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Energy 34 (2009) 2001–2008

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Energy

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Operational strategy and marginal costs in simple trigeneration systems

M.A. Lozano, M. Carvalho, L.M. Serra*

Group of Thermal Engineering and Energy Systems (GITSE), Aragon Institute of Energy Research (I3A), Department of Mechanical Engineering, Universidad de Zaragoza, CPS deIngenieros, Marıa de Luna 3, 50018 Zaragoza, Spain

a r t i c l e i n f o

Article history:Received 29 December 2008Received in revised form22 July 2009Accepted 15 August 2009Available online 19 September 2009

Keywords:CHCPTrigenerationThermoeconomicsOperational strategyLinear programming

* Corresponding author. Tel.: þ34 976 761913; fax:E-mail address: serra@unizar.es (L.M. Serra).

0360-5442/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.energy.2009.08.015

a b s t r a c t

As a direct result of economic pressures to cut expenses, as well as the legal obligation to reduceemissions, companies and businesses are seeking ways to use energy more efficiently. Trigenerationsystems (CHCP: Combined Heating, Cooling and Power generation) allow greater operational flexibility atsites with a variable demand for energy in the form of heating and cooling. This is particularly relevant inbuildings where the need for heating is restricted to a few winter months. In summer, the absorptionchillers make use of the cogenerated heat to produce chilled water, avoiding waste heat discharge. Theoperation of a simple trigeneration system is analyzed in this paper. The system is interconnected to theelectric utility grid, both to receive electricity and to deliver surplus electricity. For any given demandrequired by the users, a great number of operating conditions are possible. A linear programming modelprovides the operational mode with the lowest variable cost. A thermoeconomic analysis, based onmarginal production costs, is used to obtain unit costs for internal energy flows and final products as wellas to explain the best operational strategy as a function of the demand for energy services and the pricesof the resources consumed.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

As life quality standards increase, the demand for comfort rises,together with a higher degree of conscience towards environ-mental issues. The satisfaction of such comfort demands generallyleads to a greater consumption of energy services (for example, anincrement in the use of air conditioning) while environmentalconscience tries to compensate the greater consumption of fossilfuels and its consequences, by means of a more rational use ofenergy.

Polygeneration systems, which include appropriate energyprocess integration for the combined production of two or moreenergy services and/or manufactured products, significantlyincrease the efficient use of natural resources [1,2]. In the lastdecades cogeneration (Combined Heat and Power – CHP) hascontributed considerably to the competitiveness, environmentalprotection and security of supply in the industrial sector [3,4].Today, energy consumption of buildings in developed countriescomprises 20–40% of total energy use and is above industry andtransport figures in the European Union (EU) and USA [5]. The mainlesson learned from some European research projects [6–8] is that

þ34 976 762616.

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there is a significant technical potential for the implementation oftrigeneration in the residential and tertiary sector of countries inthe Mediterranean area. In these countries the need for heating isrestricted to few winter months, limiting the application ofcogeneration systems. There is, however, a significant need forcooling during the summer period. In addition, the rapid increase ofthe air conditioning equipment penetration has added considerableloads to electricity networks, especially during peak demandperiods [5,9]. It is essential to provide solutions and one of them isthe use of absorption chillers for cooling. Lately, absorption chillershave provided an efficient way of recovering ‘‘waste’’ heat tocooling energy [10,11]. By combining CHP with heat-drivenabsorption chillers, the energy demand covered by cogenerationcan be extended into the summer months to match cooling loads[12,13].

Fig. 1 shows a generic trigeneration system. Trigenerationsystems may be constituted of a variety of technologies [14,15]. Theefficiency of the consumed fuel (natural gas, for example) is one ofthe main benefits of the production of three types of energyservices (heating, cooling and electricity) from the same energysource. This is important, since the better use of fuel assumeseconomic savings as well as a relief to the environment (less fuelconsumed, less pollution generated). In summary, trigenerationpresents as advantages: primary energy savings, reduction ofpollutant emissions, and a lower cost of energy services [16]. To

Nomenclature

AC Absorption refrigerator,AB Auxiliary boiler,CM Cogeneration module,EC Electric driven chiller,aw Cogeneration module work efficiency,aq Cogeneration module heat efficiency,hq Auxiliary boiler efficiency,COPq Coefficient of performance of the absorption chiller,COPe Coefficient of performance of the vapor compression

chiller,Ed Electricity demand, kW,Ep Purchased electricity, kW,Er Work to electric driven chiller, kW,Es Sold electricity, kW,Fa Auxiliary boiler fuel, kW,Fc Cogeneration module fuel, kW,Qa Heat from auxiliary boiler, kW,

