Marie-Estelle Binet

Optimal local taxation and French municipal tax distortions

Keywords: E62 -Fiscal Policy|Public Expenditures, Investment, and Finance|Taxation <, H21 -Efficiency|Optimal local taxation, optimality, congestion

This article presents both theoretical and empirical …ndings in the …eld of optimal local taxation i.e neutral in locational decision. This topic should throw light on the question of tax policy and tax reform. We extend Wildasin's (1987) model, including mobile capital. In this way, we include his marginal cost of congestion taxation rule relating to mobile households (proposition 1). This extension provides us with a new rule (proposition 2): the optimal share-out of taxes among household residents and …rms in municipalities. To illustrate these results, we discuss the French municipal tax system properties and we pick out its main distortions.

Introduction

Tax incidence and the question of optimal taxation are important topics for economic analysis. (see [START_REF] Ramsey | A contribution to the theory of taxation[END_REF], Diamond and Mirlees (1971) or [START_REF] Slemrod | Optimal taxation and optimal tax-system[END_REF] concerning main traditional results). More recently, tax incidence has been the subject of many applied studies (see Sundar C. and al (2000) about taxes on capital gains, Castae ner and al (2004) or [START_REF] Thurston | Physician behavioural responses to variation in marginal income tax rates: longitudinal evidence[END_REF] on the subject of personal income tax, [START_REF] Giosa | Do government policies distort relative factor prices: evidence from Colombia[END_REF] concerning factor prices distortions resulting from trade tax policies).

Here, we focus on the valuation of local taxation. Traditionally, in tax incidence theory, territories are treated as closed. By contrast, in local public economics, the openness of the jurisdictions plays a major part. In this article, factors and population freely ‡ow across jurisdictions. Indeed, capital tax e¤ects on investment are neglected. Further, the total supply of capital in the economy is supposed …xed. Finally, we search the local tax structure causing no distortions or neutral in location decisions, i.e. which will be considered as justi…ed by mobile taxpayers. They will not have incentives to migrate.

This article is organized around two sections. The …rst section models how local government tax policy a¤ects the locational choices of mobile factors across jurisdictions, and we ask under what conditions these policies lead to an e¢ cient equilibrium. The model extends Wildasin's analysis (1987) including mobile capital across jurisdictions. In this way, we are able to distinguish between the congestion costs, due to household and production activity. First, we recall the marginal cost of congestion taxation rule, also developed by [START_REF] Wildasin | Theoretical analysis of local public economics[END_REF]. Furthermore, we are also able to discuss a major new topic: the optimal share-out of taxation among household residents and capital owners in municipalities. The second section discusses the French municipal taxes in the light of the above theoretical results.

2 Model of the locational assignment of factors.

The fundamental goal of this section is to understand how local government taxes a¤ect resource allocation. We undertake three steps. The …rst one is to specify the e¢ cient assignment of the two mobile factors to localities.

In the second one, we de…ne the corresponding equilibrium conditions. To …nish, we compare the previous results to establish the conditions under which the locational equilibrium will be e¢ cient. The e¢ ciency depends on the structure of municipal taxes: If taxes are set appropriately, they will generate no distortions and mobile factors will not have incentives to migrate in order to avoid paying taxes.

Locational e¢ ciency

Let us assume there is a …xed number of municipalities M = 2; within which factors (capital and household workers) are mobile: Each locality provides a …xed amount of homogeneous land T i : N stands for the total household 2 F o r P e e r R e v i e w population and K for the total capital supply to be invested in the economy.

We suppose that all household taxpayers are identical (in tastes and endowment), so the government need not be concerned with questions of vertical or horizontal equity. See for example [START_REF] Moreland | The rise or fall of graduated income tax rates[END_REF] upon the question of optimal income tax rates level when wages rates di¤er. Each unit of capital k i and labor n i are employed in only one locality (see ( 2) and (3)). Let F i (n i ; k i ; T i ; g i ) be a well-behaved constant returns to scale production function for the jurisdiction i to produce a single private numeraire good x i and local public services. Equation (1) establishes fullemployment of private production. Let U (x i ; g i ) be the utility function of a household resident in locality i, which depends on its private consumption and on the local public services.

Each jurisdiction provides some public services G i which are consumed by residents and jointly used with capital in the production process. Public goods exhibit congestion when the total amount provided by the local government (G) di¤ers from the level actually consumed at the individual level (g), i.e. when n > 0 and k > 0 : g = G n n k k : The degree of residential congestion is measured by n (respectively k for industrial congestion). To produce G i units of local public goods, we need C i (G i ; n i ; k i ) units of numeraire. Then, considering the mobile factor n i for instance, when @C i @n i = 0; the local public good is pure. By contrast, when the corresponding marginal congestion cost @C i @n i > 0; then the local public good is subject to congestion or impure. The term @C i @n i > 0 (or equally @C i @k i > 0) means that additional local public spending is required if public services'users marginally increase, with a given individual level g.

Several conditions must be satis…ed to reach an e¢ cient allocation of resources. As for a closed economy, markets for goods and factors must be e¢ cient as well as the public goods provision. Furthermore, in our system of open economy, mobile factors and households must also be assigned in an e¢ cient way. All in all, locational e¢ ciency conditions can be derived from the following maximization problem :

Max U 1 (x 1 ; g 1 ) u:c: 8 > > > > < > > > > : U i (x i ; g i ) = U 1 (x 1 ; g 1 ); i = 2::M (1) n P i=1 [F i (n i ; k i ; T i ; g i ) n i x i C i (n i ; k i ; G i )] = 0 (2) N P i n i = 0 (3) K P i k i = 0 3 F o r P e e r R e v i e

w

First-order conditions lead to the following equations ( ; ; and are Lagrange multipliers):

(4)

@U 1 @x 1 P i6 =1 i @U 1 @x 1 = 0 (5) i @U i @x i = 0 i = 2:::N (6) x i @C i @n i + @F i @n i = 0 (7) @C i @k i + @F i @k i = 0 (8) i @U i @g i + @F i @g i @C i @G i @G i @g i = 0 Condition (6) implies that @F i @n i x i @C i @n i = @F j @n j x j @C j
@n j and describes the e¢ cient distribution of population. E¢ ciency is achieved when the marginal productivity of labor, minus the private consumption and the marginal congestion cost that households impose, is equalized everywhere in the economy. The same condition holds where there is mobile capital used jointly with labor and public goods in the production process: the e¢ cient distribution of capital condition (7) means that the additional output, obtained by adding one more capital unit in the local production process net of the corresponding marginal crowding costs, must be equated across jurisdictions. To …nish, combining ( 5) with ( 8) leads to the condition for e¢ cient public expenditure.

We have now depicted the locational e¢ ciency conditions; we still have to discuss the competitive equilibrium conditions for this economy.

Competitive equilibrium

Here, we describe the agents'behavior in our economy. Firms maximize their pro…ts in competitive markets. Furthermore, two budget constraints (one for the local government and one for the representative household resident) are depicted.

Each jurisdiction raises revenue by levying various kinds of local taxes: tax on land rents ( r i ), on capital returns ( k i ); and residential head taxes ( n i ). Then, the balanced-budget constraint for municipality i is:

(9) C(n i ; k i ; G i ) = n i n i + r i r i T i + k i sk i
Indeed, households are equally endowed with ownership of the land and capital, outside of the locality of residence (j 6 = i). Then, in locality j, land rents denoted r j T j and net return of capital sk j are taxed at source. Then, the budget constraint for a household in jurisdiction i can be written:

(10)

x i = w i n i + P j (1 r j )r j T j N + P j s K j N = w i n i +
The individual capital share (respectively land share) is supposed to be

K j N ( T j
N ) and w i represents the gross local wage. Let us denote as the total income stemming from land and …nancial investments.

