;n% ¼ arg minff ða; n; r; nÞg

ð1Þ

x¼1;...;X

If the economic indicator has to be maximized (as for NPV or IRR), the minimization indicated in (1) has to be changed into maximization. The constraints associated to the optimization problems refer to the equipment limits in the various control strategies, as shown in Section 3.4.3 of [1]. In the optimization problem formulation, the equipment investðaÞ ment costs, namely, CAB for purchasing the AB and C CHP for purchasing the CHP system for the ath technological alternative, are considered as deterministic entries and are referred to the beginning of the period of analysis.1 1 This direct type of investment is one of the possible investment strategies, selected here for comparing the planning alternatives. More refined strategies of investment such as sequential ones, in which multiple equipment is purchased at different times, can be developed, also depending on price volatilities [18]. The analysis of these strategies is outside the scope of this paper.

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Let us denote as r the RV representing a generic economic indicator. The Cumulative Distribution Function (CDF) F ða;n;xÞ for the RV r r is obtained through Monte Carlo simulations over the total duration of the period of analysis. More specifically, let us consider the ða;n;x;yÞ annual variable cost C j corresponding to the jth Monte Carlo extraction for the cogeneration alternative a = 1,. . ., A with control strategy n = 1, . . ., N and scenario x = 1,. . ., X at year y = 1,. . ., Y, calculated as indicated in Section 3.4.5 of [1]. The definitions of the economic indicators considered as RVs are recalled here, showing the specific notation used in this paper:

x = 1,. . ., X defined in the long-term time frame is associated to a probability of occurrence p(x), identified by the operator as part of the decision-making process. Table 1 shows the load and energy price rates of increase considered in the different scenarios. The maximum annual rate of increase considered is 7% (jr = 0.07), roughly corresponding to doubling the initial amount after 10 years. Each scenario has been applied by varying the hourly loads and prices according to the relevant annual rate of increase. 3.2. Decision criteria

s Net Present Cost2: the NPC is calculated accounting for all the costs brought to the considered initial time through standard present worth analysis [17]:

ða;n;xÞ

cj

ða;nÞ

¼ C AB þ C CHP þ

ða;n;x;yÞ Y X Cj y¼1

ð1 þ dÞy

ð2Þ

s Discounted Payback Period: the DPP is defined by taking into account the difference between the total costs of the planning alternative under study and the total costs in the SP businessas-usual case. In the reference year, only the investment cost for buying the CHP system is considered as initial negative cash flow (the same number of boilers in both SP and CHP cases ensures full heat capacity reserve). Then, the discounted cash flows at the successive years are added until their sum becomes positive. For each CHP alternative a (excluding SP) with control strategy n, and for each scenario x = 1,. . ., X, in the jth Monte ða;n;x;yÞ Carlo extraction, the cumulative cash flow Uj at year y is ða;nÞ Uða;n;x;yÞ ¼ C CHP  j

ða;n;x;mÞ y ~ ðx;mÞ X Cj C j

m¼1

ð1 þ dÞm

On the basis of the n% exceeding probabilities of the RVs representing the economic indicators indicated in Section 2, the decision-maker identifies the most convenient planning alternative by applying proper decision criteria, such as:

ð3Þ

ðx;mÞ

~ where C is the annual variable cost of SP, calculated as indij ða;n;xÞ cated in Section 3.4.4 of [1]. The DPP #j is then calculated as the time at which the discounted cash flow becomes null. However, if the total cash flow remains negative for the entire period of analysis, the solution under analysis is never more convenient than the SP, and the resulting DPP is undetermined. s Internal Rate of Return: for each CHP alternative a = 1,. . ., A with control strategy n = 1, . . ., N (excluding SP) and for each scenario x = 1,. . ., X, the IRR instance at the jth Monte Carlo extraction is _ ða;n;xÞ defined as the value d j for which the cumulative cash flows become equal to zero at the end of the period of analysis. Again, the cash flows are calculated by taking the difference between the costs of the planning alternative under consideration and the corresponding SP _case. The IRR instance is obtained ða;n;xÞ by solving with respect to d j the non-linear equation

1. Minimum expected value3criterion [13]: considering the entry ða;n;xÞ F r;n% calculated from the RV r with n% exceeding probability,4 for the ath CHP alternative with nth control strategy, in the xth scenario with associated probability p(x), the objective function used in (1) is formulated as

f ða; n; r; nÞ ¼

X X

ðxÞ

ða;n;xÞ

F r;n% ¼ min

x¼1;...;X

ð6Þ

ða;n;xÞ

ða;n;xÞ

ðxÞ

Rr;n% ¼ F r;n%  F r;n%

ð7Þ

and the objective function in Eq. (1) is expressed as

f ða; n; r; nÞ ¼ max

x¼1;...;X

n

ða;n;xÞ

pðxÞ Rr;n%

o

ð8Þ

If the economic indicator has to be maximized (NPV or IRR), the (positive) regrets are obtained as a¼1;...;A

