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Energy 78 (2014) 587e603

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Energy

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Allocating resources and products in multi-hybridmulti-cogeneration: What fractions of heat andpower are renewable in hybrid fossil-solar CHP?

Gian Paolo Beretta a, *, Paolo Iora a, Ahmed F. Ghoniem b

a Department of Mechanical and Industrial Engineering, Universit�a di Brescia, via Branze 38, 25123 Brescia, Italyb Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:Received 6 May 2014Received in revised form7 October 2014Accepted 16 October 2014Available online 18 November 2014

Keywords:Cogeneration regulationRenewable energy regulationMulti-generationPrimary energy factorsAllocation methods in cogenerationHybrid power production

* Corresponding author.E-mail address: gianpaolo.beretta@unibs.it (G.P. Be

http://dx.doi.org/10.1016/j.energy.2014.10.0460360-5442/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

A general method for the allocation of resources and products in multi-resource/multi-product fa-cilities is developed with particular reference to the important two-resource/two-product case ofhybrid fossil & solar/heat & power cogeneration. For a realistic case study, we show how the methodallows to assess what fractions of the power and heat should be considered as produced from the solarresource and hence identified as renewable. In the present scenario where the hybridization of fossilpower plants by solar-integration is gaining increasing attention, such assessment is of greatimportance in the fair and balanced development of local energy policies based on granting incentivesto renewables resources.

The paper extends to the case of two-resource/two-product hybrid cogeneration, as well as to generalmulti-resource/multi-generation, three of the allocation methods already available for single-resource/two-product cogeneration and for two-resource/single-product hybrid facilities, namely, the ExRR(Exergy-based Reversible-Reference) method, the SRSPR (Single Resource Separate Production Refer-ence) method, and the STALPR (Self-Tuned-Average-Local-Productions-Reference) method. For the casestudy considered we show that, unless the SRSPR reference efficiencies are constantly updated, thedifferences between the STALPR and SRSPR methods become important as hybrid and cogenerationplants take up large shares of the local energy production portfolio.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Many industrial processes can achieve higher production effi-ciency from the integration of production lines of a mix of differentgoods and/or using a mix of different resources and/or raw mate-rials. Such facilities may be classified as either multi-resource ormulti-generation, or both. Typical examples are the combinedproduction of heat and power [1], or the production of a mix ofliquid or gaseous fuels like methanol and hydrogen from a fossilresource, such as coal or natural gas [2]. Likewise, one can combinein the same process multiple resources either fossil or renewable orboth, such as coal, natural gas, biomass, geothermal and solar en-ergy, to produce electricity, water desalination, heating, cooling andother by-products [3]. Hybridization of power plants by replacing

retta).

part of the fossil resources with renewables is currently viewed as aprominent method for future more sustainable scenarios [4]. Inorder to design regulations with the objective of stimulating thesetechnologies and to promote the use of renewable resources it isimportant to award incentives and compare efficiencies on thebasis of a fair allocation, within each power plant, of the benefits ofhybridization and cogeneration.

Several allocation methods have been proposed in the recentyears. Broadly speaking they can be divided into thermodynamicmethods and thermoeconomic (or exergoeconomic) methods. Acomprehensive set of reviews of the rationale of such allocationschemes and their relevant implications can be found with refer-ence to assessing primary energy savings [5] and carbon dioxideemissions [6] as well as in the context of combined productionpower and desalination [7]. The thermodynamic allocationmethods are based on the computation of input/output energy,entropy [8] or exergy fluxes associated to resources and products,generally by comparison with a set of conversion efficiencies, or

G.P. Beretta et al. / Energy 78 (2014) 587e603588

primary energy factors,1 assumed as references for each resource/product pair. On the other hand, the thermoeconomic methods [9]are based on the exergetic cost balance between input and outputexergy streams. They include exergetic cost theory [10], functionalapproach [11], and specific-cost exergy-costing approach [12].

In previous papers on allocation methods, we addressed twoparticular cases of multi-resource multi-generation. In Ref. [13] weaddress the allocation problem in combined heat and power pro-duction (i.e., for single-resource/multi-generation facilities), byfocusing on the problem of defining a ‘fair’ method to assess howmuch of the fuel consumption should be allocated to the produc-tion of heat and how much to the production of electricity. In suchcases, the problem is how to determine the partial efficiencies (orequivalently the primary energy factors) of heat and electricityproduction in a cogeneration facility.

The work in Ref. [13] was motivated by the fact that none of theexisting allocation approaches is yet universally accepted, thusleading to some arbitrariness in the quantification of the economic/energetic/environmental value of the different cogenerated goodsand consequently of the basis on which subsidies are awarded tosuch facilities. A drawback of the classical allocation approachesconsists of the adoption of reference efficiencies, determined bysome local authority or the regulation itself, for the separate pro-duction of heat and electricity that may be different from the actualaverage efficiencies of heat and electricity in the energy portfolio ofthe local area of interest with which the cogeneration facility ismeant to be compared. Therefore, in Ref. [13] we formulate anddiscuss (see also Ref. [14]) a novel STALPR (Self-Tuned-Average-Local-Productions-Reference) allocation method whereby the heatand electricity allocation fractions are based on the average effi-ciencies of the local area (or a prescribed comparison context) thatincludes the cogeneration plant itself.

In Ref. [15] we address the conjugate allocation problem, i.e.,that for multiple-resources/single-product facilities. We focus onhybrid fossil-solar power plants and discuss the problem ofdefining a ‘fair’ method to allocate fractions of the produced elec-tricity to the different resources consumed. The analysis in Ref. [15]refers to hybrid fossil-solar facilities because they represent afavorable solution in terms of a reliable use of solar energy, capableto mitigate in part its drawbacks due to the intrinsic high degree ofdaily and seasonal variability. For an overview of different hybriddesigns see Ref. [17], for a discussion on the economic feasibility ofa typical hybrid configuration see Ref. [18], and for an assessment ofthe potential of hybrid fossil-solar thermal plants from the point ofview of energy policy making see Ref. [19]. In particular, we addressthe question of what fraction of the electricity produced in suchfacilities is to be considered as generated from renewables (see alsoRef. [16]). In Ref. [15], we extend to this context the definition of theclassical SRSPR (Single-Resource-Separate-Production-Referenceallocation method) based on reference partial efficiencies pre-scribed by some authority and we show that it is ‘unfair’ inasmuchas the reference efficiencies are different from the actual averageefficiencies of the energy portfolio in local area with which thehybrid facility is to be compared. Thus, also for this context wepropose a STLAPR (Self-Tuned-Average-Local-Productions-Refer-ence allocation method) whereby the allocation fractions are

1 We recall that the “primary energy factor” of a given good is defined as theamount of primary energy that is consumed to produce a unit amount of that good,taking into consideration all processes in its life cycle. In the case of a power plant itequals the inverse of the conversion efficiency calculated on the basis of the overallprimary energy consumption, i.e., not just the direct consumption in the facilityitself but also in all the other processes in its life cycle. For example, for natural gasRef. [20] suggests to increment the actual consumption in the power plant by 10% toaccount for the gas extraction and pipelining consumptions.

defined by the SRSPR method based on reference partial effi-ciencies identical to the actual average efficiencies that characterizethe actual energy production portfolio (including the hybrid plantitself) of the local area of interest for the allocation.

The present paper extends and generalizes the methods ofRefs. [13,15] to address the allocation problem for the most generalcase of multi-resource/multi-generation and exemplifies its appli-cation to the relevant case of hybrid fossil-solar combined heat andpower production.

We consider our approach a contribution towards a 'fairer' allo-cation with respect to the existing procedures. Our idea stems fromthe fact that generation facilities are always part of a local productionportfolio. Thus, the overall environmental and economic advantagesobtainedby introducinga combined/renewableproductionplant inacertain local area, have a relative impact which depends on theexisting local situation andwebelieve that it is unfair to compute theallocation without taking this into proper account.

The paper is organized as follows. In Section 2, we formallyintroduce the allocation problem and define the key parameters forthe simplest case of a two-resources/two-products (fossil-renew-able/heat-electricity) hybrid cogenerator. In Section 3, we define arealistic case study based on hybrid solar-integrated heat and po-wer cogeneration facilities that we use to compare the variousallocation methods. In Section 4, we discuss the allocation resultingfrom the logic of two ‘classical’ methods usually adopted or underconsideration by regulators, namely the SRSPR (Single-ResourceSeparate-Production Reference) method and the ExRR (Exergy-based Reversible-Reference) method. In Section 5, we introduce ourcontext-dependent adaptive marginal allocation method (that wecall STALPR). In Section 6, we define the PES (Primary EnergySavings) parameters characterizing a hybrid cogenerator. In Section7, we formalize the extension of the SRSPR and STALPR methods tothe general case of multi-resource/multi-generation. In Section 8,we draw our conclusions. In the Appendix, we provide formulasthat simplify the computation of STALPR allocation fractions fortwo-resources/two-products cases.

2. Allocation problem definition for a hybrid renewable-and-fossil-fuel heat and power cogenerator

In this section we consider a hybrid facility which is also acogeneration plant, producing on a yearly basis an amount Whyb ofelectrical energy and an amount Qhyb of thermal energy at a single

level of temperature ThybQ , while consuming on a yearly basis an

amount PhybR of renewable primary energy and an amount PhybF offossil primary energy. Accordingly, we adopt symbols and sub-scriptsW, Q, and P, respectively, for work (electricity), heat (thermalenergy), and primary energy, while the subscripts R and F stand forrenewable and fossil, respectively.