Qc Cogenerated heat, kW,Qcc Consumed cogenerated heat, kW,Qd Heating demand, kW,Ql Wasted cogenerated heat, kW,Qr Heat to absorption refrigerator, kW,Rd Cooling demand, kW,Re Cooling from compression refrigerator, kW,Rq Cooling from absorption refrigerator, kW,Wc Cogenerated electricity, kW,Wcc Consumed cogenerated electricity, kW,pep Price of purchased electricity, V/kWh,pes Price of sold electricity, V/kWh,pfa Price of auxiliary boiler fuel, V/kWh,pfc Price of cogeneration module fuel, V/kWh,rql Unit cost of waste heat, V/kWh,l Marginal cost, V/kWh,HC Operational variable cost, V/h,fa Amortization factor, yr�1.

M.A. Lozano et al. / Energy 34 (2009) 2001–20082002

obtain these benefits by the optimal design of trigeneration plantsfor buildings, two fundamental issues should be addressed [17,18],i.e. the synthesis of the plant configuration (number and capacity ofequipment for each type of technology employed) and the opera-tional planning (strategy concerning operational state of theequipment, energy flow rates, purchase/selling of electricity, etc.).For new plants these issues are not separable, but for existing plantsoperational strategy is the only concern.

Synthesis of energy systems involves the search for a solutionfulfilling an objective function (e.g. cost, environmental burden,thermodynamic efficiency), which is to be minimized or maxi-mized. The variability of energy demands, as in buildings, requiresa design methodology that builds flexible utility systems whichoperate efficiently (thermodynamic target), are capable to adjust todifferent conditions (combinatorial challenge), and are able tooperate at a minimum economic cost [19]. The reviews of Chiccoand Mancarella [20] and Hinojosa et al. [21] summarize the char-acteristics of the optimization methods for polygeneration systemspresented in recent journal publications. Mixed-integer program-ming methods [22–25] fulfill the requirements and capture thecomplexities of an investment planning procedure for poly-generation energy systems by considering the superstructure of allalternatives.

To design an energy system, the following aspects must beconsidered simultaneously: (i) the technologies and equipment toinstall; (ii) the demands to be satisfied and the energy prices, and(iii) the optimal operation taking into account the possibility ofoperating the equipment at zero/partial/full load. This workdescribes a linear programming model capable of solving theoptimal operation problem of simple trigeneration systems, basedon the minimum variable cost HC, and also develops a thermoeco-nomic analysis of the operation.

p ep

Fuel F

Sold electricity Es

Purchased electricity E p

Electricity demand Ed

Heat demand Qd

Cooling demand RdMarketprices

MARKETCONSUMER CENTER

TRIGENERATION SYSTEM

Energydemandsp es

p f

Fig. 1. Trigeneration system.

According to Gaggioli [26] the objective of thermoeconomics isto explain the cost formation process of internal flows and productsof energy systems. The costs obtained with thermoeconomics canbe used to diagnose the operation and to control the production ofexisting plants, and in addition, improve the processes andsynthesis of new systems [27]. In Lozano et al. [28] three differentapproaches were used to determine the cost of internal flows andproducts: (i) analysis of marginal costs, (ii) valuation of productsapplying market prices, and (iii) internal costs calculation. Marginalcosts, in particular, have important information for operationaloptimization of energy and process systems [29–31].

Both the optimization model and the analysis methodologyapplied in this paper can be extended to consider the total annualcost, including the amortization costs for the required investmentof the new plant to be designed (depending on the power of eachtechnology installed and on the annual amortization factor fa) aswell as the turnover variable costs (depending on the demand to besatisfied in each operation period throughout the year and on themarket prices of fuel and electricity for such period). The objectivefunction will be

faXNTI

i

InvestmentiðPoweriÞ

þXNOP

j

HCj

�Demandj; Market pricesj

�(1)

where NTI is the number of technologies installed and NOP is thenumber of hourly periods of operation during the year. This paperonly studies the second part of the objective function (the first parthaving been considered constant), minimizing the costs associatedwith the operation of a previously dimensioned system in order toanalyze the operational strategy of simple trigeneration systemsinterconnected to the electric utility grid (both to receive electricityand to deliver surplus electricity). The minimization of costs shouldalso be considered as an obligatory prior step/consideration ofproject and design methodologies for new trigeneration systems.