~ ðx;yÞ Y C ða;n;x;yÞ  C X j j  y ¼ 0 _ ða;n;xÞ y¼1 1 þ dj

F r;n%

The regret felt by a decision-maker for having chosen the planning alternative x = (a, n) in the xth scenario is

ða;n;xÞ

ða;nÞ

ð5Þ

In particular, in the equal likelihood criterion [13] equal probabilities p(x) = 1/X are set in (5) for x = 1,. . ., X. 2. Minimax weighted regret criterion [9,19]: if the economic indicator r is minimized (as for NPC or DPP), the optimal planning alternative among x = 1, . . ., X is the one that minimizes the regret felt by the decision-maker, after verifying that, given the outcomes obtained, the decisions made beforehand were not the optimal ones.5 The CDF of the RV r evaluated with n% exceeding probability n ois

Rr;n% ¼ max C CHP þ

ða;n;xÞ

pðxÞ F r;n%

x¼1

n o ða;n;xÞ ða;n;xÞ F r;n%  F r;n%

ð9Þ

ð4Þ

3. Decision-making in the decision theory framework 3.1. Definition of the scenarios Each scenario is defined by setting up a specific trend of variation for the expected values and the standard deviations of the relevant RVs. For instance, an exponential type trend can be represented as illustrated in Section 3.2 of [1]. Each scenario 2 Alternatively, the Net Present Value (NPV) could be used, that is, the opposite of the NPC [16]. In this paper, the calculations are carried out by using the NPC indicator.

3 In the classical decision theory terminology, the performance of the various alternatives is evaluated by means of a so-called ‘‘cost function”. In this paper, the term ‘‘cost function” is not used to avoid ambiguities with the various costs associated to the equipment purchase and operation. Hence, the classical decision theory criterion ‘‘minimum expected cost” is called here ‘‘minimum expected value” (of the relevant economic indicator). 4 If the economic indicator is minimized, when the value of n% decreases the outputs of the economic indicator become increasingly higher, and as such are more suitable to be evaluated by a decision maker more inclined to avoid the investment risks. The values of n% exceeding probability considered in this paper are 50% (median) and 5% (representing decision-makers having a bent for risk avoidance). When the economic indicator is maximized n% = 95% exceeding probability represents risk avoidance. 5 The concept of regrets has also been used in [20] to determine the optimal size of the cogeneration equipment with uncertain energy demands by minimizing the normalized maximum regret.

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Table 1 Load and price annual rates of increase in different scenarios. Scenario

S1 S2 S3 S4 S5 S6 S7 S8 S9

Table 2 Data of the alternative technologies.

Annual rates of increase

Alternative

Electrical load (%)

Thermal load (%)

Electricity price

Gas price

1 1 1 1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0 0 0 4% 7% 4% 4% 7% 7%

0 4% 7% 0 0 4% 7% 4% 7%

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

and the optimal planning alternative is identified once again by using the objective function (8). 3. Optimist and ‘‘pessimist” criteria [21]: considering an economic indicator to be minimized, the optimist criterion is applied by determining the overall minimum value of the indicator, corresponding to the best possible outcome for each scenario. The objective function used in (1) is

f ða; n; r; nÞ ¼ min

n

x¼1;...;X

ða;n;xÞ

F r;n%

o

ð10Þ

The ‘‘pessimist” criterion is found by determining for each planning alternative the worst possible outcome (maximum value) of the indicator in the various scenarios, then searching for the minimum among the values obtained.6 The objective function used in (1) is

f ða; n; r; nÞ ¼ max

n

x¼1;...;X

ða;n;xÞ

F r;n%

o

ð11Þ

Furthermore, it is possible to use a mixed optimist–‘‘pessimist” criterion, by introducing a weighting factor a 2 ½0; 1 to represent the transition between the solutions of (10) and (11), with objective function expressed as

f ða; n; r; nÞ ¼ a min

x¼1;...;X

n o n o ða;n;xÞ ða;n;xÞ F r;n% þ ð1  aÞ max F r;n% x¼1;...;X

ð12Þ

If the economic indicator has to be maximized, the corresponding versions of the criteria are obtained from (10)–(12) by swapping all maximizations with minimizations and vice versa.

Local unit

Rated efficiencies

Technology

Size (kWe)

