EABOT – Energetic analysis as a basis for robust optimization of trigeneration systems by linear programming

Energy Conversion and Management 49 (2008) 3006–3016

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EABOT – Energetic analysis as a basis for robust optimization of trigeneration systems by linear programming A. Piacentino a,*, F. Cardona b a b

DREAM – Department of Energetic and Environmental Researches, Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy DINCE – Department of Nuclear Engineering and Energy Conversions, Univ. ‘‘La Sapienza’’, Cso. Vitt. Emanuele II, 244 Rome, Italy

a r t i c l e

i n f o

Article history: Received 12 November 2007 Accepted 18 June 2008 Available online 6 August 2008 Keywords: Trigeneration Optimization Linear programming Thermoeconomics Multi-objective

a b s t r a c t The optimization of synthesis, design and operation in trigeneration systems for building applications is a quite complex task, due to the high number of decision variables, the presence of irregular heat, cooling and electric load profiles and the variable electricity price. Consequently, computer-aided techniques are usually adopted to achieve the optimal solution, based either on iterative techniques, linear or non-linear programming or evolutionary search. Large efforts have been made in improving algorithm efficiency, which have resulted in an increasingly rapid convergence to the optimal solution and in reduced calculation time; robust algorithm have also been formulated, assuming stochastic behaviour for energy loads and prices. This paper is based on the assumption that margins for improvements in the optimization of trigeneration systems still exist, which require an in-depth understanding of plant’s energetic behaviour. Robustness in the optimization of trigeneration systems has more to do with a ‘‘correct and comprehensive” than with an ‘‘efficient” modelling, being larger efforts required to energy specialists rather than to experts in efficient algorithms. With reference to a mixed integer linear programming model implemented in MatLab for a trigeneration system including a pressurized (medium temperature) heat storage, the relevant contribute of thermoeconomics and energo-environmental analysis in the phase of mathematical modelling and code testing are shown. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction In the last decade, a growing interest has been observed for combined heat and power (CHP) or combined heat, cooling and power (CHCP) applications in buildings. This is obviously due to the high conversion efficiency of polygeneration systems and the consequent energy and pollutant-emissions savings, but also due to favourable external conditions, like the new opportunities existing in the liberalised energy market and the growth of a ‘‘small scale CHP” market, which has gradually reduced the purchase and installation cost of CHP units (typically, in the order of 600– 800 €/kWe). These promising perspectives have stimulated the efforts of scientists towards the definition of criteria for the optimization of CHCP design and operation for applications in the civil sector. Several analyses have been oriented to assess the potential benefits in terms of energy and pollutant-emissions savings [1,2] and, in some cases, some peculiar aspects were examined adopting thermoeconomic cost-accounting methods [3] or pinch analysis [4]. In order

* Corresponding author. Tel.: +39 091 236 302; fax: +39 091 484 425. E-mail address: [email protected] (A. Piacentino). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.06.015

to understand the complexity of the optimization problem, the following aspects can be remarked: – Safety of supply and flexibility are usually ensured by redundancy, i.e. the system is designed as a ‘‘facility of systems of a same product”, where different components may alternatively contribute to cover the demand of a specific energy vector. – The problem is time-dependent: the variability in energy loads and prices requires the adoption of flexible plant operation strategies; in the civil sector, discretization on hourly basis is usually pursued, resulting in a high number of decision variables as concerns plant operation. Discussions have arisen on the possibility to adopt a reduced set of ‘‘standard days” (typically defined on seasonal and ‘‘working–nonworking” bases) without loss of reliability, but this is a controversial argument which needs ad hoc considerations for each case. – The decision variables have a non-homogeneous nature, both as concerns the way they affect the objective function and the values they can assume. Either in case of profit, energy or pollutant-emission saving-oriented optimizations, the objective function depends on annual results, calculated

A. Piacentino, F. Cardona / Energy Conversion and Management 49 (2008) 3006–3016

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Nomenclature a, b B C CHCP CHP CONS COP D E FOR GA H i* LL LP, NLP LR m MP NPV PES PHR STOR T

constants in linearized equations generic flux (energy or exergy terms) cooling demand or production combined heat, cooling and power combined heat and power energy consumption coefficient of performance energy load, on hourly basis electricity demand or production feasible operation range genetic algorithms heat demand or production interest rate load level linear and non-linear programming Lagrangian relaxation mass of hot water stored market price net present value primary energy saving power to heat ratio stored energy (kWh) temperature

TES Z, z

thermal energy storage total and specific capital cost

Superscripts abs absolute conv conventional dir for direct uses rel relative Subscripts Abs absorption chiller boilmax corresponding to boiler capacity (heat peak load) del delivering el.ch. electric chiller Exch exchanged with the grid inv investment pow.plant power plant ret return Greek symbols d 0–1 binary variable g efficiency

as sum of single values obtained for each time-step. The optimization problem can be divided into three different sub-problems: (a) Synthesis: in order to optimize the plant configuration, i.e. to select what components should be installed, a starting redundant ‘‘superconfiguration” is usually adopted, that is a general CHCP scheme where several components are included and a high level of interconnections among them is assumed. The decision variables at synthesis level are 0–1 variables, each indicating the decision to include/not include a certain component. (b) Design: CHCP systems for buildings applications are usually made up by highly standardised components (gas turbines, reciprocate engines, water–lithium bromide absorption chillers, etc.), which can be regarded as black boxes and modelled by defining their part load behaviour. The absence of variables involving the thermodynamic state of working fluids (the optimization of heat exchangers represents a 2nd refinement level, not considered in this paper) makes the design optimization easier than usual. Only a relatively small number of design variables representing the size of plant components is included. (c) Operation: the optimization of plant operation is more complex than usual; in fact, CHCP systems offer several possibilities of loading the different components to cover energy requests, the optimal solution depending on efficiency figures, energy loads and prices. This optimization level involves both 0–1 variables (the on/off state for each component) and continue variables (the load level of each component). Also, the optimization routine must be applied on hourly basis, because both energy load and prices are time-dependent.