The profitability of operation depends on quantities demandedfor energy services, and on fuel and electricity prices. A linearprogramming model was formulated, whose solution included theoperational minimum variable cost, the magnitude of the energyflows involved in the productive process, and dual prices of therestrictions. The thermoeconomic analysis of the optimal solution

Table 1Technical parameters.

Unit Efficiency coefficient Nominal capacity (kW)

CM aw h Wc/Fc¼ 0.35, aq¼Qc/Fc¼ 0.40 Wc nom¼ 350AB hq h Qa/Fa¼ 0.80 Qa nom¼ 400AC COPq h Rq/Qr¼ 0.625 Rq nom¼ 250EC COPe h Re/Er¼ 5.0 Re nom¼ 250

M.A. Lozano et al. / Energy 34 (2009) 2001–2008 2003

explains the reasons for the operation mode, obtains the marginalcosts of internal flows and final products, unravels the marginalcost formation process for the products, and evaluates theeconomical impact of changes in the demand or in the operationalconditions of equipment. Therefore, it is helpful when developingoptimization methodologies to redesign existing plants and todesign new ones.

2. Simple trigeneration system

A simple trigeneration system basically consists of a cogenera-tion module and an absorption chiller. The cogeneration moduleincludes a prime mover (gas turbine, reciprocating engine, etc.)converting the fuel energy to shaft power, an alternator whichtransforms mechanical to electrical power, and a heat recoverysystem. The absorption chiller can produce cooling from therecovered heat. Trigeneration plants are distinguished by thedifferent additional equipments incorporated [14,15]. The simpletrigeneration system shown in Fig. 2 [28], that we propose toanalyze, also includes a mechanical chiller driven by electricity andone auxiliary boiler.

The purpose of the trigeneration system is to attend the demandof different energy services (electricity, Ed; heating, Qd; and cooling,Rd) of a consumer center. The simple trigeneration system analyzedin this paper consists of the following productive units: a cogene-ration module CM, which is a natural gas reciprocating engine(providing heat, Qc, and work, Wc), an auxiliary boiler AB (providingheat, Qa), a single-effect absorption chiller AC (providing cooling,Rq, and driven by heat, Qr) and a electric driven chiller EC (providingcooling, Re, and driven by electricity, Er).

The prices of the fuel consumed by the cogeneration moduleand the boiler are, respectively, pfc and pfa. The demands will alwaysbe met either by the trigeneration system productive units or withthe help of purchased electricity from the electric grid (Ep at a pricepep). The possibilities also exist that a fraction (Ql> 0) of thecogenerated heat could be wasted, and the electricity could be soldto the market (Es at a price pes).

3. Optimal operation model

In a competitive energy market scenario, the profitability of theoperation of simple trigeneration systems depends on the capacityand performance of the installed technologies, fuel and electricityprices (subject to high variability and volatility), and quantities

QFa

S

L

P

R

CM

AB

AC EC

p e p

p es

p fc

p fa

Fc

Es

Ep

Wc

Qc

Wcc

Ql

r ql

Qa

Qr Er

Ed

Qd

Rd

Rq Re

Qcc

Fig. 2. Simple trigeneration system.

TE

demanded for energy services (with great daily and seasonal varia-tions). For a given demand several operating conditions are possible.

In many literature cases, the optimal operation of the trigener-ation system is studied beginning with a set of variable loadsdescribed by using the temporal evolution of electrical, thermal,and cooling loads. The problem formulation presented in this paperconsiders single operation points, describing them in terms of loadpowers, without considering the temporal evolution of the system.The inputs required for the problem are the load powers (demands)and the data shown in Tables 1 and 2. Table 1 shows the technicaldata of the trigeneration system productive units, which area natural gas reciprocating engine (CM), an auxiliary boiler (AB),a single-effect absorption chiller (AC) and an electric driven chiller(EC). All of them can operate either at partial load or full load. Table2 presents the prices of the energy flows exchanged with themarket. Note that different fuels are consumed by the cogenerationmodule and by the auxiliary boiler, and therefore the prices are alsodifferent. Cogenerated heat could be wasted without cost (rql¼ 0),i.e. there is no cost associated with the dissipation of heat.