Electrical

Thermal

Boiler

MT MT MT ICE ICE ICE ICE ICE ICE Separate production

100 200 300 100 300 500 700 1000 1500 –

0.3 0.3 0.3 0.35 0.38 0.38 0.39 0.41 0.43 –

0.5 0.5 0.5 0.4 0.42 0.42 0.43 0.44 0.45 –

0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9

O&M cost (€/kWhe) 0.007 0.006 0.005 0.020 0.016 0.012 0.010 0.010 0.010 0

thermal load indicated in Table 1 for 15 years (leading to an overall heat demand increase of 7.8%), with a further margin of 30%. In the specific case, the rated power of the AB is 1.3 MWt. The last alternative in the list is the SP of electricity from EDS and heat from AB. Each CHP alternative other than SP can be run under the on–off operation, electrical load tracking and thermal load tracking control strategies ([1], Section 3.4.3). Concerning investment costs, a single specific cost (1500 €/kWe) is used for MTs, in the light of cluster operation, and for the AB (150 €/kWt). Decreasing specific costs for larger sizes are used for the ICEs, namely, 1200, 1100, 1000, 950, 850, and 800 €/kWe for the ICE100, ICE300, ICE500, ICE700, ICE1000, and ICE1500, respectively. CHP operation and maintenance (O&M) costs (last column of Table 2) are taken into account in proportion to the electrical energy consumption [3,24]. The O&M costs for SP are assumed to be negligible with respect to the CHP ones. ðaÞ The availability coefficient tmk [1] is considered equal to unity for SP and for the alternatives with MTs.7 For the alternatives with Internal Combustion Engine (ICE), unavailability of the ICE in the time period of annual scheduled maintenance is taken into account.8 Since the sequence of the time steps is not represented in the smallscale uncertainty model [1], the unavailability is uniformly distributed among the groups of elementary time intervals considered. ðaÞ The same value tmk = 0.92 is assumed for any short-term time interval k in the medium-term time frame m.

4. Case study application The solution of the cogeneration planning problems formulated in Section 2 is addressed with reference to the characterization of the energy loads and prices illustrated in the companion paper [1]. The time interval of observation is Y = 15 years. A fixed discount rate of 5% is assumed. 4.1. Technological alternatives Table 2 shows the data of the technological alternatives considered. Data in Table 2 represent synthesized figures from available literature and reports [22,23], and from commercial catalogues. In order to satisfy the thermal energy production at any time, an AB group is included. This AB group is sized to cover the maximum thermal load, by assuming the 0.5% annual rate of increase of the 6 The extreme pessimistic outcome would be obtained by taking the overall worst solution, with maximization in (11) and maximization rather than minimization in (1). However, according to a common practice in the literature [21], the minimum in (1) is still used rather than the maximum in order to mitigate the pessimism in the application of the criterion. For this reason, the term ‘‘pessimistic” is indicated here in quotation marks.

4.2. Monte Carlo simulation results Starting from the data set of the reference year, the load and price rates of increase in the various scenarios defined in Table 1 have been applied to calculate the average values and the covariance matrices for each year and for each technological alternative and control strategy. For each planning alternative and each scenario at each year, J = 1000 Monte Carlo extractions have been run to build the CDFs of the economic indicators of interest (NPC, DPP and IRR). The exceeding probabilities used are 50% and 5% for NPC and DPP, or 50% and 95% for IRR. 7 In the application shown in this paper, MTs are considered to be operated in clusters, assuming that the scheduled maintenance of the MTs is carried out in different periods for each unit, thus with no impact on their availability. MTs are relatively new technologies, with limited historical reliability data. The outage rates of MTs are indicated in [25] to be relatively small compared with the ones of ICEs. 8 Aspects such as efficiency, availability and maintenance costs of the alternatives could generally be considered as uncertain or variable during the period of analysis. They are treated here as constant deterministic values with the aim to highlight those aspects related to energy price evolution.

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E. Carpaneto et al. / Applied Energy 88 (2011) 1075–1083 Table 3 Weighting probabilities associated to the scenarios. Scenario

S1 S2 S3 S4 S5 S6 S7 S8 S9

Table 4 Net Present Costs with 50% exceeding probability (106 €).

Probability

Alternative

Case 1

Case 2

Case 3

Case 4

0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111

0.04 0.04 0.04 0.04 0.04 0.25 0.15 0.15 0.25

0.06 0.06 0.06 0.12 0.15 0.12 0.12 0.15 0.16

0.06 0.06 0.06 0.12 0.06 0.12 0.12 0.20 0.20

Specific cases have been defined for different probability of occurrence assigned to each scenario (Table 3):  Case 1 assumes equal weights for all scenarios,9 corresponding to apply the equal likelihood criterion;  Case 2 weights much more the scenarios S6–S9, with higher price increase, and within them especially the scenarios S6 and S9 in which there is a similar increase for the electricity and gas prices.  Case 3 is focused on giving higher importance to the scenarios with high electricity price rate of increase, providing relatively low weights to the scenarios S1, S2 and S3 in which the electricity price does not change, intermediate weights to the scenarios S4, S6 and S7 with 4% rate of increase of the electricity price, and relatively high weights to the scenarios S5, S8 and S9 with high rate of increase of the electricity price.  Case 4 has a rationale similar to Case 3, but reflects a possibly more refined reasoning of a decision-maker who trusts less the situation of scenario S5 in which the rate of increase of the electricity and gas prices is very different (reducing the weight for this scenario), and decides to increase the weights for scenarios S8 and S9 in which there is a consistently high rate of increase of the electricity and gas prices. For the sake of clarity, the full details of application of the decision theory approach are shown here for the NPC economic indicator and for a single control strategy (on–off operation) of the CHP units. The overall results are then summarized for planning alternatives with different control strategies. Finally, the use of DPP and IRR is discussed. 4.3. Application of decision theory criteria to the NPC results with on– off operation 4.3.1. Minimum expected value criterion Tables 4 and 5 show the 50% and 5% exceeding probability values of the NPC, respectively, for the individual scenarios and for each planning alternative.10 These results clearly indicate different situations of convenience, depending on the scenarios. The SP alternative is always convenient for the scenarios S2, S3 and S7, characterized by rates of increase of the gas price higher than the ones of the electricity price. In these scenarios, the NPC of the MT and ICE alternatives increases with the size, and SP is already more convenient than the smaller CHP unit sizes for MT100 and ICE100. The scenario S1, with constant electricity and gas prices, leads to a single solution slightly better than SP. Conversely, in the scenarios S4, S5 and S8, with rate of increase of the gas price equal to or lower than the one of the electricity price, the on–off control strategy makes MT 9 10