level have been assumed. This aspect heavily influences the choice of the most appropriate resolution technique. Evolutionary search, for instance, which has been extensively used in the optimization of energy systems by genetic algorithms (GA), is not suitable for our problem because of the deficiency in handling highly constrained problems; also, GA could be preferably adopted to optimize plant operation as internal routine of an iterative synthesis-design optimization [5] and this approach is not suitable for the examined problem due to the huge number of different operating conditions. Several heuristic approaches have been proposed, oriented to determine near-optimal solutions basing on ‘‘aggregate thermal load” duration curves [6] or thermoeconomics [7]; most of them, however, do not include an ‘‘integrated” optimization process, but assume a priori a sub-optimal management strategy (either ‘‘heat tracking” or ‘‘electricity tracking” operation modes). More recently, linear programming (LP) techniques have been extensively used [8,9], due to the possibility to solve large scale problems with thousands variables approaching the ‘‘multi-level” optimization problem by an ‘‘horizontal algorithm”, where synthesis, design and operational variables are threaten similarly. More refined approaches have been proposed in [10], where a robust optimization included a sensitivity analysis in LP to consider stochastic energy loads, and in [11], where an efficient algorithm was proposed, which resulted much faster than an efficient sparse simplex code. The fact that the production of the three energy vectors follows a joints characteristic makes often convenient to include thermal energy storages (TESs, i.e. hot water and/or chilled water tanks); usually, the TES is used to maximise power production during peak hours (where high value electricity is produced), storing eventual surplus heat/cooling energy to reuse it during off-peak hours. The inclusion of a TES significantly varies the structure of the optimization problem, introducing dynamic constraints; a clear overview of the techniques proposed in the literature to deal with storage constraints was provided in [12]. Let us here briefly resume the two main currencies:

The variables of the different sub-problems are not of a ‘‘same rank”; for instance, operational variables could be optimized only once fixed values for the decision variables at synthesis and design

– Decoupling the time-dependent storage constraint, a set of small-size single-period sub-problems may be solved. In [12,13] Lagrangian relaxation (LR) methods were used,

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which decompose the original problem into multiple underlying Lagrangian sub-problems. In [13], the unit commitment (where the on/off state of each component is determined) and the economic dispatch (the optimal production rates) are solved by isolating the economic dispatch problem (a unit commitment is assumed known, which is in its turn optimized by LR) and relaxing the time-coupling constraint obtaining a dual problem which does not require dynamic programming. – Treating the multi-period large-size problem as unique and solving it by Simplex or interior point methods require larger computational efforts, but a less complicate problem formulation. The first of the two above currencies has been attracting more interest among researchers and the optimization of trigeneration systems has gradually become a research field for experts in operational research. Innovative methods frequently converge to optimal solutions whose energetic consistence might be said controversial, as evident when considering that a same lay-out, optimized by different algorithms and adopting a same set ‘‘objective function + constraints + boundary conditions”, frequently converges to quite different solutions [14]. In particular, the solutions achieved are often too sensitive to slight variations in the boundary conditions, which reflects a non-comprehensive representation of energetic conversion processes. Analytical CHCP modelling should be more closely interrelated to energetics, as may be observed in [15] where reliability of results is ensured by an energo-environmental analysis of plant performance. In this paper we put into evidence that only an in-depth energetic analysis may properly reflect the ‘‘formation” of the values of the objective function, which ultimately drive any optimization algorithm towards a final solution. Hence, no enhanced efficiency algorithms are proposed; on the contrary, a common Linear-Programming Interior Point method (a variant of Mehrotra’s predictor–corrector method) was used, which is the default option for large scale optimization in MatLab. All the analyses that will be presented have been used to code a MatLab tool, EABOT (Energy Analysis-Based Optimization of Trigeneration plants), which represent a module of a larger-scale software for the optimization of CHCP-based l-grids [16] not completed yet. Ahead in the paper four main concepts are critically examined: 1. Is the definition of a few ‘‘reference days” (as concerns load profiles) necessary? At what extent selecting a small number of reference days affects the reliability of results and reduces the consumption of computational resources? 2. Once all operational constraints have been properly defined, is it possible to determine additional conditions based on advanced energetic analyses in order to make the optimization algorithm converge faster? 3. Can multi-objective optimization offer a more relevant contribute in CHCP decision making? A pure profit-oriented optimization would not properly reflect all the implications related to polygeneration: the support mechanisms in force, like tax exemption or dispatching priority, and the recent legislation for CHP eligibility give a particular focus on the energo-environmental effects, which will become more and more profitconditioning. 4. Does the testing phase of CHCP optimization tools require energo-environmental analysis? 2. Problem formulation CHCP applications in the civil sector can be considered high risky/high profit-potential investments, which require accurate viability analysis and energy audits to have been performed. Then,

considering a generic building, its requests for heat, cooling and electricity (respectively, DH, DC and DE) are assumed to be known on hourly basis. From a qualitative point of view, typical 6 °C/ 13 °C delivering/return temperature are supposed for the cold water to/from the air conditioning units; only low temperature (up to 90 °C) heat requests for space heating and sanitary hot water are considered, which could be covered by both low and high temperature heat recovery in reciprocate engines. The same deterministic approach is assumed for energy prices; accordingly to the current trend in the European Union, where net-metering mechanisms are gradually extended to new power producers, a same MPE price for electricity purchase/sell is assumed, which is obviously time-dependent. In the reference scheme assumed as CHCP superconfiguration, no reversible heat pump will be included, but a low temperature thermal energy storage (TES). 2.1. Modelling plant components The optimization model proposed in this paper includes both economical and energo-environmental analyses; hence, appropriate cost and performance figures for the plant components must be introduced. Most CHCP components have significant scale-factor in the ‘‘purchase and installation” cost; however, the cost of the CHP unit can be approximated by a linear expression:

Z CHP;inv ¼ z0;CHP  EaCHP;nom ffi aCHP;inv ECHP;nom þ bCHP;inv dCHP

ð1Þ

where ECHP,nom and dCHP, respectively, represent the electric capacity of the CHP unit and the 0–1 binary variable expressing the decision to include/not include the unit. On more than 200 CHP units from different producers [17], this approximation introduced ±0.01 variations in the correlation coefficient R2. As concerns the part load performance, a crucial aspect is now discussed, that is the use of 0–1 binary variables for the on/off state of components. Part load efficiency penalty is usually kept into account by imposing a ‘‘feasible operation range” for the jth component in the ith hour:

FORj;i ¼ ½dopj;i ¼ 0 [ ½ðdopj;i ¼ 1Þ \ ðLLj;i 2 ½LLmin ; LLmax Þ

ð2Þ

The limit expressed by Eq. (2) with a logic operator is usually integrated in LP models as follows:

LLmin;j  dopj;i 6 LLj;i 6 LLmax;j  dopj;i

ð3Þ

This apparently non-linear constraint does not influence the solution search method, being dop-j,i a 0–1 binary variable. Quadratic or linear ge(LL) equations are usually assumed as concerns part load efficiency; alternatively, linear expressions for the characteristic equations g [18] can be assumed:

Binput ¼ g i ðx; Boutput Þ ¼ ai  LLi  Boutput;nom þ bi di

ð4Þ

which are more suitable for LP models. Below in the paper, variations in the power to heat ratio (PHRCHP) are neglected and, as concerns the absorption chiller, a fixed 0.70 COPabs is assumed. Also, as concerns the auxiliary boiler and the vapour compression chiller, linear cost figures and constant gboil (0.90) and COPel.chill. (2.5) are used; both new and existing customers can be considered by including or not the capital cost of these two components in the objective function expression. The consumption of resources in mixed integer linear programming (MILP) problems strongly depends upon the number of integer variables dop-i; optimizing plant operation on hourly basis, i.e. 8760 values per year, could become very time-consuming. A simplified method is here adopted. Let us assume LLCHP and LLabs variable in the whole range [0, 1] and neglect efficiency drops at part load operation (i.e. constant ge); let us assume not to introduce binary variables for the on/off state, a significant reduction in the calculation time being expected. Is such a roughly approximate approach reliable? In order to answer this

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question, we may observe that CHP heat recovery (cold production in the absorption chiller) may be considered as ‘‘alternative” to production of heat (cold) in a constant efficiency boiler (vapour compression chiller). Consequently, in a certain hour, the optimal unit commitment and economic dispatch problem can be generalized as follows: a combined production unit 1 produces two outputs a and b, which could alternatively be produced by components 2 and 3, respectively; let us assume a fixed a/b production ratio for the unit 1 and constant efficiencies g2 and g3. Minimizing an objective function (OF, linear equation in Binput,1, Binput,2 and Binput,3):

OF ¼ const1 Binput;1 þ const2 Binput;2 þ const3 Binput;3

ð5Þ

subject to

Da ¼ Baoutput1 þ Baoutput2 ¼ LL1 Baoutput1;nom þ LL2 Baoutput2;nom ; Db ¼ Bboutput1 þ Bboutput2 ¼ LL1 Bboutput1;nom þ LL3 Bboutput3;nom

ð6:a; bÞ

is equivalent to impose null: a

a

Boutput1;nom Boutput2;nom o OF ¼ const1   const2  oLL1 ga1 ga2  const3 

Bboutput3;nom

gb3

ð7Þ

The latter approach is here adopted, expressing the TES balance by two variables, STOR (energy stored at each time-step) and HTES,i (charge/discharge rate at each time-step i):

1

 DHloss  STORTES;i  HTES;i 100

ð8Þ

where

STORi ¼ mi  cp  ðT del  T ret Þ

Operation

HCHP;i ; C Abs;i ; LLboil;i ; LLel:ch:;i ; STORi ; Q TES;i ;

ð9Þ

Evidently, these energetical variables are related to the volume VTES, that is the parameter most affecting the purchase and installation cost. Again, Eq. (1) can be formulated assuming VTES as capacity variable; a brief cost analysis suggested to assume aTES and bTES equal to 155 €/m3 and 2050 €, respectively. 2.2. MILP model: objective function and constraints In this section, the optimization problem is formulated accordingly to the assumptions proposed in Section 2.1; a multi-objective approach is adopted to keep into account all economical and energo-environmental implications. Let us assume the following set of decision variables:

for i ¼ 1 to Nhours

2.2.1. The objective function Let us start with a brief analysis of the objective function. If a profit-oriented approach is adopted, appropriate economic indicators will be needed. In [8] the payback period was assumed as decision function, resulting in a small-size CHP unit to cover the base load; in [15] a more reliable approach is used, keeping simultaneously into account different economic indicators capable to exploit the full profit-potential through the plant life cycle. Among them, the net present value (NPV) has been proven the most useful for CHP decision making [20] and is here assumed as objective function in the profit-oriented optimization (actually, its opposite ‘‘-NPV” is minimized):

min P=Aði ; nÞ  b 

– Assumption of a fixed (usually, 8–10%) energy loss between charging and discharging; – Assumption of a fixed DHloss loss per hour, in percentage (usually, 2–3%) of the currently stored energy.