To obtain the optimal operation state, a linear programmingmodel was solved. The objective function to be minimized was theoperation variable cost (in V/h):

HC ¼ pfc$Fc þ pfa$Fa þ pep$Ep � pes$Es þ rql$Ql (2)

subject to the following restrictions.

Capacity limits

cCM : Wc �Wc nom (3)

cAB : Qa � Qa nom (4)

cAC : Rq � Rq nom (5)

cEC : Re � Re nom (6)

Equipment efficiency

eCMw : aw$Fc �Wc ¼ 0 (7)

eCMq : aq$Fc � Qc ¼ 0 (8)

eAB : hq$Fa � Qa ¼ 0 (9)

eAC : COPq$Qr � Rq ¼ 0 (10)

able 2nergy prices (V/kWh).

pep pes pfc pfa

0.100 0.080 0.025 0.020

Table 3Operation modes.

Ep> 0 and Es¼ 0 Ep¼ 0 and Es¼ 0 Ep¼ 0 and Es> 0

Qa> 0 and Ql¼ 0 C1 C4 C7

Qa¼ 0 and Ql¼ 0 C2 C5 C8

Qa¼ 0 and Ql> 0 C3 C6 C9

S

L

PCM

pep

pes

pfcFc

Es

Ep

Wc

Qc

Wcc

Ql

rql

Ed

Qcc

400

1000350

400 140

0

50

350

260

M.A. Lozano et al. / Energy 34 (2009) 2001–20082004

eEC : COPe$Wr � Re ¼ 0 (11)

QFa

ABpfaQa

Qr

Qd 1000 0160 0 Er

Balance equations

S : Wc �Wcc � Es ¼ 0 (12)

R

AC ECRd

Rq Re

100

100 0

Fig. 3. Energy flows for example ExC3.

P : Wcc þ Ep � Ed � Er ¼ 0 (13)

L : Qc � Qcc � Ql ¼ 0 (14)

Q : Qcc þ Qa � Qd � Qr ¼ 0 (15)

R : Rq þ Re � Rd ¼ 0 (16)

Demand constraints (here the restrictions for example ExC7 areshown)

ED : Ed ¼ 200 (17)

QD : Qd ¼ 600 (18)

RD : Rd ¼ 100 (19)

The results presented below were obtained by utilizing thecomputer application LINGO [32] that includes an algebraiclanguage to formulate linear programming models and optimiza-tion algorithms to carry out/solve them. Given the energy demandsto be satisfied, LINGO solves the previous model and determines thefeasible operation mode with the minimum operation variable cost.

The group of feasible operation states can be classified in 9different operation modes, considering the values of purchasedelectricity (Ep), sold electricity (Es), auxiliary heat (Qa) and wasteheat (Ql). These operation modes are shown in Table 3. A summaryof optimal results (demand, flows, and hourly cost) obtained with

Table 4Energy flows and variable cost.

ExC3 ExC4 ExC7

Ed kW 400 330 200Qd kW 100 600 600Rd kW 100 100 100Ep kW 50 0 0Es kW 0 0 130Fc kW 1000 1000 1000Fa kW 0 250 250Wc kW 350 350 350Qc kW 400 400 400Wcc kW 350 350 220Er kW 0 20 20Ql kW 140 0 0Qcc kW 260 400 400Qa kW 0 200 200Qr kW 160 0 0Rq kW 100 0 0Re kW 0 100 100HC V/h 30.00 30.00 19.60

LINGO for three examples ExC3, ExC4, and ExC7 that correspond todifferent operation modes (C3, C4 and C7) is presented in Table 4.

The model described in equations (2)–(19) could be morecomplex by considering more detailed operation conditions, e.g.minimum capacity limits of the productive units or cost of heatdissipation. However, increasing the complexity of the modelwould not provide more relevant conclusions and would hide, tosome extent, the clarity of the analysis. In other words, the modeland the examples considered are simple (as stated in the title of thepaper) but clearly structured to allow for the making of interestinganalyses and conceptual interpretations.