In Table 3, the weights indicated for Case 1 are truncated to the third decimal. Bold values in Tables 4 to 9 indicate the best alternative found for each scenario.

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Scenario S1

S2

S3

S4

S5

S6

S7

S8

S9

3.10 3.19 3.35 3.21 3.22 3.28 3.34 3.49 3.76 3.12

3.63 3.89 4.25 3.72 3.99 4.35 4.74 5.36 6.40 3.51

4.16 4.58 5.13 4.25 4.75 5.41 6.11 7.19 9.00 3.90

3.41 3.29 3.25 3.53 3.18 2.86 2.54 2.12 1.46 3.63

3.72 3.40 3.15 3.86 3.13 2.44 1.76 0.77 0.81 4.14

3.95 4.00 4.14 4.05 3.95 3.93 3.93 3.98 4.11 4.02

4.47 4.69 5.03 4.57 4.70 4.99 5.31 5.83 6.75 4.41

4.26 4.10 4.06 4.38 3.90 3.51 3.16 2.66 1.84 4.53

4.77 4.80 4.94 4.89 4.66 4.57 4.52 4.50 4.44 4.93

Table 5 Net Present Costs with 5% exceeding probability (106 €). Alternative

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Scenario S1

S2

S3

S4

S5

S6

S7

S8

S9

3.20 3.30 3.49 3.31 3.34 3.44 3.55 3.80 4.20 3.22

3.75 4.02 4.42 3.84 4.14 4.55 5.00 5.72 6.96 3.63

4.29 4.75 5.33 4.38 4.93 5.65 6.42 7.61 9.65 4.02

3.53 3.41 3.40 3.65 3.30 3.02 2.76 2.46 1.98 3.75

3.86 3.53 3.30 4.00 3.26 2.61 2.01 1.16 0.24 4.29

4.08 4.14 4.32 4.17 4.10 4.13 4.22 4.39 4.69 4.17

4.62 4.87 5.24 4.71 4.88 5.23 5.64 6.28 7.44 4.56

4.40 4.25 4.22 4.54 4.05 3.72 3.46 3.06 2.47 4.71

4.93 4.97 5.14 5.05 4.84 4.82 4.86 4.99 5.23 5.12

or ICE more and more profitable when their size increases. This is due to the possibility of selling to the EDS all the electricity produced in excess of the local consumption. ICE sizes are shown up to 1500 kWe, at which size the NPC would even be negative in the ‘‘extreme” scenario S5 with high rate of increase of the electricity price and constant gas price. In the scenarios S6 and S9, with electricity and gas prices increasing at the same rate, more solutions are close to the most convenient one, with different results for 50% and 5% exceeding probability. The results for the individual scenarios provide diverse information to the decision-maker. In part of the cases the indication is to ‘‘do nothing” and maintain the business-as-usual SP, while in most of the other cases the solution suggests to install a large-size ICE. In order to assist the decision process, the scenarios can be further weighted according to the user-defined numerical weights indicated in Table 3. Table 6 shows the expected values of the various NPC cases with 50% and 5% exceeding probability corresponding to the weighted scenarios. The SP alternative never appears as the

Table 6 Expected values resulting from the weighted scenarios with 50% and 5% exceeding probability of the Net Present Costs (106 €). Alternative

Weighted scenario case and n% exceeding probability Case 1

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Case 2

Case 3

Case 4

50%

5%

50%

5%

50%

5%

50%

5%

3.94 3.99 4.15 4.05 3.94 3.93 3.93 3.99 4.10 4.02

4.07 4.14 4.32 4.18 4.09 4.13 4.21 4.39 4.71 4.16

4.21 4.25 4.40 4.32 4.17 4.13 4.12 4.15 4.22 4.31

4.35 4.41 4.58 4.46 4.33 4.35 4.43 4.58 4.87 4.47

4.03 4.03 4.13 4.15 3.94 3.82 3.72 3.63 3.49 4.17

4.17 4.18 4.31 4.28 4.09 4.03 4.01 4.03 4.11 4.32

4.10 4.12 4.24 4.22 4.04 3.96 3.90 3.87 3.83 4.22

4.24 4.27 4.42 4.35 4.19 4.17 4.20 4.28 4.47 4.37

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E. Carpaneto et al. / Applied Energy 88 (2011) 1075–1083