This can be the case of a hybrid natural-gas or coal cogenerationfacility integrated with a biomass or solar energy input. Forsimplicity, it is assumed that the hybrid cogeneration facility isincluded in a local area where all the other heat production andpower production facilities are single-resource and single-product,each producing either heat or electricity and using either arenewable or a fossil resource. A sketch of the local area is repre-sented in Fig. 1 where the energy fluxes between input and outputof each facility can be defined as follows. In case of the renewable-only electricity plant, renewable primary energy PsrR

Wis converted

into electricityWsrR according to the plant primary energy factor f

srRW.

Similar considerations are valid for the other three single-resource

Fig. 1. Schematic representation of a local area of interest with a single hybrid cogeneration plant.

G.P. Beretta et al. / Energy 78 (2014) 587e603 589

and single-product facilities shown in the upper part of Fig. 1. Forthe hybrid cogenerator sketched in the lower part of Fig.1, the input

primary energy sources PhybF and PhybR can be conceptually dividedbetween the quota used to produce electricity and heat, namely,

PhybF ¼ PhybFW

þ PhybFQ

and PhybR ¼ PhybRW

þ PhybRQ

. In turn, the produced

electricity and heat can be apportioned between the renewable and

fossil quota, namely, Whyb ¼ WhybR þWhyb

F and

Qhyb ¼ QhybR þ Qhyb

F :

The objective of the analysis in the first part of this paper is todefine methods to allocate the heat and power productions of thehybrid fossil-solar cogeneration facility among the different pri-mary resources it consumes. This is equivalent to finding the eightunknowns of the allocation problem, i.e., the four resource alloca-tion fraction a's and the four product allocation fractions b's definedas follows

ahybRW

¼PRW

hyb

PhybR

ahybRQ

¼PRQ

hyb

PhybR

ahybFW

¼PFW

hyb

PhybF

ahybFQ

¼PFQ

hyb

PhybF

bhybWR

¼ WhybR

WhybbhybQR

¼ QhybR

QhybbhybWF

¼ WhybF

WhybbhybQF

¼ QhybF

Qhyb

(1)

Thus, PhybRW

and PhybRQ

are the fair shares of the consumption PhybR

of primary renewable energy in the hybrid cogeneration facilitythat can be attributed to the produced electricity and heat,

respectively; PhybFW

and PhybFQ

the respective fair shares of the con-

sumption PhybF of primary fossil energy; WhybR and Whyb

F are the fairshares of the productionWhyb of electricity that can be attributed to

the consumption of renewable or fossil resource, respectively; QhybR

and QhybF the respective shares of the production Qhyb of heat.

Moreover, we clearly have the conditions

ahybR þa

hybR ¼ 1 a

hybF þa

hybF ¼ 1 b

hybW þb

hybW ¼ 1 b

hybQ þb

hybQ ¼ 1

W Q W Q R F R F

(2)

For example, the renewable-resource allocation fraction

ahybRW

represents the fraction of the overall consumption of the

renewable resource which is to be considered as used to produce

electricity, ahybRQ

¼ 1�ahybRW

to produce heat. Again, the electricity

allocation fraction bhybWR

represents the fraction of the overall elec-

tricity production that is to be considered as obtained from the

renewable resource, bhybWF

¼ 1�bhybWR

from the fossil resource.

The rationale of all the allocation methods considered here, isthat the allocation compares the hybrid cogenerator with aparticular reference energy production scenario, characterized byreference partial conversion efficiencies (or primary energy fac-tors) for each resourceeproduct pair (in our case fossil-electricity, fossil-heat, renewable-electricity, renewable-heat).In Section 6, we discuss some of the possible purposes of allo-cation, such as to measure primary energy savings, to comparepartial efficiencies and environmental impact, to design energypolicies, to distribute incentives, etc., and we suggest that eachdifferent purpose may require a different reference energy pro-duction scenario, to be identified by the regulator or some localauthority.

To allocate the co-consumed fossil and renewable resourcesamong the heat Qhyb and electricity Whyb produced by the hybrid-cogeneration plant, the general rule is that the two resources areallocated in proportion to the corresponding consumptions of fossiland renewable resources, respectively, that would be required inthe assumed reference scenario to produce the same yearlyamounts of heat, Qhyb, and electricity, Whyb. In terms of equationsthis is expressed by

G.P. Beretta et al. / Energy 78 (2014) 587e603590

hybfFW

x Whybhyb

fFQ

x Qhyb

aFW

¼fxFW

Whyb þ fxFQQhyb

aFQ

¼fxFW

Whyb þ fxFQQhyb

ahybRW

¼fRW

x Whyb

fxRW

Whyb þ fxRQQhyb

ahybRW

¼fRW

x Whyb

fxRW

Whyb þ fxRQQhyb

(3)

where the terms fxFW

, fxFQ, fxR

Wand fxR

Qare the reference partial pri-

mary energy factors that characterize the assumed reference sce-nario and hence the resource allocationmethod. More precisely, wemay call fxR

Wand fxR

Qthe reference “renewable partial primary en-

ergy factors” of electricity and heat, and fxFW

and fxFQ

the reference

“fossil partial primary energy factors” of electricity and heat.Clearly, the terms fxF

WWhyb and fxR

WWhyb represent respectively the

consumptions of fossil and renewable resources required in theassumed reference scenario to produce the electricity Whyb. Simi-larly, fxF

QQhyb and fxR

QQhyb are the consumptions of fossil and

renewable resources required in the reference scenario to producethe heat Qhyb.

Similarly, to allocate the cogenerated electricity and heat among

the fossil resource PhybF and the renewable resource PhybR consumedby the hybrid-cogeneration plant, the general rule is that the two-products are allocated in proportion to the corresponding pro-ductions of electricity and heat, respectively, that would be ob-tained in the assumed reference scenario from the consumption of

the same yearly amounts of fossil resource, PhybF , and renewable

resource, PhybR . In terms of equations this is expressed by

bhybWR

¼

PhybR

f RW

x

PhybR

fxRW

þ PhybF

fxFW

bhybWF

¼

PhybF

f FW

x

PhybR

fxRW

þ PhybF

fxFW

bhybQR

¼

PhybR

f RQ

x

PhybR

fxRQ

þ PhybF

fxFQ

bhybQF

¼

PhybF

f FQ

x

PhybR

fxRQ

þ PhybF

fxFQ

(4)

Clearly, the terms PhybF =fxFW

and PhybR =fxRW

represent the

productions of electricity that are obtained in the assumed refer-

ence scenario by consuming PhybF of fossil primary energy and PhybR

of renewable primary energy, respectively. Similarly, PhybF =fxFQ

and

PhybR =fxRQ

represent the productions of heat that are obtained in the

assumed reference scenario by consuming PhybF of fossil primaryhyb

energy and PR of renewable primary energy, respectively.

As anticipated in the introduction, it can be observed that we areextending the resources allocation approaches previously discussedin Ref. [13] for combined heat and power production, and theproducts allocation approaches for hybrid fossil-solar power plantsaddressed in Ref. [15]. In fact, the resource allocation fractions a’sdefined by Eq. (3) are the equivalent to the two allocation fractionsdefined by Eq. (2) in Ref. [13] for the case of a single-resource allo-cation. Clearly in the case considered here, we have four of suchparameters, due to the allocation of both the fossil and solar primaryresources. In the same fashion the four product allocation fractionsb's defined by Eq. (4) correspond to the two allocation fractionsobtained in Eq. (10) of Ref. [15] for case of single-product allocation.

Once the reference partial primary energy factors fxFW

, fxFQ, fxR

W,

fxRQ

are specified, the system of Eqs. (3) and (4) determines the eight

allocation fractions a's and b's, which represent the solution of theallocation problem.

Once the allocation fractions are found, they may be used tocompute the partial primary energy factors of the hybrid cogen-erator. Indeed, the meaning of the allocation is that of the electricalenergy Whyb produced by the hybrid cogeneration plant, the frac-

tion WhybF ¼ b

hybWF

Whyb is obtained from the fossil fuel while the

fraction WhybR ¼ b

hybWR

Whyb is obtained from the renewable resource

(this amount should qualify for subsidies reserved to renewable-to-

electricity conversion). Similarly, of the heat Qhyb produced, QhybF ¼

bhybQF

Qhyb is obtained from the fossil fuel and QhybR ¼ b

hybQR

Qhyb from

the renewable resource (this amount should qualify for subsidiesreserved to renewable-to-heating conversion). Conversely, of the

fossil-fuel primary energy consumption PhybF in the facility,

PhybFW

¼ ahybFW

PhybF is used to produce electricity and PhybFQ

¼ ahybFQ

PhybF

to produce heat (PhybFW

and PhybFQ

should be used to determine the

greenhouse gas fingerprints of the produced electricity and heat,respectively). Similarly, of the renewable-resource primary energy

consumption PhybR in the facility, PhybRW

¼ ahybRW

PhybR is used to produce

electricity and PhybRQ

¼ ahybRQ

PhybR to produce heat.

