Thermoeconomic optimization of vertical ground

Applied Energy (114), 492-503

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Thermoeconomic optimization of vertical ground-source heat pump systems through nonlinear integer programming

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Waldemar Retkowski, Jorg Thöming

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Abstract

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Vertical ground-source heat pump systems (GSHPSs) use the ground´s undisturbed relative constant temperature as a source for space heating of residential and commercial buildings. The design of GSHPSs is focused in finding the optimal depth and amount of boreholes and also the connected power requirement like the amount and size of heat pumps. In this paper a mixed-integer nonlinear programming (MINLP) approach to solve the design problem of a vertical GSHPS is presented. The resulting mathematical model includes the calculation of the total annual costs (TAC) and the coefficient of performance to obtain estimates of both economic and ecological relevance to design an optimal equipment set-up. For desired constraints the numerically optimal values of the design parameters (borehole depth, mass flow rate, number of boreholes, type and number of the heat pumps) were calculated. Two numerical solution alternatives are investigated, namely Generalized Reduced Gradient (GRG2) and evolutionary algorithm. The GRG2 approach provides a more stable and faster optimal solution. Calculated results are presented through a validation example. The evaluation of the proposed objectives and studied sensitivity effects present the applicability of the model. This method was able to improve the TAC about more than 10%.

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Accepted version

Retkowski W, Thöming J (2014)


Parameters Yearly operating hours (h y-1)

Abbreviations MINLP MILP GSHPS GSHP EED GHE Diff.

Air temperature (°C) Electrical efficiency (-) Start-up cycles (h-1) Geometrical expressions (-) Thermal resistance (m K W-1) Empirical based function Full operating time period (h) Heating season time period (h) Start-up time period (s)

Mixed-integer nonlinear programming Mixed-integer linear programming Ground-source heat pump system Ground-source heat pump Earth energy designer Ground-heat exchanger Difference

Indices f Fluid m Half borehole length g Grout gr Ground gs Ground surface b Borehole geo Geological dwn Down up Up ln Natural logarithm s Soil ∼ Dimensionless variable Mean value max min eva M i HC HP SC Q P tot

Maximal Minimal Evaporator Manufacturer Different equipment sections Heating circuit Heat pump Soil circuit Heat Power Total Heat coefficients Power coefficients

p Pipe dem Demand load Load pi Inner pipe po Outer pipe

Ampere (A) Volt (V) Thermal conductivity (W m-1 K-1) Heat load (kW) Heat flux (W m-1) Radius (m) Half shank space (m) Heat capacity (J kg -1 K -1)

C

Density (kg m-3) Viscosity (m²s-1) Costs (€)

Variables

OC

SPF

Massflow rate (kg s-1) Temperature (K) Total annual costs (€ y-1) Investment costs (€) Operating costs (€) Ground heat (kW) Length (m) Integer Number (-) Evaporator heat (kW) Condenser heat (kW) Electrical power start-up (kWh) Electrical power operating (kWh) Coefficient of performance (-) Seasonal performance factor (-) Temperature out of SC (K) Temperature into the SC (K) Disturbed Temperature (K) Flow rate (m³ h-1)

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Accepted version

Nomenclature



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1. Introduction

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One way to exploit sustainably produced electrical energy is in using a vertical ground-source heat pump system (GSHPS) to supply the heating equipment in residential and commercial buildings. The European number of these installed systems is rapidly growing. Extrapolating currently observed growth rates for Europe of 5.4 million heat pump units per year, let expect a number of 70 million installed units in Europe for 2020 [1]. Along with this increasing relevance and impact there is a rising demand for optimal designed vertical GSHPSs.

40 41 42 43 44 45 46 47 48 49 50

The research and developments of vertical GSHP technology based on various mathematical models and systems is described in detailed reviews [2-4]. Over the years different analytical [2,4], numerical [5-8] and hybrid [2,4] models have been developed especially to calculate the thermal behavior of ground-heat exchangers (GHEs). Therefore is the simulation of these systems an important tool for system design purposes. These approaches focus on the thermal ground behavior and are often time consuming techniques even for experienced users [9]. Furthermore is the optimal sizing of GHEs important because of the high drilling costs and the design challenge of sizing an optimal borehole thermal capacity with an optimal capacity of heat pumps [10]. To design competitive GSHPSs involves thorough technical also economic considerations. In these consequences arises the need to consider simultaneously the soil cycle, the heat pump cycle and their economic aspects for optimal design purposes.