Table 7 Decision matrix of the weighted regrets based on the 50% exceeding probability of the Net Present Costs and maximum weighted regrets (103 €) – Case 1. Alternative

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Scenario S1

S2

S3

S4

S5

S6

S7

S8

S9

max

0.0 9.6 27.5 11.5 13.5 19.6 26.5 42.6 73.2 1.6

13.6 42.0 81.5 23.6 53.3 93.4 136.2 204.8 320.4 0.0

28.5 75.9 137.1 38.7 94.8 167.8 246.1 365.2 567.3 0.0

217.4 204.1 199.5 230.5 191.8 155.5 119.7 74.0 0.0 241.5

503.5 468.5 440.7 519.7 438.5 361.3 285.9 175.9 0.0 550.4

2.5 7.4 23.8 13.8 2.0 0.0 0.0 6.1 19.8 10.0

6.7 31.2 68.7 17.3 32.1 64.1 99.5 157.2 259.9 0.0

268.6 251.6 247.1 282.7 229.2 185.7 146.5 91.5 0.0 299.5

37.4 39.7 55.8 50.2 24.8 14.2 8.9 6.4 0.0 54.3

503.5 468.5 440.7 519.7 438.5 361.3 285.9 365.2 567.3 550.4

Table 8 Decision matrix of the weighted regrets based on the 5% exceeding probability of the Net Present Costs and maximum weighted regrets (103 €) – Case 1. Alternative

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Scenario S1

S2

S3

S4

S5

S6

S7

S8

S9

max

0.0 11.6 32.2 12.6 15.4 26.6 39.5 67.3 111.8 2.4

13.3 43.1 87.9 23.6 56.7 102.1 152.0 232.5 369.6 0.0

30.6 81.1 146.1 39.8 100.8 181.5 266.2 399.1 625.9 0.0

172.0 159.3 157.5 185.2 146.7 115.5 87.1 53.4 0.0 197.0

455.2 418.4 392.3 470.3 388.8 316.2 249.8 154.7 0.0 503.2

0.0 6.6 25.8 9.7 1.4 5.7 15.4 33.8 67.7 9.7

6.8 34.2 76.0 16.9 35.4 74.6 119.6 191.3 320.3 0.0

214.3 198.4 194.9 230.0 175.9 139.0 110.1 65.9 0.0 249.0

12.6 16.8 35.7 26.2 2.2 0.0 5.0 18.6 45.9 33.3

455.2 418.4 392.3 470.3 388.8 316.2 266.2 399.1 625.9 503.2

most convenient one. The indication for 50% exceeding probability is to buy a large ICE when the electricity price increases much more than the gas price (Case 3 and Case 4), being in these cases convenient to sell excess electricity to the EDS. The MT alternatives exhibit increasing NPC values for increasing MT size and are never convenient. For 5% exceeding probability, all most convenient solutions are found for lower size units (up to making the MT100 slightly more convenient than the ICEs in Case 1). In fact, reducing the exceeding probability leads to introducing in the decisionmaking an increasing attitude towards avoiding the investment risks (Section 3.2). Thus, the investment risk exposure is mitigated by purchasing a CHP unit of lower size. 4.3.2. Minimax weighted regret criterion By using the minimax weighted regret criterion (8), the decision matrix of the NPC and the maximum weighted regrets for Case 1 are shown in Table 7 for the 50% exceeding probability and in Table 8 for the 5% exceeding probability. In each column, the regrets are null when they refer to the most convenient planning alternative in the corresponding scenario. The last column contains the maximum regret for each planning alternative. According to the minimax weighted regret criterion, from Table 7 the most convenient alternative becomes the ICE700. It is worth noting that the ICE700 was indicated as the most convenient solution only once, by using the minimum expected value criterion in Case 1. Thus, the solution obtained provides a trade-off encompassing the large regrets occurring both for the SP alternative in the scenarios for which the ICE1500 is most convenient, as well as for the ICE1500 alternative in the scenarios in which the SP is most convenient, while the regrets for the ICE700 solution are relatively low in most cases. Moreover, from Table 8 the use of the 5% NPC exceeding probability indicates again the ICE700 as the most convenient solution, even though the ICE700 never shows up as the best solution in the scenarios considered.