Therefore, for each resourceeproduct pair, the partial primaryenergy factors for the hybrid plant can be defined as follows

fhybRW

¼aRW

hybPhybR

bhybWR

Whyb

fhybFW

¼aFW

hybPhybF

bhybWF

Whyb

fhybRQ

¼aRQ

hybPhybR

bhybQR

Qhyb

fhybFQ

¼aFQ

hybPhybF

bhybQF

Qhyb

(5)

It is important to make clear that the partial primary energy

factors fhybRW

, fhybRQ

, fhybFW

, and fhybFQ

represent, respectively, the portions

G.P. Beretta et al. / Energy 78 (2014) 587e603 591

of the overall electricity or heat produced which we allocate toeither the renewable or the fossil resource, and they must bedistinguished from the primary energy factors of the renewableand fossil resources used by the hybrid cogeneration facility,

fhybR ¼ PhybR

EhybR

and fhybF ¼ PhybF

EhybF

(6)

as well as from the average primary energy factor of the mix ofresources it uses

fhyb ¼ PhybF þ PhybR

EhybF þ EhybR

(7)

where in Eqs. (6) and (7) EhybF and EhybR are the fuel energy (based onlower heating value) of the fossil and renewable resources,respectively.

Finally and importantly, it is now possible to define andcompute as follows the partial energy-conversion efficiencies andCOP (coefficients of performance) of the hybrid cogenerator. Theefficiency of conversion of fossil-fuel energy to electricity is

hhybFW

¼bWF

hybWhyb

ahybFW

PhybF =fhybF

¼ fhybF

fhybFW

(8)

The coefficient of performance of conversion of fossil-fuel en-ergy to heat is

COPhybFQ

¼bQF

hybQhyb

ahybFQ

PhybF =fhybF

¼ fhybF

fhybFQ

(9)

The efficiency of conversion of renewable-resource energy toelectricity is

hhybRW

¼bWR

hybWhyb

ahybRW

PhybR =fhybR

¼ fhybR

fhybRW

(10)

The coefficient of performance of conversion of renewable-resource energy to heat is

COPhybRQ

¼bQR

hybQhyb

ahybRQ

PhybR =fhybR

¼ fhybR

fhybRQ

(11)

If the method adopted for allocation is “fair”, then the aboveefficiencies and COP's represent a fair apportionment of the bene-fits of hybridization among the different resources and of cogene-ration among the different products, thus allowing fair terms ofcomparison between the efficiencies and COP's of the hybridcogenerator and the respective single-resource single-productionefficiencies and COP chosen by some local authority or regulator asthreshold values to access subsidies or to assess the sustainabilitywith respect to single-resource single-production facilities in alocal area, characterized by hsr

FW, COPsrF

Q, hsr

RW

, COPsrRQ.

In Section 4 we turn to examining how the reference partialprimary energy factors fxF

W, fxF

Q, fxR

W, and fxR

Qare specified in the

different allocation methods considered in the present paper andcompare for the case study defined in Section 3.

3. Case study

To provide a quantitative analysis and comparison of thedifferent allocation approaches in the case of a hybrid cogenerationfacility, let us refer to a local area of interest as sketched in Fig. 1,that we consider as representative of a generic mix of industrial,residential, and tertiary activities with yearly thermal consumptiontwice the consumption of electricity.

We assume that initially the annual electricity and heat demandare supplied according to the following shares:

- 90% of the electricity by fossil-only (i.e., natural-gas single-

resource) power plants operating with fsrFW

¼ fsr;FWF =hsrFW,

assuming an overall (yearly) average efficiency hsrFW ¼ 0.38 and a

fuel primary energy factor fsr;FWF ¼ 1.1 (natural gas); theremaining 10% of electricity by renewable-only (i.e., single-resource, non-hybrid solar) power plants operating with

fsrRW

¼ fsr;RWR =hsrRW, assuming an overall (yearly) average effi-

ciency hsrRW ¼ 0.153 and a primary energy factor fsr;RWR ¼ 1;

- 90% of the heat by fossil-only (i.e., natural-gas single-resource)

heat plants operating with fsrFQ

¼ fsr;FQF =COPsrFQ, assuming an

overall (yearly) average efficiency COPsrFQ

¼ 0.9 and a primary en-

ergy factor fsr;FQF ¼ 1.1 (natural gas); the remaining 10% of heat byrenewable-only (i.e., solar single-resource) heat plants operating

with fsrRQ

¼ fsr;RQR =COPsrRQ

, assuming an overall (yearly) average

efficiency COPsrRQ

¼ 0.5 and a primary energy factor fsr;RQR ¼ 1.0.

Thenweconsider that the fossil-only facilities (bothelectricityandheat plants) are progressively replaced by hybrid renewable-fossilheat-electricity cogenerators (Fig. 1) based on the same SICCS (So-lar-Integrated Combined-Cycle System) technology with parabolictrough solar field considered in the case study proposed in Ref. [15].However, since in the present case study the facility is operated incogeneration mode, it is assumed that a fraction of the steam isextracted from the low pressure section of the steam turbine. Thissolution, which is common practice in the operation of cogenerationcombined cycles, allows a production of heatwith a limited reductionin theproductionof electricity. To compute theannual energybalanceof the cogeneration SICCS, which is a necessary input of every allo-cation method, the following additional hypotheses are made:

- the input primary energies are the same as in the case of thenon-cogeneration SICCS [15], i.e. PhybF ¼ 1878 GWh andPhybR ¼ 425 GWh;

- the reference value of 867 GWh of yearly production of elec-tricity when the plant is operated in non-cogeneration mode, isreduced by 10% in cogeneration mode, resulting inWhyb ¼ 780 GWh;

- the thermal energy to reduced electrical production ratio isassumed equal to 5, i.e., the cogenerative mode yields a 1 kWh

Table 2Summary of SRSPR and ExRR allocation results for the hybrid solar & fossil/heat &electricity cogeneration case study defined in Section 3 (parameters summarized inTable 1).

SRSPR ExRR

hsrFW

¼ 0.38 hsrFW

¼ 0.55 ThybQ ¼ 95 �C ThybQ ¼ 150 �C

ahybRW

0.8545 0.8545 0.9043 0.8588

ahybRQ

0.1455 0.1455 0.0957 0.1412

ahybFW

0.8098 0.7463 0.9043 0.8588

ahybFQ

0.1902 0.2537 0.0957 0.1412

bhybWR

0.0911 0.0648 0.1739 0.1739

bhybWF

0.9089 0.9352 0.8261 0.8261

bhybQR

0.1215 0.1215 0.1739 0.1739

bhybQF

0.8785 0.8785 0.8261 0.8261

fhybRW

5.1109 7.1891 2.8339 2.6913

fhybFW

2.1451 1.9212 2.6355 2.5029

fhybRQ

1.1727 1.1727 0.5390 0.7953

fhybFQ

0.9370 1.2499 0.5013 0.7396

chybW

0.4429 0.3060 0.9300 0.9300

chybQ

0.6111 0.6111 0.9300 0.9300

FhybF

0.4222 0.6111 0.1902 0.2955

FhybR

0.3060 0.3060 0.1902 0.2955

hhybFW

0.5128 0.5726 0.4174 0.4395

COPhybFQ

1.1739 0.8801 2.1942 1.4872

hyb

Table 1Summary of assumptions made and values of some important parameters for thehybrid solar & fossil/heat & electricity cogeneration case study.

Parameters of the local areaYearly electricity to heat demand ratio, Wloc

tot=Qloctot 0.5

Renewable-only share of power production in the

local area, gsr;RWW

0.1

Renewable-only share of heat production in the

local area, gsr;RQQ

0.1

Primary energy factor of solar energy for solar-only

heat facilities in the local area, fsr;RWR , fsr;RQR

1.0

Average efficiency of the renewable-only electricityfacilities, hsr

RW

0.153

Average efficiency of the renewable-only heatfacilities, COPsrR

Q

0.5

Primary energy factor of natural gas (assumed the only fuel

used in the local area for fossil-only facilities), f sr;FWF , fsr;FQF

1.1

Average efficiency of the fossil-only electricity facilities, hsrFW

0.38 (0.55)

Average efficiency of the fossil-only heat facilities, COPsrFQ

0.90

Parameters of the hybrid-cogeneration facilities

Electric index Whyb=Qhyb 780/434

Fossil to renewable primary energy ratio, shyb ¼ PhybF =PhybR 1878/425

Fossil primary energy to electricity ratio, PhybF =Whyb1878/780

Renewable primary energy to electricity ratio, PhybR =Whyb425/780

Fossil primary energy to heat ratio, PhybF =Qhyb1878/434

Renewable primary energy to heat ratio, PhybR =Qhyb425/434

G.P. Beretta et al. / Energy 78 (2014) 587e603592

reduction in the production of electricity with respect to thenon-cogenerative mode for every 5 kWh of cogenerated heat,resulting in Qhyb ¼ 434 GWh.

We further assume that the installation of hybrid cogeneratorsproceeds in the local area until all the single-resource fossil powerplants are replaced, i.e., whenWsr

F is eventually entirely replaced byWhyb. It is worth noting that this situation represents the conditionof highest possible degree of penetration of cogeneration for theconsidered local area of interest.2

Finally, regarding the renewable single-resource facilities, it isassumed that both Wsr

R and QsrR remain fixed to their initial value.

Table 1 summarizes the assumptions considered in the case study.

4. Allocation methods based on different choices of theprescribed reference efficiencies

4.1. SRSPR (Single-Resource Separate-Production Referenceallocation method)

This allocation method assumes the following rule

fxFW

¼ frefFW

fxFQ

¼ frefFQ

fxRW

¼ f refRW

fxRQ

¼ f refRQ

(12)

where frefRW

, frefFW

, frefRQ, and frefF

Qare reference partial primary energy

factors chosen by some authority as representative of the perfor-mance of the (best available or representative average, usuallysingle-resource) power and heat production technologies that use,respectively, the same renewable resource and the same fossil fuelas the hybrid cogeneration facility.