dCHP ; dAbs ; dTES ; ECHP;nom ; C Abs;nom ; V TES



Unless the marginal cost (equivalent to the average cost, in the hypothesis of constant efficiencies) of the two alternative production methods are equal, Eq. (7) indicates OF to represent a monotone function. Minimizing OF would consequently lead to assume LL1 alternatively equal to 0 or Da =Baoutput1;nom ; if the combined production unit 1 is no oversized, for most of the year the above described simplified approach will lead to a technically-feasible operation. This is of course a quite ‘‘heavy” simplification, whose reliability will require an ex post analysis. Finally, as concerns the short-term heat storage, for pressurized steel tanks in [19] the prevalence of heat losses through the tank walls and due to the contact between the hot and the cold water was pointed out. A few energetic models are available in literature, following two main currencies:

STORTES;iþ1 ¼

Synthesis and design

NX hours

( cCHP gas

HCHP;i PHRCHP

gE;CHP

i¼1

þ cboil gas

LLboil;i Eboil;nom

   LLel:ch:  HCHP;i  PHRCHP  Eel:ch:;nom  DE;i  cE COPel:ch:

gboil

þ aCHP;inv  ECHP;nom þ bCHP;inv  dsinth;CHP þ aabs;inv  C Abs;nom þ babs;inv  dsinth;Abs þ aTES;inv  V TES þ bTES;inv  dsinth;TES

ð10Þ

In Eq. (10), the possibility to assume different gas prices for feeding the CHP unit and the boiler is kept into account; ‘‘P/A(i*,n)” represents the actualization factor. The b factor, which is related to the possibility to fix an arbitrary number of time periods Nhours to be used as a basis for the optimization, will be discussed in the next subsection. For an energy/pollutant-emissions saving-oriented optimization, maximization of annual savings can be only achieved with reference to ‘‘absolute indicators”. In the conventional definition of the primary energy saving index PES%:

PESrel % ¼

Fuel

conv

 Fuel conv Fuel

CHP

ð11Þ

Fuelconv is defined with reference to the actual CHP production rate and significantly depends on whether the plant is a net power seller/buyer along the year. Only when the annual energy saving is compared to a fixed energy consumption [15], maximising PES% is equivalent to maximise the energy saving. With reference to Fig. 1, where the CHCP superconfiguration and its energy flows are compared to the ‘‘separate production” case, let us indicate with CONSconv the energy consumption to cover the current loads by a conventional system:

CONSconv ¼

DH

gboil

þ

1

gpow:plant

  DE þ

DC COPel:ch:

 ð12Þ

The primary energy saving, expressed in kW (average value in a certain hour), is conv

PES ¼ CONS 

Hboilmax

gboil

 PHRCHP LLboil 

1

gE;CHP DE

gpow:plant

 

1

gpow:plant

!  HCHP

DC;max LLel:ch: COPel:ch: gpow:plant

ð13Þ

The need for ‘‘absolute” energy saving indicators can be put into evidence with reference to an example. Let us consider a hour where DE, DH and DC are, respectively, 70 kWe, 80 kWh and 310 kWc; let us also assume gpow.plant equal to 0.44. If we assume

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DC,max LL el.ch.,i Combined Heat, Cooling and Power

COPel.ch.

grid

Auxiliary el. chiller

DC,max LL el.ch.,i

Cabs,i COPabs Absorption chiller Cabs,i

D PHR H CHP,i - D E - C,max LL el.ch.,i COPel.ch.

Separate Production

DC

El. chiller

PHR H CHP,i

ηe,CHP

1 η pow.plant PHR H CHP,i

DE

CHP unit η boil

E

+

DC COPel.ch.

)

Power plant

DH

H CHP,i H boil,max LL boil.,i

(D

Auxiliary H boil,max LL boil.,i boiler

DH

η boil

Boiler

Fig. 1. Redundant CHCP superstructure.

to have installed a CHP unit with 450 kWe and 600 kWh capacity (ge,CHP = 0.33) and an absorption chiller with 250 kWc capacity, we have

250 þ 80 ¼ 437:1 kWh HCHP ¼ 0:7 ) PHRCHP  HCHP ¼ 291:4 kWe ; C abs ¼ 250 kWc DC;max  LLel:ch: 60 ¼ ¼ 24 kWe 2:5 COPel:ch: 60 ¼ 197:4 kWe DEexch: ¼ 291:4  70  2:5   80 1 310 CONSconv ¼ þ  70 þ ¼ 529:8 kWprim ; 0:9 0:44 2:5 291:4 CHP ¼ 883:0 ¼ Fuel 0:33   1 1  PES ¼ 529:8  291:4  0:33 0:44 70 60   ¼ 95:5 kWprim 0:44 2:5  0:44 DEexch: conv ¼ CONSconv þ Fuel 0:44 197:4 ¼ 529:8 þ ¼ 978:5 kWprim 0:44 conv CHP Fuel  Fuel PES ¼ PESrel conv conv % ¼ Fuel Fuel 978:5  883:0 95:5 ¼ ¼ ¼ 0:097 978:5 978:5 conv CHP Fuel  Fuel PES ¼ PESabs % ¼ CONStrad CONStrad 978:5  883:0 95:5 ¼ ¼ 0:180 ¼ 916:2 529:8

max

h

gH;CHP gboil

þg

gE;CHP

pow:plant

;



1

gH;CHP COPabs COPel:ch:



PHRCHP

g

 HCHP;i 

DH;max

gboil

LLboil;i  Ael:ch  LLel:ch:;i ð22Þ

where

ð14:a; bÞ

1

g

¼

1

gE;CHP



1

gpow:plant

;

DC;max ¼ Ael:ch ; COPel:ch: gpow:plant

CONStot ¼ CONSconv  ð15:a; bÞ

DE

ð23:a; b; cÞ

gpow:plant

The parameters in Eq. (23) are calculated once only, before starting the optimization routine.