Pro…t maximization by local competitive …rms leads to the following three conditions :

(11)

w i = @F i @n i (12) r i = @F i @T i (13) (1 + k i )s = @F i @k i
To reach an e¢ cient competitive equilibrium, one must satisfy the two locational e¢ ciency conditions for mobile capital (7) and labor (6). Locational e¢ ciency can be achieved for mobile households in the economy if we combine equations ( 6) with ( 10) and ( 11) to obtain :

(14) @F i @n i x i n i =
Finally, as total income investment is the same in all the jurisdictions, with a special value equal to ; the local head tax on mobile households must equal the corresponding marginal cost of congestion: n i = @C i @n i : Conversely when public services are pure, the optimal tax level on mobile households is then equal to zero. Identically, combining (7) with (13) leads to the second e¢ cient equilibrium condition:

(15) (1 + k i )s @C i @k i =
If s = is an unchanging value in all the jurisdictions, we obtain the e¢ cient capital taxation level: k i s = @C i @k i :

Finally, the conditions necessary for locational e¢ ciency are summarized in the two following propositions: When public goods exhibit congestion, the marginal cost of congestion @C i @n i is an e¢ cient taxation level to internalize congestion externality, that is to say the cost of providing the extra public good to maintain individual public services when faced by an expanded household population. With expenditure held …xed, an increase in the number of public users, necessarily mobile, causes a deterioration of public services, which must be internalized to reach an e¢ cient solution. If local public goods are pure, e¢ ciency does not require the taxation of mobile residents. Whatever the nature of local public goods, land rent taxation is justi…ed, setting r i so as to satisfy (9). Several authors including [START_REF] Arnott | Aggregate land rents, expenditures on public goods and optimal city size[END_REF], and then [START_REF] Yinger | Capitalization and the theory of local public …nance[END_REF] have shown that land rent taxation is e¢ cient because public services increase land value by a capitalization e¤ect.

Proposition 2 The optimal share-out of taxes among mobile household residents and capital owners

The same rule must be applied to mobile capital as long as production jointly employs public services and generates crowding e¤ects. All in all, an e¢ cient local taxation requires one to distinguish and measure exactly the marginal cost of congestion from the two di¤erent mobile users of public goods. In other words, we must share precisely the tax pressure between capital owners and household residents to internalize the cost of congestion induced respectively by each kind of public services user.

All in all, many kinds of taxes could potentially play the role of an e¢ cient tax to internalize congestion, irrespective of the tax base de…nition. The marginal cost of congestion taxation rule only requires consideration of the amount of tax paid. Each household and each owner of one unit of capital must pay the local authority a total amount equal to the cost of providing the extra public good to maintain individual public services when faced by an expanded public services user.

We have now established the e¢ cient local taxation rules, and we now need to discuss the properties of the French municipal taxes. The French public sector has two major hierarchical levels: central government and local (so called 'territorial') authorities. The latter can be divided into three, although interrelated, tiers whose lowest level is the municipality. We concentrate on that …nal jurisdiction here.

French municipal tax distortions

The two main forms of municipal taxation on household residents in France include Taxe d'Habitation (TH) and Taxe Foncière Bâtie (TFB) due from building occupants and landlords respectively. They are proportional to the property and renting values assessed by the land registry. The local business tax called Taxe professionnelle (TP) is paid by …rms based in the municipality. This tax is calculated on the whole value of tangible assets, that is to say the value of capital (including buildings). Lastly, the tax on land property is the Taxe Foncière non Bâtie (TFNB). To discuss the optimality of this tax system, we will …rst discuss proposition 2, to see if the French municipal tax system is consistent with an optimal spreading of taxes among mobile household residents and capital owners. Next, we examine whether the French local taxes match the marginal cost of congestion taxation rule.

Optimal spreading of taxes among mobile household residents and capital owners

Most empirical studies show that local public services exhibit a high degree of congestion (see [START_REF] Borcherding | The demand for the services of non-federal governments[END_REF]). In France, similar conclusions are reached in Guengant, Josselin andRocaboy (1995, 2002). However, to illustrate the application of Proposition 2, we must focus our attention on particular studies which include two di¤erent sources of crowding e¤ects: mobile household residents and capital or …rms'production activity. To our knowledge, this kind of analysis is rare, probably because relevant data to measure capital is not easy to collect. In France, only two contributions can be used even if they are not fully satisfactory.

In [START_REF] Guengant | L'in ‡uence des résidents et des actifs sur la congestion des biens publics locaux : un test sur données françaises[END_REF], production activity congestion is assessed by temporary labor migration in the municipality. Their study uses a set of 799 French municipalities of at least 10,000 inhabitants. They estimate a speci…cation similar to Borcherding and Deacon's formulation. Using a similar methodology, [START_REF] Guengant | Les coûts de la croissance périurbaine : l'exemple de l'agglomération rennaise[END_REF] congestion costs induced by household residents is preponderant with a congestion elasticity equal to 0.8 (respectively 0.2 for …rms activity). All in all, despite the imperfect measure of congestion from production activity, these studies show that tax on mobile household resident should be predominant.

We now focus on the share of local tax receipts for the numerous municipalities of less than 10,000 inhabitants; relevant values are shown in table 1.

Table 1. Tax receipts (per capita) by population size, 1996 share of receipts -700 700-2,000 2,000-5,000 M is the number of municipalities. TP tax receipts for the municipalities of at least 10,000 inhabitants are presented in the table below: Among these local taxes, TP ranks …rst giving the highest revenue ( more than 40% in most of these municipalities). Tax on building occupants (TH), compared to mobile household residents is smaller (between 20 and 25%), which seems to refute the optimal spreading of taxes principle.

French municipal taxes and the marginal congestion cost rule

To assess the main four French municipal taxes in the light of proposition 1, we …rst discuss their tax base. Furthermore, we also propose to work out the optimal tax paid by mobile taxpayers in a set of small French municipalities.

8

F o r P e e r R e v i e w

French municipal tax bases

According to [START_REF] Wildasin | Theoretical analysis of local public economics[END_REF] locational e¢ ciency requires the use of a land rent taxation, paid by immobile landlords. The basis on which TFB and TFNB are currently assessed on building and land values dates from land registry computation of the 1970s. These tax bases no longer re ‡ect today's land and building monetary values. Therefore, they are not proportional to the current rent return and di¤er from the e¢ cient taxation rule.

Until 1999, the business tax TP was set at 18% of total wages, plus the whole value of tangible assets (that is to say building and capital values used in the production process). The 1999 tax reform consisted in removing total wages from the tax base. From then on, only the use of capital was included in TP, excluding production activity congestion assessed by temporary labor migration in the municipality. This tax reform, on account of employment, still seems to move away from the e¢ cient taxation rule which consists in internalizing all congestion costs due to local production.

Optimal taxation of mobile household municipal taxpayers

The optimal taxation level which must be levied on mobile municipal taxpayers can easily be computed using a congestion speci…cation similar to [START_REF] Borcherding | The demand for the services of non-federal governments[END_REF], which includes all sources of congestion :

(16) f (N; A; K) = N n A A K k
where N , A and K measure respectively total household population, working people residing temporarily and capital units invested in the municipality. In this way we identify all public services users. One alternative model of congestion is often used in the literature. As suggested by Edwards (1980) or De [START_REF] Mello | Public …nance, government spending and economic growth: the case of local governments in Brazil[END_REF], the Camaraderie e¤ect means that there may be a population level where sharing congestible goods increases bene…ts to each user until these are eroded by crowding. But these studies can't be used here because the corresponding estimates only include residential crowding e¤ects.

Consequently, the public spending level actually consumed at the individual level can be written:

= N n A A K k G
Thus, we can deduce the e¢ cient taxation level upon mobile residents as the cost of providing the extra public good to maintain individual public services in the face of an expanded user population :

@G @N = n N G
This e¢ cient level can be compared to the TH (taxe d'habitation) actually paid by building occupants in a sample of 32 small French municipalities belonging to the same suburban area in 1996. Thus this constitutes a quite relevant sample of municipalities of less than 10,000 inhabitants. Data have been taken from AUDIAR (Greater Rennes Inter-Municipality Development Agency) and o¤er a detailed measurement of variables.