2 This case occurs whenever the initial value of the ratio WsrF /Q

srF is lower than

the electric index Whyb/Qhyb of the hybrid cogeneration plant (see Ref. [13]).

The solution of the allocation problem is obtained bysubstituting Eq. (12) into Eqs. (3) and (4) and solving for theresource and product allocation fractions a's and b's. Table 2 liststhe resulting values of the various parameters for the case studydefined in Section 3, for both the case of href

FW

¼ 0.38 and 0.55. It is

noteworthy that the benefits of hybridization and cogeneration aresignificant with respect to all single-resource single-production(SRSP) facilities in the case of href

FW

¼ 0.38, and for all except for

COPhybFQ

and hhybRW

for the case of hrefFW

¼ 0.55. In particular,

- the efficiency of conversion of fossil-fuel energy to electricitygoes from the 38% (55%) of the SRSP facility to the 51.28%(57.26%) of the hybrid facility,

- the coefficient of performance of conversion of fossil-fuel en-ergy to heat goes from the 90% of the SRSP facility to the 117.4%(88.01%) of the hybrid facility,

- the efficiency of conversion of renewable-resource energy toelectricity goes from the 15.3% of the SRSP facility to the 19.57%(13.91%) of the hybrid facility,

hRW

0.1957 0.1391 0.3529 0.3716

COPhybRQ

0.8528 0.8528 1.8551 1.2574

G.P. Beretta et al. / Energy 78 (2014) 587e603 593

- the coefficient of performance of conversion of renewable-resource energy to heat goes from the 50% of the SRSP facilityto the 85.28% of the hybrid facility.

SRSPR method, it is confirmed that the ExRR method provides

3 The question of how much exergy should be associated with incident solarradiation on the Earth surface is not yet fully resolved and we refer to the extensivediscussions available in the literature, in particular see Refs. [21,22] and referencestherein.

4.2. ExRR (Exergy-based Reversible-Reference allocation method)

According to this method we assume

fxFW

¼ fExFW

fxFQ

¼ fExFQ

fxRW

¼ fExRW

fxRQ

¼ fExRQ

(13)

where

fExFW

¼ PhybF

WhybF;ideal

¼ PhybF

ExhybF

fExRW

¼ PhybR

WhybR;ideal

¼ PhybR

ExhybR

fExFQ

¼ PhybF

QhybF;ideal

¼ PhybF

ExhybF

1� TenvThybQ

0@

1A

fExRQ

¼ PhybR

QhybR;ideal

¼ PhybR

ExhybR

1� TenvThybQ

0@

1A

(14)

where Tenv is the temperature of the environment,ThybQ ¼ ðhfeed � hreturnÞ=ðsfeed � sreturnÞ is the exergy-equivalent-single-heat-source delivery temperature, with hfeed, sfeed and hre-

turn, sreturn respectively the enthalpy and entropy of the feed andreturn streams with which the cogeneration facility delivers thethermal energy. Clearly, for heat delivered in the form of heating agas or liquid stream with negligible pressure drop and constant

specific heat capacity, ThybQ is the log-mean temperature

ThybQ ¼ Tfeed � Treturnð Þ=ln Tfeed=ð TreturnÞ. In writing Eq. (14) we take

into account that the electricity productions by reversible pro-

cesses, WhybF;ideal and Whyb

R;ideal, equal the respective exergies of the

consumed resources, ExhybF and ExhybR , while the heat productions

by reversible processes, QhybF;ideal and Qhyb

R;ideal, are related to the

exergies of the consumed resources by QhybF;ideal ¼ ExhybF =

�1� Tenv

ThybQ

�

and QhybR;ideal ¼ ExhybR =

�1� Tenv

ThybQ

�, respectively.

Substitution of Eq. (14) into Eqs. (3) and (4) yields

ahybRW

¼ ahybFW

¼ Whyb

Whyb þ Qhyb 1� TenvThybQ

0@

1A

ahybRQ

¼ ahybFQ

¼Qhyb 1� Tenv

ThybQ

0@

1A

Whyb þ Qhyb 1� TenvThybQ

0@

1A

bhybRW

¼ bhybRQ

¼ ExhybR

ExhybR þ ExhybF

bhybFW

¼ bhybFQ

¼ ExhybF

ExhybR þ ExhybF

(15)

It is clear that the exergy-based method is equivalent to anSRSPR method in which the authority sets as reference effi-

ciencies (for the energy conversions of fossil/renewable primaryenergies into electricity/heat) those of thermodynamicallyreversible machinery. In other words, we can say that the exergymethod is a Reversible-Reference version of the SRSPR method,hence the acronym ExRR. As such, this method will becomefairer as technological advances will get us closer to thermody-namic reversibility. But it is unfair with current technologiesbecause, as already noted in Ref. [13], current power productiontechnologies have average second-law efficiencies much closerto 100% than current heat production technologies. Thus, whencompared with the SRSPR method, the ExRR method credits thethermal energy with too high a share of the cogeneration ben-efits leaving an unfairly little share of the fuel savings to thecogenerated electricity. Moreover, as already noted in Ref. [15], itis unfair also because current fossil fuel technologies haveaverage second-law efficiencies much closer to 100% than cur-rent renewable technologies. Thus, when compared with theSRSPR method, the ExRR method credits the renewable re-sources with too high a share of the hybridization benefitsleaving an unfairly little share of the enhanced production to thefossil fuel consumption.

In other words, though based on sound thermodynamicreasoning, the exergy-based allocation method assumes as refer-ences, hypothetical efficiencies that are too distant from theaverage efficiencies of the current industrial and technologicalscenario. Therefore, if adopted as a basis of regulations, this methodwould result in market distortions which may for example give toomuch advantage to hybrid district heating systems thus improperlydiscouraging home owners that have access to such heating sys-tems from investing in energy-saving improvements such as betterbuilding and window insulation as well as individual uses of solarenergy.

Table 2 lists the resulting values of the various parameters forthe case study defined in Section 3, assuming ThybQ ¼ 368 K(¼95�C, typical yearly average value for a district heatingsystem) and ThybQ ¼ 423 K (¼150�C suitable for someindustrial processes). Fossil and renewable exergies arecomputed respectively as3 ExhybF ¼ PhybF ¼ 1878 GWhand ExhybR ¼ 0:93 PhybR ¼ 395 GWh. This choice is based on thefact that, on one hand, for most hydrocarbons ExhybF zEhybF towithin ±2.5% (see, e.g. Ref. [21]) and, on the other hand, for solarradiation it is typical to assume ExhybR z0:93 EhybR (see, e.g.Refs. [22,23]).

Results reported in Table 2 suggest the following comments:

- The fraction of primary resources allocated to the heat is higher

for ThybQ ¼ 150�C (ahybFQ

¼ ahybRQ

¼ 0.141) than for ThybQ ¼ 95�C

(ahybFQ

¼ ahybRQ

¼ 0.096) due to the higher exergy associated to the

higher temperature.- Conversely, the value of ThybQ has no effect on the product allo-cation fractions b's, as they depend only on the exergy of thefuels, according to third and fourth of Eq. (15).

- Comparing the values of COPhybFQ

and COPhybRQ

with those of the

G.P. Beretta et al. / Energy 78 (2014) 587e603594

too much credit to the thermal energy with respect to the stateof the art technology of heat production. Also the unfairly high

values of hhybRW

are a consequence of the lower second-law effi-

ciencies of the current renewable technologies with respect tothe fossil conversion, as already observed in the discussion

above.

5. STALPR allocation method

As reported in Refs. [13 and 15], one limitation of the classicalallocation criteria is that they are based on some fixed prescribedreference efficiencies of conversion for each resource-to-productpair. These reference efficiencies are to be assigned by someauthority and in general differ from the actual average ones thatcharacterize the energy portfolio of the local area in which thefacility under consideration is located or with which it is to becompared. Being fixed, the reference values are not dynamicallyinfluenced by the installation of new facilities nor by the pro-gressive penetration of more efficient technologies within thelocal area. Thus, effects due to the modification of the local en-ergy portfolio are neglected. This fact may result in distortions ofthe local energy market, unless the authority continuously up-dates the reference efficiencies by keeping into constant accountthe progressive penetration of different facilities within the localarea.

Our reasoning presumes that our definition of “fair allocation”is as follows: it must be based on reference efficiencies that arerepresentative of the actual average efficiencies of the energyproduction portfolio with which the resulting efficiencies of thefacility are to be compared. In our illustrative examples, to fix ideaswe assume that such portfolio is that of the local area where thehybrid cogenerator is located. But other relevant options, i.e., otherprescribed comparison contexts, are possible andmay characterizeparticular objectives of an energy policy. For instance, to promoteand maintain a healthy competition towards continuous efficiencyimprovements, it may be useful to take as reference portfolio the(dynamically changing) subset of existing facilities in a given re-gion or nation that adopt the same technology of the hybridcogenerator.

To overcome the drawbacks arising from fixed reference effi-ciencies, we propose a self-consistent method whereby the allo-cation is adaptive and self-tuned to the local energy productionportfolio. We call it the STALPR (Self-Tuned-Average-Local-Pro-ductions-Reference) method. The method was first applied to theallocation of fuel primary energy [13] in cogeneration plants andthen employed to define the renewable quota of the electricityproduced in hybrid fossil-solar power plants [15]. In this section,the STALPR method is extended to the case of a hybrid fossil-solarheat-power cogenerator operating in a local area where all otherelectricity and heating facilities are non-cogeneration non-hybridunits.