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As one early work Wall [11] analyzed heat pump systems and pointed out that a thermoeconomic optimization is an economic optimization in conjunction with thorough thermodynamic description of the system. In [12] they developed a method to minimize the entropy generated in a heat exchanger, whereupon an optimum u-tube length and diameter was determined. Sayyaadi et al. [13] optimized a vertical GSHP for a given cooling load. Seven temperature differences and one pipe diameter for the ground heat exchanger were chosen as design variables. They minimized a thermodynamic-, a thermoeconomic- and a multi-objective and considered the sensitivities of the interest rate, operating hours and the cost of electricity. In [14] they optimized a vertical GSHP for given heating and cooling loads. They developed a nonlinear optimization model and applied for a thermoeconomic optimization eight temperature design parameters and one nominal pipe design parameter. Li and Lai [15] provided analytical expressions for optimizing flow velocity and borehole length by applying the entropy generation minimization method for GSHPs with a single U-tube. Their analyses indicated the existence of optimum parameters based on pure heat transfer and thermodynamics ground. In [16] they proposed an algorithm for optimization of cooling tower-assisted GSHPs applying 12 decision variables and additional constraints. Their sensitivity consideration of costs showed that the product cost of all regarded systems increased due to an increasing interest rate. Lee et al. [17] used for GSHP optimization an objective function representing the initial system costs divided by the annual energy production as a measure of cost effectiveness, which should be minimized. With an emphasis on building optimization [18] used three objective function criteria, the total cost of the system, the primary energy saving and the CO2 emission costs. They applied different values for different penalty parameters and focused applying a mixed-integer linear program (MILP) on optimization robust building loads. Florides et al. [19] investigated the cost and efficiency

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75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

impact of double U-tubes, single U-tubes and parallel or serial arrangements. They concluded that the building costs of double U-tubes are 22-29% higher than of single U-tubes. While their parallel configuration is more efficient by 26-29%, while the series configuration by 4259%. The assumption of a maximization of heat pump performance due to a minimization of ground temperature changes was followed by [20]. They achieved a balanced ground cooling with the application of GHE distributed loads. A connection to heat pumps or modeled flow through GHEs had there not a special emphasis. A related work of [21] pointed out that an optimization potential lies in the regulation of the energy extraction of load per BHE individually and a thereby balanced cooling of the ground. An applied seasonally variable heat demand aroused that the maximum temperature in the ground migrated by 10-15%. They concluded that by a priori choosing strategic BHE arrangement the long-term performance of the system could be improved, also without load regulation and connected additional investments. Michopoulos and Kyriakis [22] regarded an overestimation of the electricity consumption of their heat pump of max. 3.8%, due to deviations between measured and predicted temperature values of about ±2 °C.

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Other authors focused on the energy and exergy analysis of GSHPs and promise knowledge about optimization potential. Hepbasli and Akdemir [23] analyzed the energy and exergy of a GSHP system and concluded that the most potential for improvement is in the compressor, followed by the condenser and then the expansion device. A comparison of the irreversibility’s associated with the heat transfer process in the evaporator and GHE showed greater irreversibility’s in the evaporator due to the operation at a lower temperature than the GHE. The losses of an investigated GSHPS were due to the electrical, mechanical and isentropic efficiencies and emphasize the need for paying close attention to the selection of this type of equipment. The second largest irreversibility was due to the condenser, the third the expansion device. The evaporator had the lowest irreversibility on the basis of the heat pump cycle [24]. Wall [11] emphasized that the difference of all incoming and outgoing exergy flows must be minimized. Bi et al. [25] specified the location of maximum exergy loss ratio with the compressor, while the location of minimum exergy efficiency and thermodynamic perfect degree is the ground heat exchanger. Torío et al. [26] distinguished with an extensive exergy study that a holistic view of a GSHP system and especially the interactions and components should be regarded to optimize the system.