ð16:a; bÞ

ð17Þ

ð18Þ

ð19Þ

2.2.2. Basic constraints The definition of an appropriate set of constraints requires formulating energy balances for each plant component and expressing their technical limits in operation; boundary conditions should be also expressed. All these constraints are here indicated as ‘‘basic”, because they have an intuitive nature and directly derive from a correct analytical representation of reality. In Section 3, a different kind of constraints will be described, derived from accurate energetic analyses and oriented to superimpose to the optimization routine some pre-determined results. Covering energy loads

C abs;i þ LLel:ch:;i  C el:ch:;nom ¼ Dc;i ; C abs;i þ LLboil;i Hboilmax þ Q TES;i P DH;i HCHP;i  COPAbs

ð24:a; bÞ

Production limits

ð20Þ

abs Evidently, PESrel % and PES% can significantly differ, being the latter higher when surplus power production (i.e. positive DEexch) is achieved. Also, PESrel % has an upper technical limit in the value:

PESrel % ¼ 1

PESi ¼ CONStot;i 

þ gE;CHP  g

1

i

PHRCHP  HCHP;i 6 ECHP;nom ; C abs 6 C abs;nom ; STORTES;i 6 V TES ; STORTES;i P 0 qwater cp ðT del  T ret Þ

ð25:a; b; c; dÞ

Operation/synthesis congruence

HCHP;i 6 jc  dCHP ;

C Abs;i 6 jc  dAbs ;

j HTES;i j6 jc  dTES ð26:a; b; cÞ

pow:plant

ð21Þ while PESabs % might exceed this value. EABOT assumes the PEStotal (sum of PESi, expressed in kWhprim, throughout a year), and not the PESrel % , as objective function in the energy saving-oriented optimization. In order to reduce the calculations, Eq. (13) was rewritten as follows:

where the congruence constant jc is assigned a conveniently large value (in the order of 105, if HCHP, CAbs and HTES are in kW). Other limits in the operation of plant components, expressed by Eqs. (2) and (3), were threaten as discussed in Section 2.1. The energy balance of the heat storage presented in Eq. (8) is also included. As concern the TES, another aspect should be underlined [13]: referring the optimization to a standard year which is supposed to repeat

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identically throughout plant life, the minimum level of charge of the storage capacity must be imposed for the first and the last hour of the year:

STORTES;1 ¼ STORTES;Nhours ¼

STORmin TES

ð27Þ

If Eq. (27) was not included, an unreasonably large VTES value would have been achieved (especially for a small number of days, i.e. small ‘‘Nhours”), because the optimization routine would have assumed to consume as much ‘‘zero-cost” energy (stored at the begin of the year) as possible. 3. Advances in LP modelling by energetic analysis

 MPCHP fuel

PHRCHP  HCHP

ge;CHP

 C abs Hdir CHP þ MPboil fuel  COPel:ch: gboil

P0

ð30Þ

which evidently depends on the design variables ECHP,nom and Cabs,nom; being our analysis oriented to save computational resources, i.e. to superimpose a priori the optimal management strategy (not related to the decision variables), it seems more convenient to compare two cases, the ‘‘combined heat and power” and the ‘‘combined power and cooling” (Eqs. (31) and (32), respectively):

dir CHP PHR CHP  HCHP

 MPfuel

ge;CHP



Hdir CHP

gboil

P0

ð31Þ

C abs C abs þ COPabs COPel:ch: PHRCHP C abs  P0  MPCHP fuel ge;CHP COPabs

Profit ¼ MPel  PHRCHP 



ð32Þ

Two indicators, the Total Spark Spread cooling and heating, (TSSc and TSSh), are derived (see Eq. (33)) from Eqs. (31), (32); the lower heating value of fuel is introduced in the TSS expressions to keep into account that MPfuel is usually expressed in [c€/Nm3] or [c€/kg] (for gaseous and liquid fuels, respectively) and MPE in [c€/ kWh]:

TSSh ¼

3600PHRCHP gboil HLVboil fuel



 MPboil fuel þ MPE

3600 ge;CHP HLVCHP fuel



MPCHP fuel

;

TSSc ¼

PHRCHP COPabs

þ COP1

el:ch:

3600PHRCHP ge;CHP HLVCHP COPabs fuel



 MPE

 MPCHP fuel ð33:a; bÞ

In order to favour a correct interpretation of the above expressions, TSSh and TSSc are here calculated, as an example, by assuming gE,CHP = 0.33, COPabs = 0.7, COPel.ch. = 2.5, MPE = 12 c€/kWh, MPfuelCHP = 0.35 c€/Nm3, MPfuelboil = 0.48 c€/Nm3: 36000:75

ð28Þ

– If no surplus heat:

C abs COPabs

Hdir CHP C abs

 0:48 þ 0:12  0:75 ¼ 1:25; 3600  0:35 0:3337000 0:75 1

þ 2:5  0:12 ¼ 1:71 TSSc ¼ 0:7 36000:75  0:35 0:33370000:7

TSSh ¼ 0:9037000

Being a priori unknown Hwaste CHP , the problem cannot be solved yet. However, from the fundamentals of thermoeconomics we know that cost should be only allocated to useful products; consequently, two cases can be distinguished, respectively, when some surplus heat is recovered and rejected and when not:

Hwaste HCHP ¼ Hdir CHP ¼ 0 CHP þ

A¼

boil Profit ¼ MPel  ðPHRCHP  Hdir CHP Þ þ MPfuel 

In this section the possibility to formulate additional constraints derived from energetic analyses is discussed; in particular, a thermoeconomic analysis is performed, in order to detect a priori trade-off profit conditions to be used for the optimization of plant operation. Thermoeconomics of CHCP systems has primarily been used for cost accounting [3]: assuming a rational basis such as ‘‘exergy destruction” for assigning a cost to each flow has allowed to determine a ‘‘rational value” for each product of polygeneration systems. Here, on the contrary, a marginal cost analysis is proposed without cost allocation, having been assumed impossible to vary the PHRCHP, i.e. to privilege the production of the products ensuring a higher unit profit. Let us refer to the simplified scheme in Fig. 2 (the effects of TES’ exclusion in the analysis will be discussed later), where HCHP is assumed as main operation variable. A low disaggregation level is adopted, treating the ‘‘CHP unit + absorption chiller” system as a basic unit to write the cost balance; evidently, the thermal output HCHP is represented as sum of three contributes: the heat covering thermal loads, the heat feeding the absorption chiller and, eventually (in case of non-heat-tracking operation), some surplus heat rejected. Trade-off profit conditions emerge by a comparison between the operating cost in the separate and combined power production cases:

 Profit ¼ MPE  PHRCHP  HCHP þ

The condition expressed by Eq. (28) leads to different trade-off conditions depending on the ratio:

ð29Þ

ð34:a; bÞ

Evidently, at the examined conditions, combined heat and power have lower margins for profit with respect to combined power and cooling. We may conclude that whenever, throughout a year, TSSh (the lower of the two TSS) is higher than 1, operating the CHCP plant in combined production mode (at any value of the Hdir CHP =C abs ratio) is profitable; of course, this result is valid as far as no surplus heat is produced. – If surplus heat (Hwaste CHP 6¼ 0):

Fig. 2. Zoom on energy flows involving the examined section of the CHCP scheme.

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When the CHP unit operates in ‘‘power production mode”, i.e. the heat recovery is not used to cover any useful demand, profit must be expressed considering the only useful output, as follows:

Profit ¼ MPel  PHRCHP  HCHP  MPCHP fuel

PHRCHP  HCHP

ge;CHP

P0

ð35Þ

Trade-off conditions can clearly expressed with reference to the Spark-Spread:

SS ¼

MPE 3600 ge;CHP HLVCHP fuel

 MPCHP fuel

ð36Þ

Again, SS values higher than 1 indicate the profitability of autoproducing electricity, with respect to the option ‘‘switch the engine off and buy electricity”. Obviously, Eq. (35) is a much more restrictive condition with respect to Eqs. (31), (32); consequently, in many countries the former condition is expected to be fulfilled in peak-hours only, while Eqs. (31) and (32) can be also fulfilled in medium and, eventually, offpeak hours. The above described method is very easy to implement; any optimization tool like EABOT can be equipped with a preliminary routine which calculates SS, TSSh and TSSc for any hour of the year and can thus perform a ‘‘conditioned optimization”, with a pre-fixed plant operation. Two aspects should be here underlined: 1. The auxiliary production units were neither included in Fig. 2 nor in the proposed analysis. This is why zero profit is associated with these units and their influence on the trade-off profit conditions is null. 2. We should ask what changes in the above analysis when we take into account the presence of the TES. Evidently, when SS is higher than 1 nothing changes: operating the CHP unit at full load remain the most profitable choice. Similarly, when TSS is lower than 1, the CHP unit should be switched off even in presence of the TES. The only deviation from the above described analysis concerns the hours where TSS is higher than 1 and SS lower than 1: in fact, the surplus energy eventually stored during peak hours (SS > 1) could be sufficient to cover the heat loads in other hours with SS lower than 1 and TSS higher than 1. While in absence of the TES the heat recovery could have been covering a useful demand (requiring to use the TSS expression), in presence of the TES the SS should be more properly adopted to assess the most rational operation. This analysis slightly complicates the problem; here, for sake of simplicity, the effects of the TES on the proposed operation criteria will be neglected. This approach is some way reasonable if very

large TES are excluded, being most of their stored energy consumed in off-peak hours when both SS and TSS are lower than 1; an ex-post analysis will be used validate this hypothesis. Before applying the method to real buildings, let us briefly discuss what effects additional constraints are expected to have on the optimization algorithm. With reference to Fig. 3a and b, where representations of linear and non-linear two-variables problems are given, it is evident that in LP problems the optimal solutions lie on the boundary of a feasible-region polytope (n-dimensional space enclosed by a finite number of hyperplanes), i.e. at the intersection of the binding (active) constraints; on the contrary, in nonlinear programming (NLP), solutions from both the boundary and within the feasible-region are accommodated. Consequently, large benefits could be expected when NLP algorithms are implemented, due to the reduction of the feasible-region polytope; on the contrary, as concerns LP techniques the benefits could either be relevant or negligible, depending on the search direction adopted. In particular, if an interior point method is adopted (like in our case), some reduction in computational resources could be achieved while no benefits are expected if the simplex method is used [21]. 4. Applying EABOT to two cases-study Optimizations were performed by EABOT for a 646 rooms (1146 beds) hotel and a 300 beds hospital situated in Italy, whose electricity, heating and cooling loads have been presented in [6,22]; the measured energy load peaks are presented in Table 1. As concerns energy prices, two different scenarios are considered, both assuming a four-bands electricity price structure (peak, high load, medium load and off-peak hours); the first scenario takes into account the tax exemption for CHP fuel, the second does not. The values adopted for the optimization, referring to the Italian case, are resumed in Table 2. The EABOT tool was systematically applied to the two examined buildings, assuming different values for Nhours (in any case, multiples of 24, being the analysis performed on hourly basis); five different conditions were tested, that are

Table 1 Measured energy load peaks of the two examined buildings

Hotel Hospital

Electricity (kW)

Heating (kW)

Cooling (kW)

985 241

3,134 2,655

1,129 1,548

Fig. 3. Search directions for a 2-variables optimization in case of (a) linear programming, (b) non-linear programming.