Results appear in the table given in appendix. The …rst columns give the e¢ cient tax levels with di¤erent …xed values for the congestion parameter n : G is measured according to the total expenditure in each jurisdiction used by the total household population settled in the municipality. Column TH gives the average amount paid in each municipality. However, in France more than 20 % of building occupants are tax-exempt and nearly 30% enjoy partial tax relief. In this case, CTH measures the corrected average amount paid by real taxpayers. For most municipalities, CTH is the e¢ cient tax level with n = 0:2: When the residential crowding parameter is greater than 0.3, CTH is under the optimal level. Overall, despite the imperfect measure of congestion in the empirical literature, these preliminary results seem to demonstrate that tax on mobile household residents should be higher in France in order to be e¢ cient.

Concluding remarks

This article presents both theoretical and empirical …ndings in the …eld of optimal local taxation. The model extends Wildasin's analysis (1987), including mobile capital across jurisdictions. In this way, we recall his marginal cost of congestion taxation rule. Furthermore, we are also able to discuss a major new topic: the optimal share-out of taxation among household residents and 10

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 5 The marginal cost of congestion taxation rule,[START_REF] Wildasin | Theoretical analysis of local public economics[END_REF] :

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		gets comparable results:
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Table 2 :

 2 TP tax receipts by population size, 1996

		10-20,000 20-50,000 50-100,000 100-300,000 +300,000 whole
	TP	47,9%	47,5%	50,3%	44,9%	37,2%	46,8%
	N	470	306	70	33	4	883

  Page 10 of 13 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK capital owners in municipalities. Having developed this theory, we are able to illustrate many of the distortions in the French municipal tax system.

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J Alberto Dopazo

José Fernández-Seara

Jaime Sieres

Francisco J Uhía

THEORETICAL ANALYSIS OF A CO 2 -NH 3 CASCADE REFRIGERATION SYSTEM FOR COOLING APPLICATIONS AT LOW-TEMPERATURES

Keywords: Cascade refrigeration, Compression system, CO 2 -NH 3, Natural refrigerants, Optimization

As a result of environmental problems related to global warming and depletion of the ozone layer caused by the use of synthetic refrigerants (CFC's, HCFC's and HFC's) experienced over the last decades, the return to the use of natural substances for refrigeration purposes, appears to be the best long-term alternative. In this paper a cascade refrigeration system with CO 2 and NH 3 as working fluids in the low and high temperature stages respectively, has been analyzed. Results of COP and exergetic efficiency versus operating and design parameters have been obtained. In addition, an optimization study based on the optimum CO 2 condensing temperature has been done. Results show that following both method's exergy analysis and energy optimization, an optimum value of condensing CO 2 temperature is obtained. The compressor isentropic efficiency influence on the optimum system COP has been demonstrated. A methodology to obtain relevant diagrams and correlations to serve as a guideline for design and optimization of this type of systems has been developed and it is presented in the paper.

experienced over the last decades, the return to the use of natural substances for refrigeration purposes appears to be sound practice. It must be a better solution to use naturally existing and environmentally harmless substances as alternatives refrigerants in refrigeration systems.

Amongst the natural refrigerants, Lorentzen and Petterson [START_REF] Lorentzen | A new efficient and environmentally benign system for car airconditioning[END_REF] suggested the use of carbon dioxide (CO 2 ) and seems to be the most promising one especially as the natural refrigerant [START_REF] Lorentzen | A new efficient and environmentally benign system for car airconditioning[END_REF][START_REF] Neksa | CO 2 -heat pump water heater: characteristics, system design and experimental results[END_REF][START_REF] Neksa | CO 2 -heat pump systems[END_REF][START_REF] Hwang | Experimental investigation of the CO 2 refrigeration cycle[END_REF][START_REF] Hrnjak | Transcritical CO 2 heat pump for residential application[END_REF][START_REF] Hwang | CO 2 transcritical cycles for high temperature application[END_REF]. The key advantages of CO 2 include the fact that is not explosive, non-toxic, easily available, environmental friendly and has excellent thermo-physical properties.

On the other hand, researches in Norway in 1993 showed that the refrigerant leakages coming from the commercial sector were 30% of the annual total [START_REF]Sintef Vedleggsrapport til STF11 A93051 Brukeroversikt-Kuldemedier I Norge[END_REF]. In this research, the use of a cascade system using CO 2 in the low temperature stage and NH 3 in the high temperature stage turned out to be an excellent alternative for cooling applications at very low temperatures [8][9][START_REF] Lobregt | CO 2 systems are sweeping through the Netherlands[END_REF]. Researches from Eggen and Aflekt [START_REF] Eggen | Commercial refrigeration with ammonia and CO 2 as working fluids[END_REF], Pearson and Cable [START_REF] Pearson | A distribution warehouse with CO 2 as refrigerant[END_REF] and Van Riessen [START_REF] Van Riessen | NH 3 /CO 2 Supermarket refrigeration system with CO 2 in the cooling and freezing section[END_REF] show practical examples of the use of a cascade system of CO 2 /NH 3 for cooling in supermarkets. Eggen and Aflekt [START_REF] Eggen | Commercial refrigeration with ammonia and CO 2 as working fluids[END_REF] developed research based on a prototype of a cooling system built in Norway. Pearson and Cable [START_REF] Pearson | A distribution warehouse with CO 2 as refrigerant[END_REF] showed data from a cooling system used in a Scottish supermarket line, (ASDA), and Van Riessen [START_REF] Van Riessen | NH 3 /CO 2 Supermarket refrigeration system with CO 2 in the cooling and freezing section[END_REF] carried out technical energy and economic research of a cooling system used in a Dutch supermarket.

In the same way, different researches about the performance of different cooling systems involving CO 2 have been carried out together with its reuse as a refrigerant fluid.

Lorentzen and Petterson [START_REF] Lorentzen | A new efficient and environmentally benign system for car airconditioning[END_REF] evaluated the possibility of the use of a heat exchanger in a CO 2 transcritical system. Hwang et al. [START_REF] Hwang | CO 2 transcritical cycles for high temperature application[END_REF] showed experimental results and simulation research including expansors and double stage cycles. Groll et al. [START_REF] Groll | Effect of pressure ratios across compressors on the performance of the transcritical CO 2 cycle with two-stage compression and intercooling[END_REF] carried out a numerical analysis of a double stage cycle changing the compression ratio of each compression stage.

Bhattacharyya et al. [START_REF] Bhattacharyya | Optimization of a CO 2 -C 3 H 8 cascade system for refrigeration and heating[END_REF] showed an optimization research of the CO 2 /C 3 H 8 cascade system for cooling and heating. Kim et al. [START_REF] Kim | Fundamental process and system design issues in CO 2 vapor compression systems[END_REF] summed up the improvements in the performance of systems based in CO 2 and their applications. They provided a critical review of literature, and discussed important trends and characteristics in the development of CO 2 technology in refrigeration applications.

Recently, studies including a wide range of possibilities in the use of CO 2 as refrigerant have been published. Deng et al. [START_REF] Deng | Particular characteristics of transcritical CO 2 refrigeration cycle with an ejector[END_REF] described a theoretical analysis of a transcritical CO 2 ejector expansion refrigeration cycle, using an ejector as the main expansion device instead of an expansion valve. Youngming et al. [START_REF] Youngming | Construction and testing of a wetcompression absorption carbon dioxide refrigeration system for vehicle air conditioner[END_REF] constructed and tested a wet-compression absorption carbon dioxide refrigeration system. Fernández-Seara et al. [START_REF] Fernández-Seara | Compression-absorption cascade refrigeration system[END_REF] analyzed a compression-absorption cascade refrigeration system considering CO 2 and NH 3 as refrigerants in the compression stage and the pair NH 3 -H 2 O in the absorption stage and evaluated the possibilities of powering the cascade refrigeration system by means of a cogeneration system.