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Only rare optimization approaches applied a mathematical programming considering both crucial cycles. From our knowledge none author up to now has given preferential consideration on a discrete choice analysis approach, as a MINLP framework has enabled to us, which allows system optimization of real existing components. Due to the finding of Shonder et al. [27] is the peak-load method the most consistent one compared with other tested design programs. For the purposes of this analysis, a design heating day should be determined from numerical investigations. Traditional design methods of GSHPSs involve many trials in order to meet the design specifications [9]. This can be avoided through the present design method, which considers the minimization of annual costs, or maximization of ecological benefits or the generation of Pareto-optimal designs while technical constraints are satisfied. The present study demonstrates for the first time an integer-based optimization procedure that involves the entire system, consisting of a ground and also heat pump circuit

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and allows for choosing real units. This approach helps by this to avoid oversized or undersized systems. For this purpose a new mixed-integer nonlinear programming (MINLP) approach has been developed, validated and analyzed.

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2. Problem statement

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Given are a huge range of different potential GSHPS configurations, a fixed time interval, fixed and relative investment costs, relevant prices, lower and upper bounds on equipment and physical behavior, heat pump data base, energy balances, some physical properties, a given total heat load and costs associated with the GSPHS operation. The goal of each GSHPS design is to determine the optimal technical configuration with specific components, especially heat pumps, well size, well amount and mass flow rate, fulfilling the physical and technical needs along with the planning decisions that minimize the TAC, or maximize the environmental savings, or determines the Pareto-optimal design.

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3. Assumptions and mathematical model

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3.1. Assumptions and requirements

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The proposed numerical optimization scheme has the following general assumptions:

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 

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  

the ground heat conductivity is a mean of all ground layers, which is sufficient [28]; the thermal properties of all the materials are means and remain constant as typical for steady state modeling; the design case of GHE fields neglects the thermal interaction between boreholes; special thermal effects in the ground where reasonable simplified or neglected and the fluid flow rate in each GHE tube is equal.

One general typical strategy is to use ground related simulation results as an input for design programs [27]. This approach is strictly forward and being recommended from us during an application of the new developed design algorithm. Complementary one should execute a specific thermal response test on site in advance to get proper ground properties as the undisturbed temperature, the maximum available heat flux and the heat conductivity of the ground.

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3.2. Mathematical formulation

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Mixed-integer nonlinear programming (MINLP) is a numerical optimization approach of mathematically formulated problems to simultaneously optimize structure (using discrete variables) and parameters (using continuous variables) of a system. The mathematical idealization of the thermo-physical and economical programming for generating an optimal solution of a GSHPS configuration design tasks can be expressed as described in the following sections.

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3.2.1. Heat production subsystem

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The heat production is taken over by two closed cycles: the soil circuit and the heat pump circuit (Fig.1). The main task for each cycle is to fulfill the energy balance and to generate 5

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relevant temperature information’s. Both cycles are coupled and this is also being expressed by the linking constraints.

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3.2.1.1. Soil circuit constraints

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The soil cycle reaches from the deepest point of the boreholes respective tubes up to the evaporator of the heat pump. Main task is to allocate ground-heat due to estimating the

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total needed heat exchanger length

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flux

associated with an estimated maximal possible heat

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(1)

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For simplicity one could determine the discrete chosen number of boreholes with

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supplied boreholes and a single borehole length

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total heat exchanger length.

parallel

, shown in Eq. (2), out of the requested

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(2)

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(3)

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If not other mentioned were here typical values for

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5.3.2) and

(analyzed in section

used.

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(4)

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The ground-heat through the evaporator is idealized with

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mass flow

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here a mean fixed value

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3.2.1.2. Heat pump circuit constraints

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The heat pump cycle supplies the required heat to the heating circuit. The functions

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and specific heat capacity

and carried by an optimal

and is given in Eq. (4). For simplicity was

for our proposed calculations used (Appendix A.3).

and

represent an individual heat pump and are usually not known explicitly. However, heat

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supply and power data, at variable fluid temperatures

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as a function of the capacity for a specific heat pump. The details are given in section 3.3 and Appendix A.