Following the same conceptual scheme of Section 4.3.1, the maximum weighted regrets obtained for the four cases with different weighted scenarios are shown and compared in Table 9. The results for Case 1 indicated in the last columns of Tables 7 and 8 are repeated in Table 9 for the sake of completeness. The preferred solution varies from the ICE700 to the ICE1000, thus confirming the consistency of the indication provided to the decision-maker. 4.3.3. Mixed optimist–‘‘pessimist” criterion In the application of the mixed optimist–‘‘pessimist” criterion, the solutions corresponding to different values of the weighting factor a may provide different optimal alternatives. The ‘‘extreme” optimistic solution corresponding to a = 1 leads to prefer the ICE1500 alternative. This solution can be easily identified, being the only negative NPC case occurring in the scenario S5 in both Table 4 (with 50% exceeding probability) and Table 5 (with 5% exceeding probability). Conversely, at the other extreme (‘‘pessi-

Table 9 Maximum weighted regrets (103 €) with 50% and 5% exceeding probability of the Net Present Costs. Alternative

Weighted scenario case and n% exceeding probability Case 1

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500 SP

Case 2

Case 3

Case 4

50%

5%

50%

5%

50%

5%

50%

5%

503.5 468.5 440.7 519.7 438.5 361.3 285.9 365.2 567.3 550.4

455.2 418.4 392.3 470.3 388.8 316.2 266.2 399.1 625.9 503.2

362.7 339.6 333.6 381.6 309.4 250.7 197.7 212.3 350.9 404.4

289.3 267.9 263.2 310.5 237.5 187.7 161.5 258.3 432.4 336.1

679.8 632.5 595.0 701.6 591.9 487.7 385.9 237.5 306.3 743.1

614.5 564.8 529.6 634.9 524.8 426.8 337.2 215.5 345.9 679.3

483.5 452.8 444.8 508.8 412.5 334.2 263.6 197.2 306.3 539.1

385.8 357.2 350.9 414.0 316.7 250.2 198.1 215.5 345.9 448.2

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mist” criterion, a = 0), the optimal solution obtained from (11) is the ICE300 for on–off operation, corresponding to scenario S3 from Table 4, with NPC = 4.75106 € with 50% exceeding probability, and again to scenario S3 from Table 5, with NPC = 4.93106 € with 5% exceeding probability. For values of a changing from 0 to 1, the most convenient solution passes from the ICE300 to the ICE1500 with 50% exceeding probability, whereas considering with 5% exceeding probability there are intermediate cases in which the MT100 solution is slightly most convenient than the better ICE one. 4.4. Summary of NPC results for planning alternatives with different control strategies The solution of the optimization problem in which the planning alternatives are defined by technological alternatives with the corresponding control strategies provides an overall response to identify the solution to be adopted. The variants considered in the application of the decision theory-based approach are the four cases with minimum expected value and maximum weighted regret criteria (each of which is applied to 50% and 5% exceeding probability on the NPC), and a set of values of the weighting factor a for the optimist–‘‘pessimist” criterion. The solution is obtained by constructing tables (not shown here for space reasons) similar to those reported in Section 4.3. The results are summarized in Table 10, in which ‘‘thermal” is an abbreviation to indicate the thermal load-following control strategy. It emerges that the solutions for the different cases provide consistent results for 50% and 5% exceeding probability (the only difference appears for the control strategy of the ICE1000 in Case 4 of the minimax weighted regret criterion). In the application studied, relatively large ICEs are always preferred and MTs never appear among the most convenient solutions. For the ICE control strategy, electrical load-following is never convenient.

Table 10 Most convenient alternatives and corresponding control strategies by using the NPC indicator. Criterion

Case 1

ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal

Case 3 Case 4

ICE 700 on–off ICE 1000 thermal ICE 1000 on–off ICE 1000 on–off

ICE 700 on–off ICE 1000 thermal ICE 1000 on–off ICE 1000 thermal

a=0

ICE 700 thermal

ICE 700 thermal

a = 0.1 a = 0.2 a = 0.3 a = 0.4

ICE 700 thermal ICE 700 thermal ICE 700 thermal ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal ICE 1500 on–off ICE 1500 on–off ICE 1500 on–off ICE 1500 on–off

ICE 700 thermal ICE 700 thermal ICE 700 thermal ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal ICE 1500 on–off ICE 1500 on–off ICE 1500 on–off ICE 1500 on–off

Case 3 Case 4

Optimist–‘‘pessimist” criterion

5%

ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal ICE 1000 thermal

Case 2

Minimax weighted regret

The DPP and IRR indicators are calculated by considering the differences between the results obtained from the planning alternative under analysis and the results obtained for SP. The analysis using these indicators has been carried out with the various control strategies, as for the NPC indicator. The results shown here for the on–off control strategy convey the basic message concerning applicability of these indicators in the decision theory-based context. Table 11 shows the 50% exceeding probability values of the DPP. Since the meaningful values of the DPP are limited to the duration of the period of analysis, the empty cells correspond to the undetermined situations in which the cumulative discounted cash flow remains negative in the whole period of analysis. The location of the empty cells clearly indicates that exploiting cogeneration units in all scenarios in which the gas price increases more than the electricity price (that is, the scenarios S2, S3 and S7) is not convenient. This is consistent with the NPC-based results. In addition, only a limited number of planning alternatives are convenient for the scenarios S1 and S6, and the MT300 is not convenient in the scenario S9. The IRR has been calculated by applying the Newton–Raphson method, with initial values of the discount rate set to 5%. Table 12 shows the 50% exceeding probability values of the IRR. Likewise for the DPP, relevant numerical values cannot be obtained for the entire set of scenarios. The indications are consistent with those of the DPP, showing the non-convenience of the scenarios in which the gas price increases more than the electricity price (that is, the scenarios S2, S3 and S7, as well as for most planning alternatives in the scenario S1 with constant electricity and gas prices.