4. Marginal costs and thermoeconomic analysis

Thermoeconomics launches an intensive analysis dose on thedesign and operation concepts of energy conversion systems for thepurpose of revealing opportunities of energy and cost savings[33,34]. In energy systems, resources are used up to provide certainqualities to the internal flows until the desired final products areobtained. According to Gaggioli [26] the objective of thermoeco-nomics is to explain the cost formation process throughout thesystem from the energy resources to the final products. Obtainingunit costs of internal flows and products of energy systems is

QFa

S

L

P

R

CM

AB

AC EC

pep

pes

pfc

pfa

Fc

Es

Ep

Wc

Qc

Wcc

Ql

rql

Qa

Qr Er

Ed

Qd

Rd

Rq Re

Qcc

330

100

600

1000

250

350

400

200

0

0

0

350

0 20

0 100

400

Fig. 4. Energy flows for example ExC4.

QFa

S

L

P

R

CM

AB

AC EC

pep

pes

pfc

pfa

Fc

Es

Ep

Wc

Qc

Wcc

Ql

rql

Qa

Qr Er

Ed

Qd

Rd

Rq Re

Qcc

200

100

600

1000

250

350

400

200

0

130

0

220

0 20

0 100

400

Fig. 5. Energy flows for example ExC7.

Q

S

L

P

R

CM

AB

AC EC

pep

λ (Ql)

0.100

0

0

0.100

λ (Ed)

λ (Qd)

λ(Rd)

0

Fig. 6. Marginal costs for ExC3.

M.A. Lozano et al. / Energy 34 (2009) 2001–2008 2005

a cornerstone of several thermoeconomic approaches that havebeen presented in the literature [27,35–40]. Thermoeconomicmethods are powerful tools for the analysis [41–44], diagnosis [45–48] and optimization [49–52] of energy conversion systems.

In a previous paper [28] the authors substantiated threedifferent approaches to determine the cost of internal flows andproducts: (i) analysis of marginal costs, (ii) valuation of productsapplying market prices, and (iii) internal costs calculation. Thispaper analyzed the marginal costs more deeply; in particular, it willbe shown that marginal costs have important information for theoperational optimization of energy systems.

It is important to mention that the term thermoeconomics hasa broader scope and covers more aspects then those presented inthis paper, which focuses on an economic evaluation of the resultsof the optimal operation of a trigeneration system. As explained inthe next subsection, the interpretation of the dual prices as sensi-tivities is independent of the type of physical system and of themodel employed. In this respect the approach presented in thispaper could be considered to be based more on economics than onexergoeconomics. The analysis presented here could also beextended to Second Law variables, providing sensitivity informa-tion on the exergetic cost of flow streams and irreversibilities inproductive units.

4.1. Marginal costs and operation modes

Thermoeconomic analysis was carried out for the optimaloperation examples (ExC3, ExC4 and ExC7) shown in Table 4. Theenergy flows that correspond to the operation with minimumvariable cost for each example are shown in Figs. 3–5.

The LINGO solution report for the model presented in theprevious section also gives a dual price figure for each constraint.The dual price can be interpreted as the amount by which theobjective function would increase as the right-hand side (constant

Table 5Marginal costs for the final products (V/kWh).

lEd lQd lRd

ExC3 0.100 0 0ExC4 (Ed Rd þ) 0.100 0.025 0.020ExC4 (Ed Rd �) 0.080 0.025 0.016ExC7 0.080 0.025 0.016

term) of the constraints is increased by one unit. If a constraintexpresses the produced quantity of a flow, then its dual price can beinterpreted as the marginal cost of this flow. Dual prices are alsocalled shadow prices, because they indicate how much one iswilling to pay for an additional unit of a specific resource. Table 5shows the marginal costs for the final products (dual prices of thedemand restrictions: equations (17)–(19)).

Dual prices and marginal costs information are important fortwo reasons: (i) to identify which operation constraint could bechanged to improve the solution and (ii) to react automaticallywhen external operational circumstances (prices of resources andproduct demands) change. Marginal costs have important infor-mation for design and operation optimization of energy systems.Figs. 6–8 show the marginal costs associated with operation modesExC3, ExC4 and ExC7.

Fig. 6 graphically explains the direction (origin) of the marginalcosts obtained for the final products in example ExC3; that is, howthe equipment will operate to produce an additional unit of the finalproducts. In this case, the cogeneration module operates at full loadand electricity is purchased; therefore, if an additional unit ofelectricity is required, it can only be obtained by purchasing it fromthe electric grid at a price of lEd¼ pep. A part of the cogenerated heatis wasted (Ql> 0), but it could be utilized at no cost (lQd¼ lRd¼ 0) tosatisfy directly the additional demand of heat and indirectly,through the absorption chiller, the additional cooling demand.