The closure of the problem according to the STALPR method isgiven by following rules

fxFW

¼ f locFW

fxFQ

¼ f locFQ

fxRW

¼ f locRW

fxRQ

¼ f locRQ

(16)

where we have introduced the local partial primary energy factors,i.e., the average local renewable or fossil primary energy factors ofheat and electricity, defined as follows

PRsr þ PR

hybPFsr þ PF

hyb

f locRW

¼ W W

WsrR þWhyb

R

f locFW

¼ W W

WsrF þWhyb

F

f locRQ

¼PRQ

sr þ PRQ

hyb

QsrR þ Qhyb

R

flocFQ

¼PFQ

sr þ PFQ

hyb

QsrF þ Qhyb

F

(17)

Using Eq. (1), these may conveniently be rewritten in terms ofthe allocation fractions a's and b's as follows

f locRW

¼PRW

sr þ aRW

hybPhybR

WsrR þ b

hybWR

Whyb

flocFW

¼PFW

sr þ aFW

hybPhybF

WsrF þ b

hybWF

Whyb

f locRQ

¼PRQ

sr þ aRQ

hybPhybR

QsrR þ b

hybQR

Qhyb

f locFQ

¼PFQ

sr þ aFQ

hybPhybF

QsrF þ b

hybQF

Qhyb

(18)

so that again, by substituting in Eqs. (3) and (4), we can solve for the

eight unknowns ahybRW

, ahybRQ

, ahybFW

, ahybFQ

, bhybWR

, bhybQR

, bhybWF

, bhybQF

, once the

productions WsrR , W

srF , Q

srR , Q

srF , Whyb, Qhyb and the primary energy

consumptions PsrRW

, PsrFW

, PsrRQ, PsrF

Q, PhybR , PhybF are known.

Figs. 2e5 show the results of the analysis by showing for thevarious parameters the ratios of values obtained with the STALPRmethod to the corresponding values obtained with the SRSPRmethod, plotted as functions of the degree of penetration of hybridcogeneration in the local area, represented by the variablex ¼ Whyb=ðWsr

F þWhybÞ. To obtain the actual value of each variableon the various curves, it is therefore necessary to associate thecorresponding value of the SRSPR method reported in the secondand third columns of Table 2. The reference primary energy factorsfor the SRSPR method are taken equal to the average factors thatexisted in the local area before the introduction of hybrid cogene-ration, therefore, STALPR and SRSPR allocations coincide for x ¼ 0.In each Figure we compare the results for two values of the refer-ence efficiency for (single-resource single-production) power pro-duction from fossil-fuels, namely, a conservative value hsr

FW ¼ 0.38and a more up-to-date value hsr

FW ¼ 0.55.In general, it can be observed that the differences between the

STALPR and SRSPR methods grow as hybrid cogeneration takes onhigher shares of the local energy production portfolio and becomesimportant in areas where their level of penetration is relativelyhigh. For instance, as shown in Fig. 5b the highest deviation of the

COPhybRQ

from the SRSPR method is þ58% at x ¼ 1, while the differ-

ence remains below 10% if the penetration of hybrid cogenerationremains below 30%.

Nonetheless, it is important to remark that the results shown inFigs. 2 to 5 are specifically referred to the particular values chosenfor the parameters that characterize the local area assumed in thiscase study (they are recalled for clarity in each chart). Significantchanges in both the values and profiles are indeed to be expectedfor different local area conditions that can be found in other real-istic energy portfolios.

(a)

(b)Fig. 2. Ratio of the STALPR to the SRSPR values for the partial primary energy factors

fhybFW

, fhybFQ

, fhybRW

, fhybRQ

of the hybrid cogeneration facility, plotted as functions of the

hybrid penetration parameter Whyb=ðWsrF þWhybÞ for the parameters listed in Table 1

and (a) hsrFW

¼ 0.38, (b) hsrFW

¼ 0.55. The SRSPR values are given in Table 2 and are

computed assuming reference values equal to the primary energy factors of therenewable-only fossil-only power plants in the area.

(a)

(b)Fig. 3. Ratio of the STALPR to the SRSPR values for the resource allocation fractions

ahybFW

, ahybFQ

, ahybRW

, ahybRQ

of the hybrid cogeneration facility, plotted as functions of the

hybrid penetration parameter Whyb=ðWsrF þWhybÞ for the parameters listed in Table 1

and (a) hsrFW

¼ 0.38, (b) hsrFW

¼ 0.55. The SRSPR values are given in Table 2 and are

computed assuming reference values equal to the primary energy factors of therenewable-only fossil-only power plants in the area.

G.P. Beretta et al. / Energy 78 (2014) 587e603 595

In principle, a better understanding of the effects of the variousparameters of the STLARP method could be obtained following asystematic analytical study [of the system of twelve equations intwelve unknowns defined by Eqs. (3), (4) and (18)] along the samelines of the analysis we developed for the simpler cases inRefs. [13,15]. Here, however, the additional mathematicalcomplexity of the two-resources/two-goods problem prevented usso far from obtaining useful general considerations on the effects ofthe various parameters. Nevertheless, the mathematical

complexity can be reduced to some level as shown in the Appendix,where the system of twelve equations in twelve unknowns thatdefines the STALPR method is reduced to a much more tractablesystem of three equations in three unknowns that can be easilyimplemented numerically, even in a spreadsheet equipped with anumerical solver.

The method we propose could be used as a basis for regulationsaiming at providing an allocation scheme that remains permanentlyfair as the result of it being adaptive and dynamically tied to theevolving local area production scenario, through the (evolving)

(a)

(b)Fig. 5. Ratio of the STALPR to the SRSPR values for the partial energy conversion ef-

ficiencies hhybFW

, COPhybFQ

, hhybRW

, COPhybRQ

of the hybrid cogeneration facility, plotted as

functions of the hybrid penetration parameter Whyb=ðWsrF þWhybÞ for the parameters

listed in Table 1 and (a) hsrFW

¼ 0.38, (b) hsrFW

¼ 0.55. The SRSPR values are given in

Table 2 and are computed assuming reference values equal to the primary energyfactors of the renewable-only fossil-only power plants in the area.

(a)

(b)Fig. 4. Ratio of the STALPR to the SRSPR values for the product allocation fractions

bhybWF

, bhybQF

, bhybWR

, bhybQR

of the hybrid cogeneration facility, plotted as functions of the

hybrid penetration parameter Whyb=ðWsrF þWhybÞ for the parameters listed in Table 1

and (a) hsrFW

¼ 0.38, (b) hsrFW

¼ 0.55. The SRSPR values are given in Table 2 and are

computed assuming reference values equal to the primary energy factors of therenewable-only fossil-only power plants in the area.

G.P. Beretta et al. / Energy 78 (2014) 587e603596

average conversion efficiencies of the local area, assuming these areknown on a yearly basis. Such an implementation, of course, posesthe important question of how to provide the utilities that mustadopt the STALPR scheme with reliable data about the referenceefficiencies of their local area, which are needed by the allocationscheme. To our knowledge, such reference conversion efficienciescan be extrapolated for each country from official energy statisticsdata, like those computed yearly for instance by the InternationalEnergy Agency [24] or British Petroleum [25]. On amore local scale,almost every country provides official data about its own internalenergy balances (for instance in Italy, the electricity production by

fuel is computed on a yearly basis by Terna [26]). Such data couldrepresent a proper framework to characterize the local area of in-terest. Nonetheless, the objective of some energy policies may be tofocus on more specific smaller areas (for instance, a region, aprovince, a county, a town, a district) affected by a lack of officialreference data. Such cases may therefore require a more directinput by some local authority, for example, by collecting from everyutility and energy consumer above a certain size the necessary dataon their yearly consumptions and productions, so as to compute theaverage conversion efficiencies in an unambiguousway and publishthe results so as to allow the implementation of the STALPR allo-cation method.

G.P. Beretta et al. / Energy 78 (2014) 587e603 597

6. Partial primary energy savings and incentive policies

The allocation analysis discussed so far, may provide a soundframework to address incentive policy issues, because it yields the

values of the efficiencies and coefficients of performance hhybFW

,

COPhybFQ

, hhybRW

, COPhybRQ

of the hybrid cogenerator. Within the process

of quantifying the incentives for such facilities, these values areimportant because they can indeed be compared to the corre-sponding reference efficiencies that regulators may decide to set asthreshold values, to be either prescribed periodically by some au-thority or taken to coincide with the yearly average values definedby the STALPRmethod for a local area or other reference portfolio towhich the hybrid cogenerator belongs.

Notably, in this context it is convenient to introduce the partialprimary energy savings that can be achieved by employing hybridcogenerators instead of single-resource single-production facilitiesthat use same fuel and renewable resource, operating with a pre-scribed set of references efficiencies and coefficients of perfor-

mance hrefFW

, COPrefFQ, href

RW

, COPrefRQ.