Lee et al. [START_REF] Lee | Thermodynamic analysis of optimal condensing temperature of cascade-condenser in CO 2 /NH 3 cascade refrigeration systems[END_REF] carried out a thermodynamic analysis of optimal condensing temperature of cascade-condenser in CO 2 /NH 3 cascade refrigeration systems. In this work, effects of isentropic compressors efficiencies and practical limit of the compressors discharge temperatures were not taking into account and the exergetic efficiency was not evaluated.

The scope of the present research is focused on the analysis of the parameters of design and operation of a CO 2 /NH 3 cascade cooling system and their influence over the system's COP and exergetic efficiency. The statistical significance of each of the parameters evaluated has been analysed. Moreover, optimization research of these parameters has been included in order to show highest COP. Finally, a discussion about the effect of the compressors' isentropic efficiency on the optimum system COP is presented.

SYSTEM DESCRIPTION

A schematic diagram of the cascade system is shown in figure 1. The cascade refrigeration system is constituted by two single stage systems connected by a heat exchanger (cascade heat exchanger). The low temperature system with CO 2 as refrigerant is used for cooling. The high temperature system with NH 3 as refrigerant is used to condensate the CO 2 of the low temperature system.

In the evaporator, the CO 2 at the evaporating temperature absorbs the cooling duty 2 CO Evap Q • from the cooling space (at T F temperature), then is compressed in the CO 2 compressor and condensed in the cascade heat exchanger at a condensing temperature of T Cond CO2 , and then sent to the expansor from which the evaporator is supplied.

In the condenser, the heat flow

MATHEMATICAL MODEL

Model

To calculate the compressors' powers, heat transfer rates and energetic and exergetic efficiencies, each cascade system component is considered as a control volume at stationary flow. Mass balance, energy balance and exergy balance have been performed.

The following assumptions are made in the analysis: e. The difference between the refrigerated space temperature (T F ) and the evaporation temperature (T Evap CO2 ) is constant and equal to 5 ºC.

Taking into account the assumptions previously made, the mass, energy and exergy balances are given by Eqs. 1, 2 and 3, respectively.

Mass balance

• • = out in m m ( 1
)
Energy balance

• • • • ⋅ - ⋅ = - in out h m h m W Q (2)
Exergy balance

• • • • • ⋅ - ⋅ + - ⋅ - = out in out j j 0 Lost m m W Q T T 1 X ψ ψ (3)
In Table 1 specifics equations for each system's components are summarized.

Compressor isentropic efficiency was calculated using the Eqs. 4 and 5 for each case.

( ) (

)

1 2 1 2 2 h h h h s CO Comp s - - = η (4) ( ) ( ) 5 6 5 6 3 h h h h s NH Comp s - - = η (5)
Overall compressor's efficiency has been determined with the Eq. 6.

Elec Mec s η η η η ⋅ ⋅ = Comp (6)
The lowest temperature difference (DT) in the cascade heat exchanger is formulated by Eq. 7. The system's COP has been calculated by the Eq. 8.

NH3 2 2 Comp CO Comp CO Evap W W Q COP • • • + = (8)
The system's exergetic efficiency is given by Eq. 9. respectively [START_REF] Yumrutas | Exergy analysis of vapour compression refrigeration systems, Exergy[END_REF].

The system's reversible power input is given by Eq. 10.

- ⋅ = • • 1 2 0 2 Rev CO Evap CO Evap T T Q W (10)

Parametric study

The equations of the mathematical model reveal that both the system's COP and it's exergetic efficiency can be expressed as a function of six design/operating parameters, as shown in Eq. 11.

( ) ( )

NH3 CO2 3 2 , , , , , , 2 Comp Comp CO Cond NH Cond CO Evap II DT T T T f COP η η η = (11)
To determine the influence of each design/operating parameter on the system's COP and it's exergetic efficiency a parametric study was done. The ranges of values for each parameter comprise the specific intervals of interest. T Evap CO2 , T Cond NH3 and T Cond CO2 were varied from -55 to -30 ºC, from 25 to 50 ºC and from -25 to 5 ºC, respectively, at intervals of 5 ºC. The temperature difference in the cascade heat exchanger was varied from 3 to 6 ºC. Moreover, a prototype cascade refrigeration system (in construction at this moment in our laboratory) was taken into account as reference. This prototype has been supplied with a semihermetic compressor to compress the CO The isentropic efficiency of each compressor is considered equal to the volumetric efficiency and it is estimated according to Eq. 12.

RC 04 . 0 1 s ⋅ - = η ( 12
)
To solve each case derived from the parametric study specific software developed by Sieres and Fernández-Seara [START_REF] Sieres | Simulation of compression refrigeration systems[END_REF] has been employed. The thermodynamic properties of CO 2 and NH 3 were calculated from REFPROP [START_REF] Lemmon | Referent Fluid Thermodynamic and Transport Properties (REFPROP) V 7.0[END_REF].

Model validation

Reference [START_REF] Sawalha | CO 2 in Supermarket Refrigeration, CO 2 Project Report Phase I[END_REF] reports experimental data from a NH 3 /CO 2 cascade system prototype for refrigeration in supermarkets. This prototype includes a CO

RESULTS AND DISCUSSION

Exergy losses

To evaluate the exergy losses of each system's components and the exergy loss rate of the whole system a parametric study was applied at prototype specified design conditions: the system's cooling capacity is 9 kW, the CO 2 evaporating temperature (T Evap CO2 ) -50 ºC, the NH 3 condensing temperature (T Cond NH3 ) 40 ºC and the cascade heat exchanger temperature difference (DT) 5 ºC. Figure 3 shows the exergy loss rate of each system's components (left axe) and the exergy loss rate of the whole system (right axe). It can be clearly appreciated that the highest exergy loss rate occurs at the lowest CO 2 condensing temperature, and decreasing trends are present in all the NH 3 system's components. In the CO 2 compressor case, the CO 2 throttle and the cascade heat exchanger present an increasing tendency with T Cond CO2

increments. The exergy loss rate in the CO 2 evaporator tends to be constant. In some of the system's components the exergy loss rate increases whilst in others the exergy loss rate decreases with T Cond CO2 increments. This behaviour has been observed after evaluating the whole system whereby there existed initially decreasing trends up to a minimum exergy loss rate at T Cond CO2 -5 ºC approximately, and then increasing trends appear. Therefore an optimum value of T Cond CO2 is found.

System COP and exergetic efficiency trends

To evaluate the influence of the operating parameters on both the system's COP and exergetic efficiency, a statistical procedure has been used to analyse the parametric study results obtained considering the ranges of values indicated in section 3.2 This statistical procedure is the Anova Multifactorial [START_REF] Mendenhall | Statistics for Engineering and the Sciences[END_REF]. In Table 2 the results of the "P-Value" statistical factor can be observed. This factor tests the statistical significance of each of the operating parameters evaluated. Since all P-values are less than 0.05, all the evaluated parameters have a statistically significant effect at the 95% confidence level. Therefore all the parameters considered in Eq. ( 11) should be included in the analysis and none can be discarded. shown in figure 4 reveal that the T Evap CO2 has a greater effect than T Cond NH3 on the system COP; meanwhile the influence of DT is small. In figure 4 the exergetic efficiency trends are also depicted. There is not a clear overall tendency in the case of T Evap CO2 . (a). In this case there is an initial increasing tendency up to -40 ºC then the change tends to decrease for average values, while maximum values increase and minimum values decrease continuously. However, the exergetic efficiencies show a decreasing tendency with increases in T Cond NH3 (b) and DT (c), around 45% and 9%, respectively. The influence of DT on the U•A product in the cascade heat exchanger is also included in figure 4c. The results are given as increments in percentage with respect to the case of DT=5 ºC. If a specific type of heat exchanger is selected, these results will evaluate the increase in size, and consequently in cost, of the cascade heat exchanger.

Figure 5 shows both COP and exergetic efficiency behaviour versus T Cond CO2

variations. In both cases increasing trends are present initially up to -5 ºC, after which the change tends to decrease. These trends occur for the average, maximum and minimum values.