Figs. 7 and 8 explain the marginal costs for example ExC4, inwhich Ep¼ Es¼ 0. As can be seen in Fig. 7, if an additional unit ofelectricity is required, it must be obtained through the purchasedelectricity (lEd¼ pep), because the cogeneration module is oper-ating at full load. The additional heating will be produced by theauxiliary boiler (lQd¼ pfa/hq), and the additional cooling will beproduced by the vapor compression chiller driven by purchasedelectricity (lRd¼ pep/COPe). Fig. 8 explains how a decrease in thedemand of electricity or cooling allows the sale of the electricity notrequired to the grid (lEd¼ pes and lRd¼ pes/COPe, respectively).

Fig. 8 also explains how the additional demand units will besatisfied in the operation example ExC7, in which surplus electricityis produced and sold to the electric grid. An additional unit ofelectricity can be consumed if one less unit is sold to the market,therefore the marginal cost is the selling price (lEd¼ pes). Anadditional unit of heat will be produced by the auxiliary boiler(lQd¼ pfa/hq). To produce an additional unit of cooling, 1/COPe unitsless of electricity are sold to the market and therefore used to drivethe vapor compression chiller (lRd¼ pes/COPe).

Q

S

L

P

R

CM

AB

AC EC

pep 0.100

0.020

0.025

0.100

λ (Ed)

λ (Qd)

λ(Rd)

pfa

0.020

Fig. 7. Marginal costs for ExC4 (Ed Rd þ).

Table 6Optimal operation in function of the electricity demand (Qd¼ 600 kW,Rd¼ 100 kW).

Ed (kW) Operationmode

Wc (kW) Es (kW) Ep (kW) Wcc (kW) HC (V/h)

0 C7 350 330 0 20 3.60100 C7 350 230 0 120 11.60200 (ExC7) C7 350 130 0 220 19.60300 C7 350 30 0 320 27.60330 (ExC4) C4 350 0 0 350 30.00400 C1 350 0 70 350 37.00500 C1 350 0 170 350 47.00600 C1 350 0 270 350 57.00

M.A. Lozano et al. / Energy 34 (2009) 2001–20082006

4.2. Operation modes versus variable demands and energy prices

Example ExC4 belongs to the special operation mode C4 (Ep¼ 0,Es¼ 0) which represents the discontinuity between modes C1

(Ep> 0, Es¼ 0) and C7 (Ep¼ 0, Es> 0). When Qd¼ 600 kW andRd¼ 100 kW, and if the electricity demand was preciselyEd¼ 330 kW, the optimal solution would not correspond to eitherpurchase or sale of electricity. Given the market and demandconditions, the cogeneration module is operating at full load inoptimal mode; therefore, an increase in the demand or consump-tion of electricity is covered by purchasing from the electric grid,while a decrease would allow the sale. Since all of the cooling isproduced by consuming electricity in the vapor compressionchiller, an additional unit of cooling implies the purchase of elec-tricity, while a decrease in the demand allows the sale of theelectricity not required.

The close relationship that exists between the marginal cost ofproducts and the operation mode of the simple trigenerationsystem is therefore clearly exhibited. Table 6 and Fig. 9 show theresults corresponding to the optimal operation when the demandsof heat and cooling are fixed, Qd¼ 600 kW and Rd¼ 100 kW, andthe electricity demand Ed is modified, from 0 to 600 kW. Table 6

Q

S

L

P

R

CM

AB

AC EC

pes 0.080

0.016

0.025

0.080

λ (Ed)

λ (Qd)

λ(Rd)

pfa 0.020

Fig. 8. Marginal costs for ExC7 and ExC4 (Ed Rd �).

also presents the variable energy flows (the remaining flows are thesame as indicated for ExC7 in Table 4 and Fig. 5).

In the previous examples the optimal operation corresponds tothe full load operation of the cogeneration module. In these casesthe cogenerated heat is used to cover the heating and coolingdemands and the reason is simple: when heat is wasted due toa lack of demand, it is more profitable to produce electricity in thecogeneration module at a unit cost of pfc/aw¼ 0.0714 V/kWh,which is lower than pes¼ 0.080 V/kWh and pep¼ 0.100 V/kWh.This makes the optimization labor somewhat trivial to some extentin this case, as the results presented in Table 4 could be determinedwithout solving the optimization model with LINGO.