Following this approach, for a hybrid cogenerator we can definethe following partial primary energy savings for the conversion offossil-fuel energy to electricity

PEShybFW

¼fhybF bW

F

hybWhyb�hrefFW

� ahybFW

PhybF

fhybF bhybWF

Whyb�hrefFW

¼ 1�hrefFW

hhybFW

(19)

for the conversion of fossil-fuel energy to heat,

PEShybFQ

¼fhybF b

hybQF

Qhyb�COPrefF

Q� a

hybFQ

PhybF

fhybF bhybQF

Qhyb�COPrefF

Q

¼ 1�COPrefF

Q

COPhybFQ

(20)

for the conversion of renewable-resource energy to electricity,

PEShybRW

¼

fhybR bhybWR

Whyb�hrefRW

� ahybRW

PhybR

fhybR bhybWR

Whyb�hrefRW

¼ 1�

hrefRW

hhybRW

(21)

and for the conversion of renewable-resource energy to heat,

PEShybRQ

¼

fhybR bhybQR

Qhyb�COPrefR

Q� a

hybRQ

PhybR

fhybR bhybQR

Qhyb�COPrefR

Q

¼ 1�COPrefR

Q

COPhybRQ

(22)

Moreover, we can define the overall primary fossil-fuel energysavings,

PEShybF ¼1� PhybF

fhybF bhybWF

Whyb�hrefFW

þfhybF bhybQF

Qhyb�COPrefF

Q

(23)

the overall renewable primary energy savings,

PEShybR ¼ 1� PhybR

fhybR bhybWR

Whyb�hrefRW

þ fhybR bhybQR

Qhyb�COPrefR

Q

(24)

the fossil þ renewable primary energy savings of electricityproduction,

PEShybW ¼ 1�aFW

hybPhybF þ aRW

hybPhybR

fhybF bhybWF

Whyb�hrefFW

þ fhybR bhybWR

Whyb�hrefRW

(25)

and the fossil þ renewable primary energy savings of heatproduction,

PEShybQ ¼ 1�aFQ

hybPhybF þ aRQ

hybPhybR

fhybF bhybQF

Qhyb�COPrefF

Qþ fhybR b

hybQR

Qhyb�COPrefR

Q

(26)

Thus, for instance, policies intended to promote an efficientutilization of either the fossil or renewable sources should prescribepositive values (or above a certain threshold) of PEShybF and PEShybRrespectively.

It is easy to verify that the above definitions are interrelated asfollows

0BBBBB@bWF

hybWhyb

hrefFW

þbQF

hybQhyb

COPrefFQ

1CCCCCAPESF¼

bWF

hybWhyb

hrefFW

PESFW

þbQF

hybQhyb

COPrefFQ

PESFQ

(27)

0BBBBB@bWR

hybWhyb

hrefRW

þbQR

hybQhyb

COPrefRQ

1CCCCCAPESR¼

bWR

hybWhyb

hrefRW

PESRWþbQR

hybQhyb

COPrefRQ

PESRQ

(28)

0BBBB@

fhybF bWF

hyb

hrefFW

þfhybR bW

R

hyb

hrefRW

1CCCCAPESW ¼

fhybF bWF

hyb

hrefFW

PESFW

þfhybR bW

R

hyb

hrefRW

PESRW

(29)

0BBBB@

fhybF bQF

hyb

COPrefFQ

þfhybR bQ

R

hyb

COPrefRQ

1CCCCAPESQ ¼

fhybF bQF

hyb

COPrefFQ

PESFQ

þfhybR bQ

R

hyb

COPrefRQ

PESRQ

(30)

Table 3 lists the resulting partial energy savings calculatedaccording to Eqs. (19e26) as resulting from the SRSPR and ExRR

Table 3PES (Primary energy savings) as resulting from the SRSPR and ExRRmethods, for thehybrid solar & fossil/heat & electricity cogeneration case study defined in Section 3(parameters summarized in Table 1).

SRSPR ExRR

hsrFW

¼ 0.38 hsrFW

¼ 0.55 ThybQ ¼ 95�C ThybQ ¼ 150�C

PEShybFW

0.2590 0.0395 �1.3958 �1.2753

PEShybFQ

0.2333 �0.0226 �1.3969 �1.2762

PEShybRW

0.2182 �0.0999 �1.6353 �1.5027

PEShybRQ

0.4137 0.4137 �1.6366 �1.5038

PEShybF0.2542 0.0244 �1.3961 �1.2755

PEShybR0.2542 0.0248 �1.6351 �1.5026

PEShybW0.2514 0.0137 �1.4368 �1.3142

PEShybQ0.2667 0.0580 �1.4375 �1.3149

G.P. Beretta et al. / Energy 78 (2014) 587e603598

methods. It can be observed that the SRSPR approach is capable tofeature the benefits of both hybridization and cogeneration, fromthe point of view of the efficient conversion of the renewable andfossil fuel (positive values of PEShybR and PEShybF ) as well as of theefficient overall production of electricity and heat (positive valuesof PEShybW and PEShybQ ). On the contrary, the negative valuesof PES obtained with the ExRR method (which takes as referencethe efficiencies of thermodynamically reversible machinery)confirm that this approach is unfair in the present technologicalcontext.

7. Generalization of the SRSPR and STALPR allocationmethods to multi-resource/multi-generation

In the present section we generalize the SRSPR and STALPRmethods to a local-area scenario including multi-generation fa-cilities producing a mix of different goods by consuming a mix ofresources or groups of resources. Since our main interest is in en-ergy production and consumption, typical examples of goods canbe: electric power, process heat at different temperature levels,district heating, compressed air, chilling or refrigeration, waterdesalination as well as other energy-intensive productions such ascement, steel, aluminum, other materials or flows. Similarly in theenergy production and consumption context, examples of re-sources can be: oil, natural gas, coal, nuclear energy fuels, biomass,hydroelectric, solar, wind, tidal and geothermal energy, while ex-amples of homogenous groups of resources can be the renewableand non renewable energy resources but the same reasoning canbe applied as well to materials processing or other industrial fa-cilities where in addition to energy resources we must also ac-count for materials (limestone, iron ore, bauxite and other primaryraw materials).

We consider a local area of interest that includes multi-generation facilities that use and transform a mix of different re-sources. Let the generic term primary impact identify the particulareffect or impact that we wish to quantify and allocate in a fair wayamong the various goods produced in a local area by quantifying foreach good the relative contribution of impact that is to be associ-ated with each different resource or group of resources. Examplesof primary impacts are:

- primary energy consumption- greenhouse gas (CO2 equivalent) emissions- exergy consumption (by irreversibility)

As a result, particularly interesting in multi-resource CHP (heatand power cogeneration) are the following primary impact-goods-resources combinations:

- (primary energy consumption)-(heat and electric power)-(renewable and non-renewable resources)

- (CO2 equivalent emissions)-(heat and electric power)-(renew-able and non-renewable resources)

but many more combinations may also be important.

7.1. General formulation

Let us consider the k-th production facility in a local area of

interest. We denote by Gfacility kproduct j the “amount of good (product)” of

j-th type it produces and by Rfacility kresource i the “amount of resource” of i-

th type it consumes. We use different letters to denote the“amounts” of goods and resources because we are free to selectdifferent methods of accounting (and hence valueing). Indeed, theenergy resources (fossil fuels, solar energy, etc.) need not benecessarily expressed and accounted for only in terms of theirenergy content. For example, they could also be valued in terms oftheir exergies or their economic market values, or the amount ofprimary energy consumption or greenhouse gas production asso-ciated with their consumption on a well-to-waste full-life-cyclebasis. Similarly, the raw materials (iron ore, etc.) as well as theproduced goods (electricity, heat, steel, cement, potable water, etc.)need not be necessarily expressed and accounted for only in termsof their amounts in the most natural units (chilowatthours, tons,etc.), but we could also value them in terms of their exergies ortheir economic market values, or some measure of the environ-mental fingerprint associated with their well-to-waste life cycle.

The quantities gfacility kproduct j, r

facility kproduct j, a

facility kresource iproduct j

, bfacility kproduct jresource i

, 4facility kresource iproduct j

,

4locresource iproduct j

defined below are independent of this choice of ac-

counting method as long as all facilities use the same accountingunits for the corresponding resources and goods. Thus, thesequantities represent a ‘universal’ aspect of the allocation problemwhich is independent of such choice. However, the quantities

ffacility kresource iproduct j

, f localresource iproduct j

which are needed as intermediate variables in

the solution of the allocation problem do depend on the chosen setof accounting units, and so do the results as well as the verymeaning of the allocation problem itself. For instance, if for thehybrid renewable& fossil/heat& electricity cogenerator consideredin the previous section, instead of accounting resources in terms oftheir respective primary energy impacts we do it in terms of theirrespective greenhouse-gases fingerprints, we obtain an entirelydifferent allocation problem. However, it can be noted that the twoproblems are related through the ‘universal’ part of the solution ofthe allocation problem which we will see below depends only on

the market shares gfacility kproduct j and r

facility kproduct j which characterize the

local-area scenario.We begin by defining the local market share of facility k with

respect to the production of good j

gfacility kproduct j ¼

Gfacility kproduct j

Gloc;overallproduct j

where Gloc;overallproduct j ¼

Xn

Gfacility nproduct j (31)

G.P. Beretta et al. / Energy 78 (2014) 587e603 599

and the local market share of facility k with respect to theconsumption of resource i

rfacility kresource i ¼

Rfacility kresource i

Rloc;overallresource i

where Rloc;overallresource i ¼

Xn

Rfacility nresource i (32)

Solving the allocation problem for these facilities means to

identify the quantities Rfacility kresource iproduct j

and Gfacility kproduct jresource i

, where Rfacility kresource iproduct j

is

the fair share of consumption of resource i in facility k that can be

attributed to the production of good j and Gfacility kproduct jresource i

is the fair share

of production of good j in facility k that can be attributed to theconsumption of resource i. Of course,

Rfacility kresource i ¼

Xj

Rfacility kresource iproduct j

and Gfacility kproduct j ¼

Xi

Gfacility kproduct jresource i

(33)

and Rfacility kresource iproduct j

¼ Rfacility kresource i if Nproducts ¼ 1, Gfacility k

product jresource i

¼ Gfacility kproduct j

ifNresources ¼ 1. These quantities, for Rfacility k

resource is0 define the resourceallocation fractions

afacility kresource iproduct j

¼Rresource iproduct j

facility k

Rfacility kresource i

(34)

and for Gfacility kproduct js0 the product allocation fractions

bfacility kproduct jresource i

¼Gproduct jresource i

facility k

Gfacility kproduct j

(35)

where, clearly, afacility kresource iproduct j

¼ 1 if Nproducts ¼ 1 and bfacility kproduct jresource i

¼ 1 if

Nresources ¼ 1. It is also be convenient to conventionally set

afacility kresource iproduct j

¼ 0 for every j whenever Rfacility kresource i ¼ 0, i.e., when facility

k does not consume resource i, and bfacility kproduct jresource i

¼ 0 for every i

whenever Gfacility kproduct j ¼ 0, i.e., facility k does not produce good j.