, are often given by the manufacturer

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(5)

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(6)

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With Eq. (5) and Eq. (6) one can determine the discrete chosen number of heat pumps

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the specific produced heat and estimated electrical demand of each heat pump. With the sum of produced thermal heat one could guaranty the requested amount of heat . The

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Accepted version

from the ground. The formulation is shown in Eq. (1).


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heat pump start-up process, modeled in Eq. (7) and Eq. (8), requires pre-defined start-up cycles and asks for total additional power to be supplied.

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(7)

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(8)

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with

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The shown start-up heat pump power supply calculation considers a three-phase current basis, results for a certain as typical for the considered heat pumps. A total electrical demand

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operating time interval

and can be calculated with Eq. (9).

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(9)

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The total chosen heat pumps

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expressed in Eq. (10).

should contain at least one unit; this is

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(10)

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It is necessary to calculate the from the ground to be provided heat

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should be converted into thermal energy; this efficiency can be expressed with the factor

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The about the electrical power to heat efficiency factor reduced heat pump electrical power demand should be managed with the heat provided by the heat pumps as shown in

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Eq. (11). This couples the heat pump circuit with the soil circuit and gives the heat demand to be supplied by the soil circuit.

. The electrical power

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.

(11)

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Related to the specific heat pump

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define a maximum number of heat

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realized in Eq. (12). Also the maximal requested specific electrical power is limited by as shown in Eq. (13).

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and to the individual characteristic curve one can the selected heat pump could supply, which is

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(12)

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(13)

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(14)

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The current mass-flow rate

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a manufacturer´s minimum design flow rate

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the controller due to a lack of adequate fluid. These parameter values can be taken from

should be converted to flow rate

and not fall below

. This ensures unwanted switching of by

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Accepted version

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manufacturer´s data sheets. To calculate

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the temperature dependent density

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3.2.1.3. Temperature constraints

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The mean fluid temperature

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simplified calculation can be realized by building the mean of the soil fluid inlet

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fluid outlet

simply one could divide the design variable given in Appendix A.

as crucial variable of the soil circuit has to be calculated. A and soil

(15)

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These temperatures depend strongly on the estimated or measured maximal heat flux

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and the so-called mean borehole resistance

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(17).

, which can be defined as in Eq. (16) and Eq.

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(16) with

, where

is assumed with

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(17)

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A mean temperature

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with an undisturbed temperature fraction

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specific ground gradient

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values vary about approx. 0.01 °C m-1 up to 0.05 °C m-1. Typical values for Germany vary between approx. 0.025-0.035 °C m-1. One could calculate the local ground gradient dependency as in [5] upon the depth with

located at the half of a single borehole depth is being calculated and a depth-dependent fraction related to the

. The value depends strongly on the soil properties. Often used

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(18)

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It is assumed that the average boundary temperature at the ground surface

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higher than the average annual air temperature

is approx. 1-2 °C

.

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(19)

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(20)

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The mean fluid temperature depends in particular on the existing specific thermal borehole resistance. Florides et al. [29] highlighted the great importance of the proper borehole filling. A classical thermal resistance arrangement of a borehole is indicated in Fig. 2. Lamarche et al. [30] have published an overview and evaluation of common methods to calculate the borehole resistance in GSHP systems. They recommended the multipole method proposed by Bennet et al. [31] to solve the problem and mentioned that with the solution in the form of an infinite 8

Accepted version

temperatures, or as shown in Eq. (15).

221

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by


series (multipole expansion) one can compute pipe related steady-state conductive heat flows. The calculation of the thermal resistance is given by Eqs. (21)-(23).

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(21) ;

;

(22)

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(23)

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Lamarche et al. [30] pointed out that the first term in Eq. (23) is known as line-source formula and the second term was proposed as first-order multipole correction [30-32] and that this expression is also used in the EED design software [33] and GLHEPRO 4.0 [18]. The range of values for is approximately between 0.01-0.8 K m-1 W-1 with typical values for

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Germany between 0.10-0.35 K m-1 W-1. For several test cases Lamarche et al. [34] have calculated a maximal error between analytical and numerical solutions of only 0.9%. The maximal error of the other tested methods varied between 18% and 47%. The overall estimate for all tested methods is between 43% and 150%.