Fig. 10 and Table 7 show the variation of minimum operationcost when the price of fuel pfc is increased, being the demandthe same as in example ExC3 (Ed¼ 400 kW, Qd¼ 100 kW andRd¼ 100 kW). The values in Fig. 10 and Table 7 were obtainedutilizing the prices of electricity shown in Table 2(pes¼ 0.080 V/kWh and pep¼ 0.100 V/kWh) and by consideringthat the difference between the prices of the fuels consumed inthe cogeneration module and auxiliary boiler remainedconstant: pfc� pfa¼ 0.005 V/kWh. In ExC3 electricity waspurchased at pep¼ 0.100 V/kWh, and therefore only when pfc ishigher than aw$pep¼ 0.035 V/kWh it makes sense to considerthe possibility of operating the cogeneration module at partialload. In fact, as shown in Fig. 10 and Table 7, when the value ofpfc¼ 0.035 V/kWh is reached, there is no waste of heat and thecogeneration module operates at partial load. But when the fuelcost is additionally increased, being higher than (awþ aq$COPq/COPe) pep¼ 0.040 V/kWh, the production of cooling utilizing thecogenerated heat is not profitable and the cooling demand iscovered by the electrical chiller. Finally, when the fuel cost

0 100 200 300 400 500 600 7000

10

20

30

40

50

60

Ed

(kW)

HC (€/h)

Optimal operation modeC1

C7

C4

330

ExC4

ExC7

λ (Εd

) = 0.08 €/kWh

λ (Εd

) = 0.10 €/kWh

Optimal operation mode

Optimal operation mode

(Electricity selling)

(Electricity purchasing)

Fig. 9. Hourly cost versus the electricity demand (Qd¼ 600 kW, Rd¼ 100 kW).

Fig. 10. Hourly cost versus the price of fuel (Ed¼ 400 kW, Qd¼ 100 kW, Rd¼ 100 kW).

Table 8Dual prices of the restrictions for ExC7.

Restriction l (V/kWh)

Capacity limitscCM Wc�Wc nom �0.037cAB Qa�Qa nom 0cAC Rq� Rq nom 0cEC Re� Re nom 0Equipment efficiencyeCMw aw$Fc�Wc¼ 0 0.043eCMq aq$Fc�Qc¼ 0 0.025eAB hq$Fa�Qa¼ 0 0.025eAC COPq$Qr� Rq¼ 0 0.016eEC COPe$Er� Re¼ 0 0.016Balance equationsS Wc�Wcc� Es¼ 0 0.080P Wccþ Ep� Ed� Er¼ 0 0.080L Qc�Qcc�Ql¼ 0 0.025Q QccþQa�Qd�Qr¼ 0 0.025R Rqþ Re� Rd¼ 0 0.016Demand constraintsED Ed¼ 200 0.080QD Qd¼ 600 0.025RD Rd¼ 600 0.016

M.A. Lozano et al. / Energy 34 (2009) 2001–2008 2007

fulfills the condition pfc> aw$pepþ (aq/hq) pfa, cogeneration isnot profitable (not even to cover the heat demand) and it ismore interesting to produce heat with the auxiliary boiler. Thisis the case when pfc> 0.065 V/kWh, in which the cogenerationmodule is not operating.

4.3. Internal costs

The marginal costs of the internal flows of the simple trigener-ation system can be obtained by interpreting the dual pricescorresponding to restrictions (3)–(16) of the optimization model.Table 8 shows the dual prices obtained by LINGO for the linearprogram, minimizing the operation variable cost corresponding toexample ExC7.

According to the optimization theory, if f(x) is the objectivefunction of the program and g(x)¼ b is an active restriction at theoptimal point, the dual price l of the restriction is interpreted as thederivative of the objective function f regarding the parameter b ofthe active restriction. That is

l ¼�

df *=db�

(20)

The super-index * in the previous equation expresses that f*(b)corresponds to the trajectory of the value of the objective functionfor the optimal solutions when b varies. In this optimizationproblem of the operation of a simple trigeneration system, f is thehourly cost HC in V/h, and all the restrictions g express energy flowsin kW; therefore, the dual prices are expressed in V/kWh.