Next, we define the partial resource-i-to-good-j factors for fa-cility k

f facility kresource iproduct j

¼Rresource iproduct j

facility k

Gfacility kproduct jresource i

¼aresource iproduct j

facility k Rfacility kresource i

bfacility kproduct jresource i

Gfacility kproduct j

¼aresource iproduct j

facility krfacility kresource iR

loc;overallresource i

bfacility kproduct jresource i

gfacility kproduct jG

loc;overallproduct j

(36)

and we normalize them by the ratio of the overall consumption ofresources to the overall production of goods in the local area

ffacility kresource iproduct j

¼Gloc;overallproduct j

Rloc;overallresource i

f facility kresource iproduct j

¼aresource iproduct j

facility krfacility kresource i

bfacility kproduct jresource i

gfacility kproduct j

(37)

Similarly, we define the local-area-averaged partial resource-i-to-good-j factors for the local area

flocresource iproduct j

¼

PkRresource iproduct j

facility k

PkGfacility kproduct jresource i

(38)

and we normalize them in the same way

flocresource iproduct j

¼Gloc;overallproduct j

Rloc;overallresource i

f locresource iproduct j

(39)

Finally, according to the STALPRmethod, the allocation fractionsbased on the local-area-averaged resource-i-to-good-j ratios areadopted as follows:

� the key assumption for the resource allocation fractions is

afacility kresource iproduct j

¼fresource iproduct j

loc:ave: Gfacility kproduct j

Pm

floc:ave:resource iproduct m

Gfacility kproduct m

(40)

meaning that the consumption of resources in each facility areallocated among the different goods it produces based on therelative proportions of the consumptions of the respectiveresource that would be required if the same amounts of goodswere to be produced in single-resource separate-productionfacilities operating with resource-to-product ratios equal tothe respective average resource-to-product ratios character-izing the local area of interest; and

� the key assumption for the product allocation fractions is

bfacility kproduct jresource i

¼

Rfacility kresource ifresource iproduct j

loc:ave:

Pn

Rfacility kresource n

f loc:ave:resource nproduct j

(41)

meaning that the production of goods in each facility areallocated among the different resources it consumes based onthe relative proportions of the productions of the respectivegood that would be obtained if the same amounts of resourceswere to be used in single-resource separate-production facil-ities operating with resource-to-product ratios equal to therespective average resource-to-product ratios characterizingthe local area of interest.

By combining the above equations, we readily obtain thefollowing system of (2Nfacilities þ 1)NresourcesNproducts independentcoupled equations

G.P. Beretta et al. / Energy 78 (2014) 587e603600

floc:ave:resource iproduct j

¼

Pk

aresource iproduct j

facility krfacility kresource i

Pkbfacility kproduct jresource i

gfacility kproduct j

afacility kresource iproduct j

¼fresource iproduct j

loc:ave:gfacility kproduct j

Pm

floc:ave:resource iproduct m

gfacility kproduct m

if rfacility kresource is0

bfacility kproduct jresource i

¼

rfacility kresource i

fresource iproduct j

loc:ave:

Pn

rfacility kresource n

floc:ave:resource nproduct j

if gfacility kproduct js0

afacility kresource iproduct j

¼ 0 if rfacility kresource i ¼ 0

bfacility kproduct jresource i

¼ 0 if gfacility kproduct j ¼ 0

(42)

which, for given values of the local market shares gfacility kproduct j and

rfacility kresource i which characterize the local-area scenario, can be solved

numerically for the (2Nfacilities þ 1)NresourcesNproducts STALPR un-

knowns 4loc:ave:resource iproduct j

, afacility kresource iproduct j

, bfacility kproduct jresource i

. Of course, once the values of

these unknowns are obtained, all other quantities may becomputed, for example, the partial resource-i-to-good-j factors forfacility k are obtained from Eq. (36), i.e.,

f facility kresource iproduct j

¼aresource iproduct j

facility k rfacility kresource i

bfacility kproduct jresource i

gfacility kproduct j

Rloc;overallresource i

Gloc;overallproduct j

(43)

where it can be noted that the first fraction on the rhs is the ‘uni-versal’ part of the solution whereas the second fraction brings inthe particular set of accounting choices which characterize theallocation problem.

It is finally noteworthy that since the system of Eq. (42) must besolved numerically, it may be convenient to adopt as initial guessthe solution of the SRSPR method for prescribed reference valuesequal to the average primary energy factors of the single-resourcesingle-product facilities in the local area, i.e., by setting

frefresource iproduct j

¼ fSRSPfacilitiesresource iproduct j

. If for some given resource/product pair ij

there is no single-resource/single-product facility in the local area,

then we need to adopt some reasonable reference frefresource iproduct j

. In the

multi-resource/multi-product case, the nonzero resource andproduct SRSPR allocation fractions are

a

SRSPRfacility kresource iproduct j

¼fresource iproduct j

ref Gfacility kproduct j

Pm

frefresource iproduct m

Gfacility kproduct m

if Rfacility kresource is0

b

SRSPRfacility kproduct jresource i

¼

Rfacility kresource i

fresource iproduct j

ref

Pn

Rfacility kresource n

frefresource nproduct j

if Gfacility kproduct js0

(44)

Therefore, a convenient initial guess for the STALPR system ofEq. (42) is, assuming for simplicity that the local area has at least asingle-resource single-product facility for each resourceeproductpair ij,

f

GUESSloc:ave:resource iproduct j

¼Gloc;overallproduct j

Rloc;overallresource i

fSRSPfacilitiesresource iproduct j

a

GUESSfacility kresource iproduct j

¼fresource iproduct j

GUESSloc:ave: g

facility kproduct j

Pm

f

GUESSloc:ave:resource iproduct m

gfacility kproduct m

if rfacility kresource is0

b

GUESSfacility kproduct jresource i

¼

rfacility kresource i

fresource iproduct j

GUESSloc:ave:

Pn

rfacility kresource n

f

GUESSloc:ave:resource nproduct j

if gfacility kproduct js0

(45)

8. Conclusions

In this paper, we consider a multi-resource multi-generation fa-cility and address the problem of allocating each resource among thedifferent products and each product among the different resources.The aim of the allocation problem is to provide a reasonable andgeneral criterion to define what quota of each the resourcesconsumed by the facility is to be attributed to each of the goods itproduces and what quota of each of the goods it produces is to beattributed to each of the resources it consumes. For a hybridcogenerator producing heat and power by consuming a mix of arenewable resource and a fossil-fuel the allocation problem providesanswers to questions such as: how much of the produced electricityand heat are renewable? What is the efficiency of production ofelectricity from fossil fuel in the cogenerator?What is the coefficientof performance of the production of heat in the cogenerator? Thesequestions represent important aspects of the considerations aboutenergy savings, avoided greenhouse emissions, efficient use andpreservation of fossil resources, and promotion of renewable re-sources that regulators must take into account when designing en-ergy policies and defining incentives in current scenarios where

G.P. Beretta et al. / Energy 78 (2014) 587e603 601

increasing and renewed attention is reserved to the use of renewableresources in hybrid cogeneration technologies.

Thepresentpaper completes andgeneralizes twoprevious studiesthat we published on allocation, namely Ref. [13], where we focus onthe (single-resource/two-products) allocation of the primary energyconsumption in a cogeneration facility among its heat and powerproduction, and Ref. [15] where we focus on the (two-resources/sin-gle-product) allocation of the electricity production ina hybridpowerplant among its solar energy and fossil fuel consumption.

To illustrate quantitatively the allocation problem, we refer tothe realistic case of hybrid cogeneration facilities based on thetechnology of SICCS (Solar-Integrated Combined-Cycle System)operated in cogeneration mode, included in a generic local areawith other traditional single-resource separate-production heatand power facilities.

We first extend to hybrid cogeneration the main classical allo-cation methods, namely, the SRSPR (Single Resource SeparateProduction Reference method) and the ExRR (Exergy-basedReversible-Reference method). It is shown that the SRSPR can beconsidered a fair approach, capable in principle to allot the benefitsof hybridization and cogeneration in a fair way among the re-sources and products, provided the reference single-resource/single-production efficiencies on which it is based are not toodifferent from the actual average efficiencies of the energy portfolioin the local area with which the hybrid cogenerator is to becompared. On the contrary, the ExRR (which takes as referencesingle-resource/single-production efficiencies those of thermody-namically reversible machinery) is unfair in the present techno-logical context because, as we already noted in Refs. [13,15], currentfossil-fuel and power production technologies have averagesecond-law efficiencies much closer to 100% than current heatproduction and renewable technologies. Thus, compared with theSRSPR method, the ExRR method credits the thermal energy withtoo high a share of the cogeneration benefits leaving an unfairlylittle share of the primary energy savings to the cogenerated elec-tricity. Moreover, it credits the renewable resources with too high ashare of the hybridization benefits attributing an unfairly littleshare of the production to the co-consumed fossil fuel.