Table 11 Discounted Payback Period with 50% exceeding probability (years). Planning alternatives referred to SP. The empty cells correspond to situations with non-convenient investment.

n% exceeding probability 50%

Minimum expected value

4.5. Application of decision theory criteria with the DPP and IRR indicators

Case 1 Case 2

a = 0.5 a = 0.6 a = 0.7 a = 0.8 a = 0.9 a=1

Alternative

Scenario S1

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500

S2

S3

13.4

S4

S5

S6

7.5 8.8 10.1 10.5 8.4 8.1 8.1 8.0 8.0

6.3 7.0 7.7 7.8 6.6 6.4 6.3 6.2 6.2

9.8 13.8

S7

12.5 12.9 13.3 14.5

S8

S9

7.2 8.4 9.8 9.6 7.7 7.7 7.8 7.8 7.8

8.6 11.3 13.1 9.9 10.3 10.7 11.3 12.1

Table 12 Internal Rate of Return with 50% exceeding probability. Planning alternatives referred to SP. The empty cells correspond to situations with non-convenient investment. Alternative

Scenario S1

MT100 MT200 MT300 ICE100 ICE300 ICE500 ICE700 ICE1000 ICE1500

0.062 0.013

S2

S3

S4

S5

S6

0.175 0.148 0.124 0.119 0.161 0.170 0.175 0.179 0.179

0.235 0.212 0.194 0.197 0.235 0.245 0.249 0.261 0.262

0.113 0.063 0.013 0.021 0.077 0.072 0.068 0.055 0.043

S7

S8

S9

0.194 0.164 0.136 0.145 0.189 0.191 0.190 0.194 0.196

0.145 0.097 0.047 0.075 0.125 0.117 0.111 0.099 0.089

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4.6. Comments on the decision theory results The structure of the outputs shown in the tables of the previous subsections is remarkably synthetic and easy to read, thus providing a useful picture to explain the comparative effect of each planning alternative in each scenario. The results obtained in the specific application shown in this paper (Table 10) suggest the ICE1000 solution with thermal loadfollowing control strategy in most NPC cases using the minimum expected value or the minimax weighted regret criterion for different cases of weighted scenarios, as well as with moderately optimistic criteria. This solution can be thus generally indicated as the overall outcome of the decision theory analysis in the specific application. However, an optimist decision-maker could decide to install an ICE of even larger size, to score profits from selling excess electricity to the grid. In terms of applicability of the decision theory-based approach, the NPC indicator has no limitation. On the other hand, undetermined outcomes could result by using the DPP and IRR indicators (Tables 11 and 12). If undetermined values are found only for some planning alternatives in all scenarios, it is possible to discard these alternatives and carry out the analysis by considering the other planning alternatives. However, when in some of the scenarios identified by the decision-maker the undetermined values refer to all the planning alternatives, it is not possible to evaluate any alternative on the basis of a complete set of determined values. In the specific application case, this fact makes the decision theory-based analysis with the DPP and IRR indicators inapplicable, as it would be unreasonable to eliminate the scenarios lacking of solution. The DPP and IRR indicators are thus only viable when it is possible to evaluate a set of planning alternatives in all scenarios.

5. Concluding remarks This paper has provided additional formulations, analyses and discussions to complete the conceptual framework for cogeneration planning under uncertainty in the multiple time frame approach identified in [1]. The analyses illustrated in this paper have specifically addressed the long-term time frame, based on large-scale uncertainty concepts and on the definition and assessment of scenarios of variation of energy loads and prices identified by the decision-maker. The proposed approach, based on decision theory concepts, deploys the results obtained by using the probabilistic representation of the correlations among load and energy price data determined from the actual data gathered for a reference year on a short-term and medium-term basis [1]. The difficulty to cope with large-scale uncertainty is overtaken by evaluating a set of evolution scenarios describing envisioned trends of increase of loads and prices. Different economic indicators have been tested with various decision theory criteria (minimum expected value, minimax weighted regrets, and mixed optimist–‘‘pessimist”). The NPC indicator has emerged to be fully suitable to identify situations of major convenience, providing numerical values for any planning alternative and scenario. Possible limitations of using the DPP and IRR indicators have been discussed. The results of the application case have shown that ICE technology is preferred with respect to MTs, notwithstanding the hypotheses made for the MTs included full availability and reduced maintenance costs compared to the ICEs. These results are based on purchasing the CHP technologies in the present technological context. Possible future reductions in the capital costs of MT equipment could change the terms of convenience of the CHP solutions. The possible presence of further benefits (e.g., fiscal discounts on the gas prices for the cogeneration sector) could generally increase

the convenience of adopting cogeneration solutions with respect to separate production of electricity and heat, and could change the specific planning outcomes. The overall approach defined to deal with large-scale uncertainty aims at offering, more than ‘‘the” solution, a set of outcomes depending on concepts that could represent individual ways of thinking of the decision-maker (regrets, optimism/pessimism). A key advantage of the decision theory framework is indeed the possibility of effectively involving the decision-maker in the planning process, from the definition of the scenarios to the interpretation of the results. In this respect, the decision theory approach illustrated and used in this paper becomes a useful tool that can be easily customized by the decision-maker to define and test further scenarios, changing the relative weights for handling the scenarios to get the economic indicator outcomes. Future work will report on the application of the overall framework developed to distributed multi-generation systems [26] supplying additional types of energy loads and interacting with different energy networks.