Thus, these results indicate that there is a T Cond CO2 optimal value. The COP of the NH 3 stage increases with T Cond CO2 whereas the COP of the CO 2 stage decreases. Consequently, a maximum COP exists that corresponds to the T Cond CO2 optimal value. This optimal value also corresponds to the minimum found in section 4.1 for the exergy loss rate of the whole system, as can be seen in figure 3.

Optimization

From the results obtained a clear viewpoint of the method to obtain a maximum COP value is proven.

The CO 2 evaporating temperature has to be as high as possible in order to obtain the highest COP. This temperature depends on the environmental temperature (T F ) that has to be cooled (system operating parameter) and the difference between temperatures established as a design parameter in the evaporator, between the refrigerant fluid and the environmentally cooled temperature. It is therefore clear that for a specific application, optimization has to be the maximum reduction of this temperature difference.

On the other hand, the NH 3 condensing temperature has to be as low as possible to increase the COP to the maximum. This temperature is a function of the condensing environment and the difference of the temperatures between the condensing environment (T 0 )

and the NH 3 temperature established as a design parameter of the condenser. If the environmental condensing temperature can not be modified because the system is subjected to a particular situation, the only remedy is to decrease the temperature differences in order to obtain the higher COP increase.

As for temperature differences in the cascade exchanger, DT, results always show that for a lower DT a higher COP is obtained.

Differing from the previous parameter, the CO

- ⋅ - ⋅ + - ⋅ + - = (14)
The proposed correlations (Eqs. 13 and 14) explain 99.45% and 99.21% of the variability (R 2 statistic) in T Cond CO2 Opt and COP, respectively. The mean absolute errors between the raw data and the predictions are 0.3 and 0.025 for the T Cond CO2 Opt and COP, respectively.

Eqs. 13 and 14 contribute important information for a cascade system such as the one studied. It is important to emphasize that Eq. 13 gives the optimum value of the CO 2 condensing temperature for specific values of the CO 2 evaporating temperature and the NH 3 condensing temperature which can be replaced (previously defining the values of the temperature differences in the evaporator and condenser) by the cooled environment temperature and the condensed environment temperature, respectively. On the other hand, the maximum possible COP that the system can reach, operating per a specific situation, is determined by Eq. ( 14). This maximum value of COP obviously corresponds to a CO 2 optimum condensing temperature value. Using Eqs. 13 and 14, the lines corresponding to the iso-values of the CO 2 optimum condensing temperature and the system's maximum COP have been drawn, for DT = 5 ºC, figures 6 and 7, respectively. A shadowed area is included in both figures, which corresponds to the operating system's conditions where the CO 2 or the NH 3 compressors discharge temperature is higher than 120 ºC. If the discharge temperature is above this practical limit, usability and stability of the associated lubricant oil may become a severe problem after a long-term operation [START_REF] Rozhentsev | Some design features of a CO 2 air conditioner[END_REF].

From figure 6, it is obvious that the CO 2 optimum condensing temperature increases with the CO 2 evaporating temperature and the NH 3 condensing temperature increases. In figure 7 it can be observed that the system's maximum COP increases with the CO 2 evaporating temperature increase, and the NH 3 condensing temperature decrease.

Effect of the compressor isentropic efficiency on system COP

For all the cases previously studied, the compressor isentropic efficiency was considered as a function of the compression ratio of each one using Eq. 12. Different empirical correlations to calculate the compressor's isentropic efficiency can be found in existing literature. Eqs. 15, 16 and 17; Eqs. 18 and 20; and Eq. 19 were obtained from [START_REF] Chen | The optimum high pressure for CO 2 transcritical refrigeration systems with internal heat exchangers[END_REF], [START_REF] Lee | Thermodynamic analysis of optimal condensing temperature of cascade-condenser in CO 2 /NH 3 cascade refrigeration systems[END_REF] and [START_REF] Tarique | Performance and economic study of the combined absorption/compression heat pump[END_REF], respectively. Moreover, if a specific installation or prototype is considered the isentropic efficiency equation for each compressor will be determined from the manufacturer data.

For an NH 3 compressor m.

2 s RC 0013378 . 0 RC 0366432 . 0 976695 . 0 ⋅ + ⋅ - = η (19)
n.

2 s RC 00097 . 0 RC 01026 . 0 83955 . 0 ⋅ - ⋅ - = η (20)
In figure 8 the system's COP evolution versus T Cond CO2 depicts the use of different correlation combinations to calculate the isentropic efficiency of each compressor. Again, a shadowed area is included to represent the operating system's conditions when the CO 2 or the NH 3 compressors discharge temperature is higher than 120 ºC. It can be appreciated that in each case, a different value of CO 2 optimum condensing temperature exists. These values are included in a range from -25 ºC (a-m correlations combination case) up to -5 ºC (initial correlation considered -Eq. 12-in both compressors). Results in figure 8 clearly show the significant influence of the compressors isentropic efficiency on the optimal CO 2 condensing temperature. Therefore, it is concluded that these efficiencies should be evaluated as accurately as possible and taken into account in the analysis of a cascade refrigeration system.

CONCLUSIONS

In this paper, the analysis of the parameters of design and operation of a CO 2 /NH 3 cascade cooling system and their influence over the system's COP and exergetic efficiency is reported. The analysis was carried out based on a general mathematical model that was validated using experimental data found in the literature. The system's COP and it's exergetic efficiency can be expressed as a function of six design/operating parameters. To determine the influence of each parameter a parametric study was done considering the specific intervals of interest. A statistical procedure has been used to analyse the parametric study results obtained.

The analysis reveals that all the evaluated parameters have a statistically significant effect and should be taken into account.

The results show that the COP increases 70% when the T Evap CO2 varies from -55 ºC to -30 ºC. As T Evap CO2 increases, the influence of the other parameters on the COP also increases. The COP diminishes 45% when the T Cond NH3 increases from 25 ºC to 50 ºC. The system COP disminishes 9% when DT varies from 3 ºC to 6 ºC. The exergetic efficiency decreases around 45% and 9% with the increases indicated above in T Cond NH3 and DT, respectively. In the case of T Evap CO2 there is an initial increasing tendency up to -40 ºC then the change tends to decrease. For a specific system and operating conditions, results show that following both, exergy analysis and energy optimization methods, an optimum value of condensing CO 2 temperature is obtained. The CO 2 optimum condensing temperature value increases when CO 2 evaporating and/or NH 3 condensing temperatures increase. Diagrams and correlations to demonstrate the evaluations of the maximum COP and CO 2 optimum condensing temperatures for the specific conditions considered in the analysis have been developed. On the other hand, it was established that the isentropic compressors efficiency influences the maximum COP and the CO 2 optimum condensing temperature calculations. Different empirical correlations to calculate the compressor's isentropic efficiency that can be found in existing literature were used to demonstrate the influence of these parameters on the maximum COP and CO 2 optimum condensing temperature. Based on the results shown in the paper, it is concluded that, if a specific installation or prototype is considered, the isentropic efficiency for each compressor in the cascade system should be determined as accurately as possible from the manufacturer or experimental data in order to obtain reliable values for the optimum CO 2 condensing temperature and maximum COP: Table 1. Balance equations for each system component.