A. Piacentino, F. Cardona / Energy Conversion and Management 49 (2008) 3006–3016 Table 2 Energy prices assumed for the optimization

Electricity Peak hours High load hours Medium load hours Off-peak hours Natural gas No tax exemption With gas exemption

Prices

SS and TSS (if tax exemption)

SS and TSS (if no tax exemption)

0.16 (€/kWhe) 0.105 (€/kWhe) 0.07 (€/kWhe)

1.27 and 1.89 0.84 and 1.45 0.56 and 1.17

1.02 and 1.51 0.67 and 1.16 0.45 and 0.94

0.045 (€/kWhe)

0.36 and 0.97

0.29 and 0.78

3

0.50 (€/Nm ) 0.40 (€/Nm3)

(a) Profit-oriented optimization, with tax exemption for CHP fuel. (b) Profit-oriented optimization, with no tax exemption for CHP fuel.

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(c) Profit-oriented optimization with tax exemption for CHP fuel and an additional constraint ensuring the solution to achieve a minimum PESrel % (as required by the Directive 2004/8/EC). (d) Primary energy saving-oriented optimization (max PESabs % , regardless of economic viability). (e) Profit-oriented optimization with tax exemption and a prefixed operation strategy (see Section 3) superimposed. The results obtained in terms of ECHP,nom, Cabs,nom, VTES, NPV and annual PESabs % are plotted in Fig. 4 versus the number of days used for the optimization; in Fig. 5 the calculation time, referred to an ordinary performance machine (3 GHz processor, 1024 GB RAM), is also represented. Let us observe that – All the main synthesis and design variables converge to quite constant solutions when the number of days used for the optimization is higher than 15–20. – The consumption of resources (i.e. the calculation time) rapidly increases when the number of days is higher than 30;

Fig. 4. Optimal ECHP, Cabs,nom,VTES, NPV and PESabs % versus the number of days used as a basis for the optimization.

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The comparative analysis revealed that a number of days ranging between 24 and 30 already ensures a high reliability of results; then, the results shown in the following of this paper will be referred to the assumption of 24 days (Nhours equal to 576). As concerns the energetic results:

Fig. 5. Time consumed for the optimization as a function of the number of days.

even considering 60 days, however, the calculation time is quite small due to the relevant simplifications introduced in the model.

– Comparing the results of the options ‘‘a” and ‘‘b” (the option ‘‘c” is not represented in the figure), the relevance of tax exemption to make trigeneration viable is evident. Very small CHP units resulted in the no incentives case, leading to exclude the absorption chiller from the plant lay-out and consequently to achieve quite small energy and money savings. – The PES-oriented optimization (option ‘‘d”) leads to oversize the combined production unit and the TES, as could be expected. In fact, being any economic index excluded from the decision function, a convergence towards negative NPVs may be observed. As concerns the PESabs % , high values are achieved (in the optimization of the CHCP plant for the hos-

Fig. 6. Optimal operation for three consecutive winter days and dependence on energy prices.

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pital, not shown in the figure, values higher than 0.35 also resulted!), which can be explained by the analysis presented in the Section 2.2.1. – As concerns all the 5 monitored parameters, the optimization options ‘‘a” and ‘‘e” converged to the same solutions; hence, at synthesis-design level the superimposed operation strategy does not affect the reliability of results (with the additional constraint, a convergence to the same ‘‘optimal solution” as in the ‘‘no additional constraint” case can be observed in a lower computation time).

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In order to put into evidence the rational of the results obtained, a zooming representation of results is preferable to a full scale one. Consequently, in Figs. 6 and 7 the results are shown for three consecutive days in the winter and four consecutive days in the summer period, respectively; representing the results for all the optimized 24 days would lead to a non-clear plot, while zooming the results for one single days would make us loose any possibility to detect the mutual influence of TES charge between two consecutive days.

Fig. 7. Optimal operation for four consecutive summer days and dependence on energy prices.