References [1] Carpaneto E, Chicco G, Mancarella P, Russo A. Cogeneration planning under uncertainty. Part I: multiple time frame approach. Applied Energy 2010. [2] Biezma MV, San Cristóbal JR. Investment criteria for the selection of cogeneration plants – a state of the art review. Appl Therm Eng 2006;26:583–8. [3] Al-Mansour F, Kozˇuh M. Risk analysis for CHP decision making within the conditions of an open electricity market. Energy 2007;32:1905–16. [4] Yoshida S, Ito K, Yokoyama R. Sensitivity analysis in structure optimization of energy supply systems for a hospital. Energy Convers Manage 2007;48:2836–43. [5] Chicco G, Mancarella P. From cogeneration to trigeneration: profitable alternatives in a competitive market. IEEE Trans Energy Convers 2006;21:265–72. [6] Matos MA. Decision under risk as a multicriteria problem. Eur J Oper Res 2007;181:1516–29. [7] Carpinelli G, Celli G, Mocci S, Pilo F, Russo A. Optimisation of embedded generation sizing and siting by using a double trade-off method. IEE Proc Gener Transm Distrib 2005;152:503–13. [8] El-Khattam W, Hegazy YG, Salama MMA. An integrated distributed generation optimization model for distribution system planning. IEEE Trans Power Syst 2005;20:1158–65. [9] Miranda V, Proença LM. Why risk analysis outperforms probabilistic choice as the effective decision support paradigm for power system planning. IEEE Trans Power Syst 1998;13:643–8. [10] Dahlgren R, Liu CC, Lawarrée J. Risk assessment in energy trading. IEEE Trans Power Syst 2003;18:503–11. [11] Carpaneto E, Chicco G, Mancarella P, Russo A. A risk-based model for cogeneration resource planning. In: Proceedings 10th international conference on probabilistic methods applied to power systems (PMAPS 2008), Rincón, Puerto Rico; May 25–29, 2008 [paper 083]. [12] Zafra-Cabeza A, Ridao MA, Alvarado I, Camacho EF. Applying risk management to combined heat and power plants. IEEE Trans Power Syst 2008;23:938–45. [13] Anders GJ. Probability concepts in electric power systems. New York: Wiley; 1990. [14] Carpaneto E, Chicco G, Mancarella P, Russo A. A decision theory approach to cogeneration planning in the presence of uncertainties. In: Proceedings IEEE powertech 2007, Lausanne, Switzerland; July 1–5, 2007 (paper no. 592). [15] Kaikko J, Backman J. Technical and economic performance analysis for a microturbine in combined heat and power generation. Energy 2007;32:378–87. [16] Tapiero C. Risk and financial management: mathematical and computational methods. Hoboken (NJ): John Wiley & Sons; 2004. [17] Willis HL, Scott WG. Distributed power generation: planning and evaluation. New York: Dekker; 2000. [18] Siddiqui AS, Maribu K. Investment and upgrade in distributed generation under uncertainty. Energy Econ 2009;31:25–37. [19] Miranda V, Proença LM. Probabilistic choice vs. risk analysis – conflicts and synthesis. IEEE Trans Power Syst 1998;13:1038–43. [20] Yokohama R, Ito K. Optimal design of energy supply systems based on relative robustness criterion. Energy Convers Manage 2002;43:499–514. [21] French S. Decision theory, an introduction to the mathematics of rationality. Chichester, UK: Ellis-Horwood; 1989. [22] Danny Harvey LD. A handbook on low-energy buildings, district energy systems: fundamentals, techniques, and examples. London, UK: James and James; 2006.

E. Carpaneto et al. / Applied Energy 88 (2011) 1075–1083 [23] EDUCOGEN, The European educational tool on cogeneration; December 2001 (July 2010). . [24] Aabakken J, editor. Power technologies energy data book, 4th ed. National Renewable Energy Laboratory Technical Report NREL/TP-620-39728, Golden, CO; March 2006 (July 2010). .

1083

[25] Pipattanasomporn M, Wellingham M, Rahman S. Implications of on-site distributed generation for commercial/industrial facilities. IEEE Trans Power Syst 2005;20:206–12. [26] Chicco G, Mancarella P. Distributed multi-generation: a comprehensive view. Renew Sustain Energy Rev 2009;13:535–51.