Tables

Component

Mass Energy Exergy CO 2 -Compresor

1 2 • • = m m ( ) CO2 1 2 1 2 Comp s CO Comp h h m W η - ⋅ = • • ( ) 1 2 1 2 2 ψ ψ - ⋅ - = • • • m W X CO Comp Comp CO Lost NH 3 -Compresor 5 6 • • = m m ( ) NH3 5 6 5 3 Comp s NH Comp h h m W η - ⋅ = • • ( ) 5 6 5 NH3 3 ψ ψ - ⋅ - = • • • m W X Comp NH Comp Lost CO 2 Exp. Device 3 4 • • = m m 3 4 h h = ( ) 4 3 1 2 ψ ψ - ⋅ = • • m X Exp CO Lost NH 3 Exp. Device 7 8 • • = m m 7 8 h h = ( ) 8 7 5 3 ψ ψ - ⋅ = • • m X Exp NH Lost Evaporator (CO 2 ) 4 1 • • = m m ( ) 4 1 1 2 h h m Q CO Evap - ⋅ = • • ( ) 1 4 2 CO Evap F 0 2 CO Evap Lost m Q T T 1 X ψ ψ - ⋅ + ⋅ - = • • • Condenser (NH 3 ) 6 7 • • = m m ( ) 6 7 5 NH3 h h m Q Cond - ⋅ = • • ( ) 7

  of environmental problems related to global warming and depletion of the ozone layer caused by the use of synthetic refrigerants (CFC's, HCFC's and HFC's)

  the NH 3 at condensing temperature of T Cond NH3 to condensing medium (at T 0 temperature). The NH 3 is expanded, then evaporated at an evaporating temperature of T Evap NH3 in the cascade heat exchanger, and then compressed in the NH 3 compressor and discharged into the condenser.

Figure 2

 2 Figure 2 shows the process evolutions for both the CO 2 and NH 3 cycles in a logP-h diagram. Saturation lines are included.

  a. Refrigerants at the cascade heat exchanger outlets, condenser outlet and evaporator outlet, are saturated. b. Pressure losses in connecting pipes and heat exchangers have been neglected. c. Cascade heat exchanger and pipes are perfectly isolated. d. The dead state (ambient) is T 0 = 25 ºC and P 0 = 1 atm.

W

  are the system's reversible power input and actual power input,

  2 and an open type compressor was selected to compress the NH 3 . Due to the compressors selection, were considered 0.9 and 0.95 the values of mechanical and electrical efficiency )

  2 circuit at medium temperature, thus, in this case the COPs of the low and high stages are compared separately. The experimental data for the CO 2 stage are: T Evap CO2 = -26 ºC with 7 ºC superheat at the evaporator outlet, T Cond CO2 = -9 ºC assuming no subcooling, s =0.21 due to a deficient compressor operation, Mec =0.93 and Elec =0.8. Experimental data for the NH 3 stage are: T Evap NH3 = -11 ºC, T Cond NH3 = 32 ºC, no superheating and subcooling are assumed, s = 0.76, Mec = 0.93 and Elec = 0.8. The cooling duty at the medium temperature level is considered at this stage. The model results provide a relative error with respect to the experimental COPs of 1.8% and 7.1% for the CO 2 and NH 3 stages, respectively. The results of the model validation with the experimental data are considered satisfactory.

Figures 4 and 5

 5 Figures 4 and 5 depict the system's COP and exergetic efficiency average values and the ranges of values resulting from the parametric studies for each one of the evaluated parameters. Therefore, these figures show the maximum and minimum values of the COP and exergetic efficiency for the entire region analysed, as well as their average value for each parameter when varying all others simultaneously. Thus, the results and conclusions obtained from this analysis have applicability over the entire region studied.

For

Figure captions Figure 1 .

 captions1 Figure captions Figure 1. Schematic diagram of the CO 2 /NH 3 cascade refrigeration system.

Figure 2 .

 2 Figure 2. LogP-h diagram of the CO 2 and the NH 3 thermodynamic cycles.

Figure 3 .

 3 Figure 3. Exergy lost rates of each system's components (left axe) and of the whole (right axe) as function of T Cond CO2 .

Figure 4 .

 4 Figure 4. Average value and range of values of the system's COP and exergetic efficiency as function of (a) T Evap CO2 , (b) T Cond NH3 and (c) DT. Figure (c) also includes the relative U•A increments in the cascade heat exchanger as function of DT.

Figure 5 .

 5 Figure 5. Average value and range of values of the system's COP and exergetic efficiency as function of T Cond CO2 .

Figure 6 .

 6 Figure 6. Iso-T Cond CO2 Opt contours plotted on T Evap CO2 -T Cond NH3 plane.

Figure 7 .

 7 Figure 7. Iso-COP Max contours plotted on T Evap CO2 -T Cond NH3 plane.

Figure 8 .

 8 Figure 8. System COP vs. T Cond CO2 using different combinations of correlations to calculate the compressors' isentropic efficiency.

  Figure 2

Figure 7 s

 7 Figure 7

Table 2 .

 2 ANOVA Results of COP and Exergetic Efficiency.

		Sum of Squares	Df		Mean Square	F-Ratio	P-Value
	Source
		COP	η II	COP	η II	COP	η II	COP	η II	COP	η II
	A: T Evap CO2	55.404 0.06310 5	5	11.0808 0.01262 824.79	23.41	0.000 0.000
	B: T Cond NH3	71.291 5.67083 5	5	14.2583 1.13417 1061.30 2103.82 0.000 0.000
	C: DT	2.120	0.16968 3	3	0.7066	0.0566	52.60	104.92	0.000 0.000
	D: T Cond CO2 26.939 2.03396 6	6	4.48981 0.3390	334.19	628.82	0.000 0.000
	RESIDUAL 13.274 0.53263 988	988 0.0134	0.00054
	TOTAL				100
		169.03 8.47019	100
	(CORR.)


D Dobersek

D Goricanec

Optimisation of tree path pipe network with nonlinear optimisation method

Keywords: District heating, Pipe network, Optimisation, Non-linear programming, Simplex method

In this article the optimisation of pipe network with hot water is presented. The mathematical model, consisting of the non -linear objective function and system of non -linear equations for the hydraulics limitations is developed. On its basis the computer program for determination optimal tree path with the use of simplex method was solved. For economic estimation the capitalized value method, which consider all costs of investment and operation was used. The results are presented for real case study network with 24 nodes and 33 pipe sectors.

Introduction

Nowadays efforts connected to energy savings demand the search for new technical scientific expertise in the field of heating techniques [1 -4]. The focus of research is on better and more efficient use of primary energy [5].

If we limit ourselves to district heating energy systems, we may state that these systems ensure savings in the process of use of primary energy and are acceptable from the ecological point of view. The whole system is represented as a non-linear target function by non-linear equations of hydraulic limitations and by minimizing the non-linear function the optimal design and dimensions of the pipe network are defined [START_REF] Goricanec | Dimensioning and design of tree path transport pipe networks[END_REF][START_REF] Dobersek | Calibration of pipe networks for district heating using the nonlinear optimisation method[END_REF].

One method for modelling pipe networks is by linear programming, consisting of optimisation with limitations [START_REF] Saul | Linear Programming Methods and Aplications[END_REF], meaning the search for the best possible, optimal solution of the stated problem within given conditions.

Nowadays two methods for solving linear programmes are used, both of them interactive, searching for a gradually better solution, until the optimum value is achieved.

Mostly, simplex methods focused on searching for permissible solutions within the monotonous defined extreme point of the convex polyhedron of possible solutions and a defined base of the vector space, are used.

District heating systems

District heating systems are intended for the distribution of heat energy by a fluid from a heating source to different users [START_REF] Levenspiel | Engineering Flow and Heat Exchange[END_REF][START_REF] Garbai | The Economic Optimum Of The Setting Up Of Loop District Heating Networks[END_REF][START_REF] Krope | Zmanjšanje tla nih izgub v vro evodnih cevnih mrežah = The reduction of friction losses in district-heating pipelines[END_REF]. The pressure losses of the system are defined by the non-linear Darcy-Weisbach equation [START_REF] Roberson | Engineering fluid mechanics[END_REF]:

2 ij v ij 4 2 ij v 0.81 q K d L d q p j i p ⋅ = + ⋅ ⋅ ⋅ ⋅ = - ρ (
≥ = ≥ ≤ ⋅ + + ⋅ + ⋅ = ≥ ≤ ⋅ + + ⋅ + ⋅ = ≥ ≤ ⋅ + + ⋅ + ⋅ ⋅ + + ⋅ + ⋅ x x x x x x x x x x x x x x x (2)
The mathematical model ( 2) is soluble with a simplex algorithm [START_REF] Sierksma | Linear and integer programming: Theory and practice[END_REF].

The conditions in the form of non-equations are transferred into equations by introducing additional variables.