Nonetheless, the SRSPR method still has a limit in that it re-quires a set of prescribed reference primary energy conversion ef-ficiencies to be defined every now and then by some authority, butit does not specify criteria for the choice of such efficiencies, withthe problem that if they differ from the actual values that charac-terize the energy conversion portfolio of the local area where thehybrid cogenerators are included they result in unfair comparisonswhich in turn may induce possibly distorted incentive policies thusdefeating their objectives.

Therefore, we formulate and extend to the present two-resources/two-products problem, the novel more consistentSTALPR (Self-Tuned-Average-Local-Productions-Reference)method introduced in Refs. [13,15], whereby the allocation isadaptive and self-tuned to the actual efficiencies of the local energyscenario, with no need for prescribed reference efficiencies.

By applying the STALPR method to the considered hybridcogenerator case study, it turns out that the differences with theSRSPRmethod growas hybrid cogeneration technologies take highershares of the local energy production portfolio and become impor-tant in areas where their penetration reaches relatively high levels.

Acknowledgment

The authors gratefully acknowledge the Cariplo–UniBS–MIT-MechE faculty exchange program co-sponsored by the Universit�a diBrescia (UniBS) and the CARIPLO Foundation, Italy under grant2008-2290.

Appendix

A general formulation can be obtained by recasting the alloca-tion problem defined in Section 2 in terms of the following newvariables.

shyb ¼ PhybF

PhybR

εhyb ¼ Whyb

Qhyb(A.1)

chybW ¼

aRW

hyb

f RW

hyb

aFW

hyb

fhybFW

chybQ ¼

aRQ

hyb

f RQ

hyb

aFQ

hyb

fhybFQ

(A.2)

FhybF ¼

f FQ

hybbQF

hyb

fhybFW

bhybWF

FhybR ¼

f RQ

hybbQR

hyb

fhybRW

bhybWR

(A.3)

so that using Eqs. (1) and (5) the allocation fractions may berewritten as follows

ahybRW

¼ εhyb

εhyb þ FhybR

and ahybRQ

¼ FhybR

εhyb þ FhybR

(A.4)

ahybFW

¼ εhyb

εhyb þ FhybF

and ahybFQ

¼ FhybF

εhyb þ FhybF

(A.5)

bhybWR

¼ chybW

shyb þ chybW

and bhybWF

¼ shyb

shyb þ chybW

(A.6)

bhybQR

¼chybQ

shyb þ chybQ

and bhybQF

¼ shyb

shyb þ chybQ

(A.7)

and the partial primary energy factors as follows

fhybRW

¼ εhyb

chybW

shyb þ chybW

εhyb þ FhybR

PhybRWhyb

(A.8)

fhybFW

¼ εhyb

shyb

shyb þ chybW

εhyb þ FhybF

PhybFWhyb

(A.9)

fhybRQ

¼ FhybR

chybQ

shyb þ chybQ

εhyb þ FhybR

PhybRQhyb

(A.10)

fhybFQ

¼ FhybF

shyb

shyb þ chybQ

εhyb þ FhybF

PhybFQhyb

(A.11)

The above formulation makes it apparent that in order to obtainthe allocation fraction and the partial primary energy factors we

G.P. Beretta et al. / Energy 78 (2014) 587e603602

must specify the values of chybW , chyb

Q , FhybF , and F

hybR . Such specifi-

cation constitutes the closure of the allocation problem and char-acterizes the allocation method. In particular according to thisformulation each allocation method defined in Section 4 and 5assumes the following general closure conditions.

chybW ¼ cx

W with cxW ¼

fFW

x

fxRW

(A.12)

chybQ ¼ cx

Q with cxQ ¼

fFQ

x

fxRQ

(A.13)

FhybF ¼ Fx

F with FxF ¼

fFQ

x

fxFW

(A.14)

FhybR ¼ Fx

R with FxR ¼

fRQ

x

fxRW

(A.15)

where for the SRSPR, ExRR and STALPR method, respectively, thesuperscript “x” in Eq. (A.12eA.15) stands for “ref”, “Ex” and “loc”.

It is noteworthy that for STLARP allocation, the problem can besimplified to a system of three equations in three unknowns chosenamong c

hybW , chyb

Q , FhybF , Fhyb

R as follows. First, we define the localmarket share of the single-resource separate-production facilitiesand of the hybrid facility with respect to the productions of elec-tricity and heat.

gsrFW

¼ WsrF

WsrF þWsr

R þWhyb; gsrR

W¼ Wsr

RWsr

F þWsrR þWhyb

; ghybW

¼ Whyb

WsrF þWsr

R þWhyb

(A.16)

gsrFQ

¼ QsrF

QsrF þ Qsr

R þ Qhyb; gsrR

Q¼ Qsr

RQsr

F þ QsrR þ Qhyb

; ghybQ

¼ Qhyb

QsrF þ Qsr

R þ Qhyb

(A.17)

Then we define the local market share of the single-resourceseparate-production facilities and of the hybrid facility withrespect to the consumption of fossil fuel and renewable energy.

rsrFW

¼PFW

sr

PsrFW

þ PsrFQ

þ PhybF

; rsrFQ

¼PFQ

sr

PsrFW

þ PsrFQ

þ PhybF

;

rhybF ¼ PhybF

PsrFW

þ PsrFQ

þ PhybF

(A.18)

rsrRW

¼PRW

sr

PsrRW

þ PsrRQ

þ PhybR

; rsrRQ

¼PRQ

sr

PsrRW

þ PsrRQ

þ PhybR

;

rhybR ¼ PhybR

PsrRW

þ PsrRQ

þ PhybR

(A.19)

Finally, we substitute Eqs. (A.4)e(A.7) into Eq. (18), using theabove definitions and the STALPR closure relations, to obtain aftersome rearrangements the following four equations in the un-knowns chyb

W , chybQ , Fhyb

F , FhybR ,

clocW

sloc¼

rFW

sr þ rhybF

εhyb

εhybþFlocF

rsrRW

þ rhybR

εhyb

εhybþFlocR

gRW

sr þ ghybW

clocW

shybþclocW

gsrFW

þ ghybW

shyb

shybþclocW

(A.20)

FlocF

εloc¼

rFQ

sr þ rhybF

FlocF

εhybþFlocF

rsrFW

þ rhybF

εhyb

εhybþFhybF

gFW

sr þ ghybW

shyb

shybþchybW

gsrFQ

þ ghybQ

shyb

shybþclocQ

(A.21)

clocQ

sloc¼

rFQ

sr þ rhybF

FlocF

εhybþFlocF

rsrRQ

þ rhybR

FlocR

εhybþFlocR

gRQ

sr þ ghybQ

clocQ

shybþclocQ

gsrFQ

þ ghybQ

shyb

shybþclocQ

(A.22)

FlocR

εloc¼

rRQ

sr þ rhybR

FlocR

εhybþFlocR

rsrRW

þ rhybR

εhyb

εhybþFlocR

gRW

sr þ ghybW

clocW

shybþclocW

gsrRQ

þ ghybQ

clocQ

shybþclocQ

(A.23)

Only three of these equations are independent of one anotherbecause of the identity

clocW Floc

F

clocQ Floc

R

¼ 1 (A.24)

Therefore, we can use this identity to eliminate one of the four

unknowns chybW , chyb

Q , FhybF , Fhyb

R from three of the equations above

and so we are left with a system of three equations in threeunknowns.

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Glossary

COPkij

and COPkij: partial energy-conversion efficiency of facility k from resource i to

product jE: energyEx: exergyf: primary energy factor

f : average primary energy factor

Gfacility kproduct j : amount of product of j-th type produced by facility k

Gfacility kproduct jresource i

: amount of product of j-th type produced by facility k considered as pro-

duced from the i-th type of resourceP: primary energyPESk

ij

: partial primary energy saving of facility k for the conversion from resource i to

product jQ: heat

Rfacility kresource i: amount of resource of i-th type consumed by facility k

Rfacility kresource iproduct j

: amount of resource of i-th type consumed by facility k considered as used

to produce the j-th type of goodT: temperatureW: electricity

Superscripts

Ex: exergyhyb: hybridloc: local area of interestref: referencesr: single resourcex: generic allocation method

Subscripts

env: environmentF: fossilhyb: hybridsr: single resourceQ: heatR: renewableW: electricity

Greek symbols

a: resource allocation fractionb: product allocation fractiongsr;RWW : fraction of the overall electricity produced in the local area that comes from

the renewable-only electricity facilities

gsr;RQQ : fraction of the overall heat produced in the local area that comes from therenewable-only heat facilities

gfacility kproduct j: dimensionless local market share detained by facility k in the local area

scenario with respect to the production of good j, defined by Eq. (31)εhyb: electric index, defined in Eq. (A.1)

4facility kresource iproduct j

: dimensionless partial resource-i-to-good-j factor for facility k, defined by

Eq. (37)

FhybF and F

hybR : dimensionless parameters defined by Eq. (A.3)

hkij

and hkij: partial energy-conversion efficiency of facility k from resource i to

product j

rfacility kresource i: dimensionless local market share detained by facility k in the local area

scenario with respect to the consumption of resource i, defined by Eq. (32)shyb: fossil to renewable primary energy ratio, defined in Eq. (A.1)

chybW and c

hybQ : dimensionless parameters defined by Eq. (A.2)