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Both figures, referring to the optimization option ‘‘a” (profit oriented with incentives), allow us to recognize that the spontaneous convergence of the optimization reflects the operation strategy superimposed in the option ‘‘e”. In Fig. 6, the CHP unit is operated at full load when TSS is higher than 1 and switched off in other cases. However, this occurs just because a useful heat demand exist, as evident when looking at the HTES curve, which shows that any heat excess is used to charge the tank; in fact, being SS lower than 1 (no peak, just high load hours are present in winter days), operating the unit at an as high load level as to reject surplus heat would not be convenient. Evidently, even including the TES, superimposing the operation strategy in peak (SS > 1, engine at full load operation) and off peak (TSS < 1, engine switched off) hours allows the LP algorithm to converge to the optimal solution, being validated the hypothesis made in Section 3 to neglect TES’ influence. Also, looking at the STORTES we may observe that the presence of long peak-hours periods leads to very high peaks of the stored energy; in Fig. 6, almost 8,000 kWh, 5 times more than in the summer period (see Fig. 7) where the energy price profile is more irregular. In Fig. 7 the cooling balance is also plotted; it is evidently influenced by the heat recovery rate HCHP, which ultimately depends on the energy price. During peak hours the CHP unit operates at full load regardless of the existence of a useful heat demand; however, again no heat is rejected due to the sufficient capacity VTES of the storage tank. Let us observe that a note in the PESabs % curve (see Fig. 7) indicates the scarce energetic relevance of its singular points; in fact, being it artificially PESabs % calculated on hourly basis, depending on whether the TES is charging or discharging, very high (positive or negative) energy saving rates could result, which have no relation with the actual energy saving potential in the examined hour. Actually, a detailed analysis which cannot be presented here would detect minimum differences in the optimal operation achieved according to the options indicated as ‘‘a” and ‘‘e”; however, the reduction in the consumption of resources can be significant (depending on the algorithm adopted) and it absolutely justifies the use of additional constraints. The discussion of results (Figs. 4, 6 and 7) presented in this subsection ensures the system to have converged to ‘‘energetically meaningful” solutions; the experience of the designer is decisive in this phase of code testing. 5. Conclusions Starting from the assumption that large progresses have been made in the use of efficient algorithms for the optimization of synthesis, design and operation of a CHCP system including thermal energy storages, a new approach was proposed in this paper, focusing the attention on the energetic analysis of the plant. Considering that MILP optimization of trigeneration systems represents a quite approximate approach due to the linearization of components’ behaviour and cost figures, some simplifications were introduced, like the exclusion of binary variables for the hour by hour unit commitment problem which significantly reduces the consumption of computational resources. An in-depth analysis of trade-off profit conditions was proposed, partially based on thermoeconomics, allowing us to formulate an artificial (i.e. non-physical) constraint which essentially superimposes a pre-fixed operation

strategy. Implementing the proposed approaches into a tool allowed us to perform multi-objective optimizations for two large buildings in the civil sector and to derive a few conclusions about the optimal number of days to be used for the optimization (which resulted from a compromise between the objectives of results’ reliability and fastness of the optimization) and the consistence of the hypotheses introduced. This paper was only intended to offer a new perspective on the problem of the improvement of MILP techniques for CHCP optimization; such analysis must be evidently coupled with those most typically proposed by operational research specialists. References [1] Hernández-Santoyo J, Sánchez-Cifuentes A. Trigeneration: an alternative for energy savings. Appl Energy 2003;76:219–27. [2] Ziher D, Poredos A. Economics of a trigeneration system in a hospital. Appl Therm Eng 2006;26:680–7. [3] Temir G, Bilge D. Thermoeconomic analysis of a trigeneration system. Appl Therm Eng 2004;24:2689–99. [4] Teopa Calva E, Picón Núñez M, Rodrı´guez Toral MA. Thermal integration of trigeneration systems. Appl Therm Eng 2005;25:973–84. [5] Dimopoulos GG, Kougioufas AV, Frangopoulos CA. Synthesis, design and operation of a marine energy system. Energy. doi:10.1016/ j.energy.2007.09.004. [6] Cardona E, Piacentino A. A methodology for sizing a trigeneration plant in Mediterranean areas. Appl Therm Eng 2003;23:1665–80. [7] Cardona E, Piacentino A. Optimal design of CHCP plants in the civil sector by thermoeconomics. Appl Energy 2007;84:729–48. [8] Kong XQ, Wang RZ, Huang XH. Energy optimization model for a CCHP system with available gas turbines. Appl Therm Eng 2005;25:377–91. [9] Oh SD, Lee HJ, Jung JY, Kwak HY. Optimal planning and economic evaluation of cogeneration systems. Energy 2007;32:760–71. [10] Gamou S, Yokoyama R, Ito K. Optimal unit sizing of cogeneration systems in consideration of uncertain energy demands as continuous random variables. Energy Convers Manage 2002;43:1349–61. [11] Rong A, Lahdelma R. An efficient linear programming model and optimization algorithm for trigeneration. Appl Energy 2005;82:40–63. [12] Rong A, Lahdelma R, Luh PB. Lagrangian relaxation based algorithm for trigeneration planning with storages. Eur J Oper Res. doi:10.1016/ j.ejor.2007.04.008. [13] Dotzauer E, Holmström K, Ravn HF. Optimal unit commitment and economic dispatch of cogeneration systems with a storage. In: Proceedings of the 13th PSCC conference, Trondheim, Norway; 1999. [14] Piacentino A. CHP and CHCP applications in buildings: energetic, exergetic and economic analysis of the possible solutions. PhD thesis, Palermo, January 2005 [in Italian]. [15] Arcuri P, Florio G, Fragiacomo P. A mixed integer programming model for optimal design of trigeneration in a hospital complex. Energy 2007;32: 1430–47. [16] Piacentino A, Cardona F. Integrated optimization of synthesis, design and operation in CHCP-based l-grids – Part I. Description of the method. In: Proceedings of ECOS 2007, Padova. Italy: SGE Pub.; June 2007. p. 575–84. [17] Cardona E, Piacentino A. DABASI-WWW promotion of energy saving by CHCP plants – database and evaluation. SAVE II Project. Contract No. 4.1031/Z/02060, Bruxelles; 2005. [18] Valero A, Serra L, Lozano MA. Structural theory of thermoeconomics. In: ASME book-AES, Thermodynamics and the design, analysis, and improvement of energy systems, vol. 30; 1993. p. 189–98. [19] Kostowski W, Skorek J. Thermodynamic and economic analysis of heat storage application in co-generation systems. Int J Energy Res 2005;25:177–88. [20] De Paepe M, Mertens D. Combined heat and power in a liberalised energy market. Energy Convers Manage 2007;48:2542–55. [21] Venkataraman P. Applied optimization with MATLAB programming. New York: Wiley; 2002. [22] Piacentino A, Cardona F. On thermoeconomics of energy systems at variable load conditions: integrated optimization of plant design and operation. Energy Convers Manage 2007;48:2341–55.