≥ ⋅ + + ⋅ + ⋅ x x x (3)
The possible surplus is subtracted and the non-equation becomes an equation:

1 1 n n 1n 1 b a 11 a = - ⋅ + + ⋅ + x x x ( 4
)
The additional variable (x n+1 ) was added and we now have to consider this with conditions, but it has no influence on the objective function due to the fact that its coefficient (c n+1 ) is zero.

This modified linear programme is complemented with artificial variables (x n+m+1 ) to find the primary interior point where the simplex calculations are to begin.

1 1 m n 1 n n 1n 1 11 b a a = + - ⋅ + + ⋅ + + + x x x x ( 5
)
We complement the objective function with a new variable and ascribe to it the coefficient (c n+m+1 ):

1 m n 1 m n m n 1 n 1 n n n 1 1 c c c c c m n c + + + + + + + ⋅ + ⋅ + + + ⋅ + ⋅ + + ⋅ = + x x x x x (6)
The artificial variables serve for determination of the base allowed solution. In the last step the objective function is transferred from the minimum type to the maximum type:

= ⋅ - = - n 1 j j j c c x ( 7
)
Due to the fact that the first result is not necessarily optimal, we execute the optimality test. If the solution is not optimal a new base is required.

Optimisation of pipe network

For an effective and accurate description we handle the pipe network as a linear, directed graph, where the fluid entrance points are named as supplied nodes, fluid exit points are user nodes and the nodes where no external inflow or outflow is present as fictive nodes [START_REF] Garbai | Optimizing Problems Of Loop District Heating Networks[END_REF][START_REF] Garbai | Flow In Energy Pipe Network Systems Muszaki Könyvkiadó[END_REF].

If we define the number of pipes as N c and N v as the number of nodes for the tree pipe network then the number of nodes is always less than the number of pipes for the structure with loops.

The fluid flow in the net has the direction from the node with higher pressure to the node with lower pressure and is from i to j positive and in the other direction negative. With that we have to consider the first Kirchoff`s law which says that the sum of all flows into a node equals the sum of all outflows.

The optimal path with minimal transport costs can be defined by two methods of linear programming, the transport method and the simplex method.

Garbai and Krope present a graph-theoretical method to set up a network with optimum operational costs. Their method is based on the Bellmann optimum principle and the application of discreet dynamic programming [START_REF] Bellmann | Dynamic Programming[END_REF].

The mathematical model for defining the optimal tree path consists of the objective function of capitalised costs (  )

C(q v ) that is
c v vij v c N j vi vij v N i vi v N j N i q N i q q q q ,..., 3 , 2 , 1 ..., 3 , 2 , 1 0 ,..., 3 , 2 , 1 0 0 1 2 1 = = ≥ = = ± ± = - = = (9)
Figure 3 shows the nonlinear relationship between capitalised costs and volume flow. ( )

i C i L B B CC n 0 0 0 0 1 1 + - + - + = ( 10
)
And the equation ( 10) became equation ( 11):

( ) i C i L B L CC n 0 0 0 0 1 1 + + - - + = - ( 11
)
Regarding capitalised costs the annuities are defined, based on a discount rate and composed of the sum of all and revenues transposed to equal yearly shares.

Within defining the annual costs they start from the start of the pipe network operation and end with the closure of the pipe network operation and are presented at the end of each year in equal portions. Between the annuity costs and capitalised costs the following dependence is valid:

i CC AC ⋅ = ( 12
)
Defining the fluid transport costs over the pipe network we have to consider that the initial investment costs are very high. In the first step we define the costs of insulated pipes that are a function of the diameter with the following equation:

2 1 d C d B A C ⋅ + ⋅ + = ( 13
)
The conversion on standard diameters is done with the use of economic method and is presented in [START_REF] Pristovnik | Two-phase flow in pipe line systems. Heating and airconditioning of buildings[END_REF].

The investment value for pumps depends on the utilisation rate and the pump price calculated per watt of power, the flow volume and the pressure drop that arises with the increase in pump capacity.

p q C P C C ∆ ⋅ ⋅ = ⋅ = v p p 2 (14)
The pumping costs depend on the electrical energy price, the pump power and the operational lifetime of the pipeline.

t P C C ⋅ ⋅ = e 3 ( 15
)
The construction costs depend on pipe diameter in defined pipe sectors and on the construction site environment of the pipe network.

2 4 d F d E D C ⋅ + ⋅ + = ( 16
) 4 3 2 1 C C C C C + + + = ( 17
)
Figure 4 shows the relationship between the pipe costs, pump and pumping costs and costs for constructions.

The Optimisation of the District Heating System

The simplex method and the capitalised costs method were used for the optimisation of the district heating system (Figure 6).

Results and discussion

By using the developed computer programme, which enables optimization of the pipe network using the simplex method two optimal branch trees was defined.

The first optimal branch tree, Figure 7 branch tree path are defined by the cost capitalisation method and are presented for each pipe in table 6. The second optimal branch tree, Figure 8 and tables 7, 8, 9 was received by Version 2. This is for the case when at passage on standard diameters we consider pipe costs while in objective function we consider only C 1 , C 2 , C 3 . Of some interest are the influences of price changes of construction work, pumps and electrical energy regarding the course and characteristics of the branch tree pipe network.

With equal construction and pump costs, if the costs of electric energy increase by 50% we get by the first version the third path (Figure 9) and by the second version second path.

Conclusion

Determination by the optimal tree branch path was executed using the simplex method. The basic factors for determination of the path are economy and functionality of the tree pipe network. Because the solution of nonlinear systems requires the use of computers, a computer programme was developed, which considers all the limitations and requests of the given case.

The programme enables, after the determination of the optimal branch tree path and capitalised costs, also variations of different influential parameters and the evaluation of their influence on the district heating system as a whole. The optimal path, defined by the programme is not necessarily the shortest path, due to the fact that the branch tree pipe network is a function of the minimising both the investment and operational costs.

Within defining the optimal branch tree path the results

show the importance of which costs do we consider into objective function and which costs do we use at passage on standard pipe diameters, also the construction and operating costs of the system have important role. The path itself varies with the changing of construction costs, pump and electric energy costs. With their increase the investment, operational and maintenance costs also go up.

Therefore using the simplex method for each sole case the solution, representing the most reasonable fluid transport path from entering nodes to exiting nodes, was selected.

1 )Figure 1 : 2 .

 112 Figure 1: Simplified flow diagram for optimal thermal insulation thickness (c s -insulation costs)

Figure 2 :insulation thickness 3 .

 23 Figure 2: The difference between the static and dynamic economic methods by determining the economic thermal insulation thickness

  minimised. Capitalised costs of each pipe CC j are the sums of discounted values of investment in pipes, pumps, building expenses and operational costs. As the capitalised costs depend on the pipe diameter, which is discreet variable, and the diameters depend on the flow, the object function is non-linear. elements of the mathematical model are nonlinear equations of hydraulic limitations that represent the continuity of the pipe network and limitations of the simplex method representing the required no-negativity of the simplex variables.

Figure 3 :

 3 Figure 3: Relationship between CC and q v

Figure 4 :Figure 5 Figure 5 :

 455 Figure 4: Relationship between all annuity costs vs. pipe diameter for the q v = 0.04 m 3 /s and L = 1 m

  The network consists of 33 pipeline sectors and 24 nodes. Hot water flowing at high pressure enters the system in the node TOM1 and exits in T3, T4, T8, T9, T10, T12, T13, T15, T20 and T22, other nodes on figure3are virtual, and represent the branching of pipes.Data needed for the optimisation of the pipe network are presented in tables 1, 2 and 3.

Figure 6 :

 6 Figure 6: Loop pipe network of all possible routes of the district heating system

and tables 4, 5 , 6 ,

 56 was received by Version 1. This is for the case when at passage on standard diameters we consider only pipe costs while in objective function we consider all costs (C 1 , C 2 , C 3 and C 4 ).