A few examples of interpretation of dual prices are shown next.Restriction (3) cCM: Wc�Wc nom corresponds to the cogenera-

tion module, and in case ExC7 is the only active capacity restrictionin the optimum. Rewriting restriction (3) in the form g(x)¼ b,

Table 7Optimal operation in function of the price of fuel (Ed¼ 400 kW, Qd¼ 100 kW,Rd¼ 100 kW).

pfc (V/kWh) Operationmode

Wc

(kW)Ql

(kW)Qa

(kW)Ra

(kW)Ep

(kW)HC(V/h)

0.025 (ExC3) C3 350 140 0 100 50 30.000.035� C3 350 140 0 100 50 40.000.035þ C2 227.5 0 0 100 172.5 40.000.040� C2 227.5 0 0 100 172.5 43.250.040þ C2 87.5 0 0 0 332.5 43.250.065� C2 87.5 0 0 0 332.5 49.500.065þ – 0 0 100 0 420 49.500.070 – 0 0 100 0 420 50.125

results cMC: Wc¼ 350, being 350 kW the nominal production ofthe motor. If for any reason (variation in environmental conditions,degradation of lubricating oil, etc.) the electricity productioncapacity of the engine decreases by 2 kW, the hourly cost wouldincrease by approximately 0.074 V/h:

DHC*ylðcMCÞ$DbðcMCÞ ¼ �0:037$ð � 2Þ ¼ 0:074 (21)

Interpretation: the cogeneration module would produce 2 kWless electricity and 2.29 kW less heat (2aq/aw). The decrease in fuelconsumption by the motor creates savings of 0.143 V/h (2pfc/aw),but a decrease of 0.160 V/h (2pes) in the sale of electricity. The heatis produced by the boiler at a cost of 0.057 V/h (2.29pfa/hq). Theresulting total cost is therefore 0.074 (0.160þ 0.057� 0.143).

Restriction (8) eAB: hq$Fa�Qa¼ 0 corresponds to the produc-tion of the auxiliary boiler. If because of poor insulation, 5 kW of theproduced heat is lost, the restriction should be written as eAB:hq$Fa�Qa¼ 5, meaning that the boiler would increase itsconsumption of fuel to compensate such a loss. From the shadowprice of the restriction, the cost can be estimated as 0.125 V/h:

DHC*ylðeABÞ$DbðeABÞ ¼ 0:025$5 ¼ 0:125 (22)

Finally, it was observed that the shadow prices of the restric-tions corresponding to the energy balances can be immediatelyinterpreted as the marginal costs of the demanded energy services.

5. Conclusions

This work showed the characteristics of the different operationmodes of a simple trigeneration system. The linear programmingmodel developed allows the determination of the optimal opera-tion mode corresponding to the minimum variable cost. The resultscorresponding to different demands of energy services and oper-ation modes were presented and analyzed. This paper analyzes andexplains, by using thermoeconomic analysis, the reason for theoptimal production mode and also obtains the marginal cost ofinternal flows and final products, unravels the marginal costformation process of products, and evaluates the economic impactof changes in the demand or operational condition of the equip-ment. Thermoeconomic analysis also aids in the development ofeffective methodologies for the design of new plants and theretrofit of existing plants to new demand and market price

M.A. Lozano et al. / Energy 34 (2009) 2001–20082008

conditions. An appropriate distribution of the minimum variablecost to the final products must consider the nature of the optimaloperation mode, in order to promote rational and efficient energyservices production and consumption.

This paper dealt with the analysis of the operation of simpletrigeneration systems only. The cost of satisfying the energy servicedemand of the consumer center was minimized using an optimaloperation mode which exchanged energy flows at market pricesand utilized the productive capacity of the installed equipment. Thefindings would not change, when considering a different objectivefunction to optimize or more complex trigeneration systems.Similarly, a greater sophistication of the model, using non-linearproduction restrictions and binary variables limiting both theminimum load of the productive units and the on/off status, wouldprovide more precise results but in general, the above conclusionsprevail.

Acknowledgments

This work has been developed in the framework of researchproject ENE2007-67122, included in the Spanish National Plan forScientific Research and Technological Development and Innovation(R&D&I), funded in part by the Spanish Government (Energyprogram) and the European Union (FEDER program).

Monica Carvalho is supported by the EU Program of High LevelScholarships for Latin America (Alban Scholarship No.E06D100314BR).

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