Figure 7 :

 7 Figure 7: Optimised branch tree path of the district heating system

5 4 5 Figure 8 :

 558 Figure 8: Optimised branch tree path of the district heating system by second version

Figure 9 :

 9 Figure 9: The third optimised branch tree path

Figure 1 :

 1 Figure 1: Relationship between CC and q y

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 23456 Figure 2: Relationship between all annuity costs vs. pipe diameter Figure 3: Loop pipe network of all possible routes of the district heating systemFigure 4: Optimised branch tree path of the district heating systemFigure 5: Optimised branch tree path of the district heating system by second version Figure 6: The third optimised branch tree path

Table 1 :

 1 Pipe data

	Pipe	L [m]
	1	210	10
	2	400	5
	3	150	5
	4	200	15
	5	200	5
	6	220	5
	7	100	5
	8	160	5
	9	210	15
	10	90	5
	11	140	5
	12	210	5
	13	150	15
	14	320	5
	15	110	10
	16	130	10
	17	160	5
	18	180	15
	19	90	5
	20	140	15
	21	230	5
	22	310	5
	23	90	10
	24	340	5
	25	180	5
	26	180	10
	27	150	5
	28	170	15
	29	120	10
	30	160	5
	31	150	15
	32	120	10
	33	80	5

Table 2 :

 2 Input and output data in nodes

	Node	q v ×10 -3 [m 3 /s]
	1	47.50
	2	0
	3	-2.20
	4	-7.60
	5	0
	6	0
	7	0
	8	-9.8
	9	-7.3
	10	-3.4
	11	0
	12	-4.8
	13	-5.2
	14	0
	15	-2.5
	16	0
	17	0
	18	0
	19	0
	20	-3.0
	21	0
	22	-1.7
	23	0
	24	0

Table 3 :

 3 Physical and economical data

	Input pressure	10 6 Pa
	Input fluid temperature	110°C
	Output fluid temperature	70°C
	Density	934.8 kg/m 3
	Kinematics viscosity	0.226×10 -6 m 2 /s
	Operating time	8760 h/a
	Interest rate	0.1
	Price of electrical energy	7.1×10 -5 EUR/Wh
	Pump price	0.15 EUR/W
	Pumps lifetime	10 year
	Pumps efficiency	0.75
	Pipe network lifetime	40 year
	Pipe roughness	0.4 mm
	Coefficient of pipe costs polynomial:
	A	18 EUR
	B	291 EUR/m
	C	229 EUR/m 2
	Coefficient of construction costs polynomial:
	D	287 EUR
	E	310 EUR/m
	F	1275 EUR/m 2

Table 4 :

 4 Output pressure in nodes

	Node	p [Pa]	Node	p [Pa]
	1	1000000	13	929678.
	2	989448.	14	911975.
	3	899868.	15	921361.
	4	932760.	16	934913.
	5	947095.	17	954457.
	6	972226.	18	923717.
	7	967123.	19	901952.
	8	959943.	20	875753.
	9	944151.	21	851575.
	10	988733.	22	858976.
	11	969521.	23	891215.
	12	956542.	24	921451

Table 5 :

 5 Optimisation results by the first version

	Pipe	D [m]	qv [m 3 /s]	v [m/s]	p [Pa]
	1 0.2191	0.03160	0.838	10552.
	3 0.0570	0.00220	0.862	32892
	6 0.1937	0.03160	1.072	17222
	7 0.2191	0.03160	0.838	5103
	8 0.2191	0.03160	0.838	7180
	9 0.1778	0.02180	0.878	15792
	10 0.11143 0.00980	0.955	11391
	11 0.0825	0.00470	0.879	20434
	15 0.0635	0.00250	0.789	19544
	17 0.1524	0.01590	0.872	11267
	18 0.1330	0.01250	0.900	19212
	19 0.0635	0.00250	0.789	15064
	20 0.1270	0.01000	0.789	12979
	21 0.0889	0.00520	0.838	26864
	26 0.0700.	0.00300	0.780	26199
	27 0.0825	0.00470	0.879	21764
	28 0.0510	0.00170	0.832	42976
	33 0.0635	0.00250	0.789	13552

Table 6 :

 6 Capitalized costs

	Pi.	CC1 [EUR]	CC2 [EUR]	CC3 [EUR]	CC4 [EUR]	CC [EUR]
	1	19920	110	2760	89360	112150
	3	5420	20	600	47370	53410
	6	18660	180	4500	88840	112180
	7	9480	50	1330	42550	53410
	8	15180	70	1880	68080	85210
	9	16530	110	2850	82120	101610
	10	4990	40	920	31210	37160
	11	6240	30	790	45990	53050
	15	4210	20	400	35080	39710
	17	11070	60	1480	59530	72140
	18	11180	80	1990	64570	77820
	19	3440	10	310	28700	32460
	20	8400	40	1070	49670	59180
	21	10740	50	1160	76350	88300
	26	7270	30	650	57970	65920
	27	6680	30	850	49280	56840
	28	5810	20	600	53220	59650
	33	3060	10	280	25510	28860
		168280	960	24420	995400	1189060

Table 7 :

 7 Output pressure in nodes by second version

	Node	p [Pa]	Node	p [Pa]
	1	1000000	13	929678
	2	989448.	14	911975
	3	900614	15	921361
	4	933506	16	934913
	5	949518	17	954457
	6	972226	18	914753
	7	965478	19	893868
	8	952674	20	967669
	9	934341	21	869284
	10	988733	22	876685
	11	969521	23	891215
	12	956542	24	914182

Table 8 :

 8 Optimisation results by the second version

	Pipe	D [m]	qv [m 3 /s]	v [m/s]	p [Pa]
	1 0.2191	0.03160	0.838	10552
	3 0.0570	0.00220	0.862	32892
	4 0.1270	0.00980	0.774	16012
	5 0.1143	0.00980	0.955	22708
	6 0.1937	0.03160	1.072	17222
	7 0.1778	0.02180	0.878	6748
	8 0.1683	0.02180	0.9801	12804
	9 0.1270	0.01030	0.813	18333
	11 0.0700	0.00300	0.780	19588
	13 0.0510	0.00170	0.832	38492
	15 0.0635	0.00250	0.789	19544
	17 0.1524	0.01590	0.872	11267
	18 0.1330	0.01250	0.900	19212
	19 0.0635	0.00250	0.789	15064
	20 0.1270	0.0100	0.789	1279
	21 0.0889	0.00520	0.838	26864
	26 0.0700	0.00300	0.780	26199
	27 0.0700	0.00300	0.780	20886
	30 0.0510	0.00170	0.832	37497
	33 0.0635	0.00250	0.789	13552

Table 9 :

 9 Capitalized costs by the second version

	Pi.	CC1 [EUR]	CC2 [EUR]	CC3 [EUR]	CC4 [EUR]	CC [EUR]
	1	19920	110	2760	0	22790
	3	5420	20	60	0	5500
	4	12000	50	1300	0	13350
	5	11100	70	1840	0	13010
	6	18660	180	4500	0	23340
	7	7870	50	1220	0	9140
	8	12020	90	2310	0	14420
	9	12590	60	1560	0	14210
	11	5650	20	490	0	6160
	13	5130	20	540	0	5690
	15	4210	20	400	0	4630
	17	11070	60	480	0	11610
	18	11180	80	990	0	12250
	19	3440	10	310	0	3760
	20	8400	40	70	0	8510
	21	10740	50	160	0	10950
	26	7270	30	650	0	7950
	27	6060	20	520	0	6600
	30	5470	20	530	0	6020
	33	3060	10	280	0	3350
		181260	1010	20970	0	203240

Table captions Table 1 :

 captions1 Pipe data

Table 2 :

 2 Input and output data nodes

Table 3 :

 3 Physical and economical data

Table 4 :

 4 Output pressure in nodes

Table 5 :

 5 Optimisation results by the first version

Table 6 :

 6 Capitalized costs

Table 7 :

 7 Output pressure in nodes by second version

Table 8 :

 8 Optimisation results by the second version

Table 9 :

 9 Capitalized costs by the second version