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To avoid frozen layers close to the GHE and damage of equipment German engineer standards [35] restrict as shown in Eq. (24) the returning heat carrier fluid for a peak load with and for a constant load with .

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(24)

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Additionally one should restrict the lowest fluid temperature depending on the binary mixture, in our case approx. -10 °C [44]. Ensuring a proper heat transfer through the evaporator one should set a lower bound of approx. 2 °C and an upper bound of approx. 7 °C

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as shown in Eq. (25) which are derived from experience.

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(25)

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3.2.1.3. Linking constraints

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A crucial linking constraint is formulated in Eq. (26). With this constraint should the heat balance from the soil up to the heat pumps been guaranteed.

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(26)

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This ensures with the given heat extraction capacity shown in Eq. (27) the fulfillment of the heat demand derived from the parallel executed heat pumps and soil capacity. The second crucial equality constraint guarantees the proper generated heat provided by the optimal

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heat pump configuration.

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(27)

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Accepted version

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3.2.1.4. Performance indicators

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The efficiency and therefore the environmental impact under certain conditions might be expressed as coefficient of performance (COP) as shown in Eq. (28).

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(28)

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To get an approx.

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run the annual average heat flux, calculated as shown in Eq. (29) or directly derived from measurements, to get according model outputs with as the hours of an specific year. This

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approach is equal to a simplified averaged COP, where in Europe often an averaged COP is called SPF. Here is COP used as for one operating point valid, regarding the input it is especially the peak heat flux, instead is the SPF connected to a longer term, expressed as averaged heat flux.

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(29)

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3.2.2. Economic constraints

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The economic sub-model is based on two simplified main economic factors which are the investment cost (IC) and the operating cost (OC). The sum of these components should be minimized. IC includes the cost of heat pumps, heat exchangers and an average amount for connecting these components, namely additional drilling cost. For this value one can

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assess in Germany between approx. 25%-35% of the borehole drilling costs

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additional investment installation costs

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the electricity required on site is occupied to determine the OC. The economical sub-model is expressed as follows in Eqs. (30)-(33):

and gets the

(includes PE pipes, filling material, etc.). Mainly

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(30)

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(31)

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(32)

300

(33)

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3.3. Heat pump data basis

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The from manufacturer´s provided technical data sheets are containing information’s about characteristic curves of specific heating capacity and required electrical power. The through regression resulting functions are related to the fluid temperature. Note that depending on the type of the characteristic curve it may fit better to apply the model based fluid temperature .

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And due to specific restrictions on measuring system may some additional restrictions be required. Out of this empirical data were coefficients (Appendix A.1) and functions in the general form of Eq. (34)-(35) determined. The used data bases contained three or four different heat pumps, each valid for a heating circuit temperature of 35 °C. The temperature 10

Accepted version

for a specific year of operation one could apply after an optimization


of 35 °C is a typical water temperature level used in floor heating systems which promise a good efficiency for heat pump systems. These coefficients produce a bridge to the real problems a designer is being faced: the selection of a proper heat pump and circuit pump fulfilling the specific constraints and boundary conditions.

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(34)

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(35)

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The specific upper bounds are respectively given by the maximum thermal and electrical properties and attend as inequality constraints as shown in Eqs. (12)-(14). It should be taken care of the constraints that the maximal or minimal known allowed heap pump specific values are not violated. Note that related on the type of characteristic curve and resp. heat pump one might restrict the temperature spread in Eq. (25) as well for a constant level of 3 K, which dependents on the preconditions of the specific heat pump.

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3.4. Objective functions

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Three different objective functions were applied to illustrate the capability of the new method to generate powerful solutions. The main focus was taken on the investigation of the technical design by a minimization of the costs in the first year, which is realized with Eq. (37). With the presented formulation of Eq. (38) it might be possible to get a compromise between a thermo-economic design and a focus on the crucial ecological impact. And with the Eq. (39) it is possible to maximize the crucial performance indicator COP. All optimizations were undertaken with the constraints given in Eqs. (1)-(14), Eqs. (16)-(28) and Eqs. (30)-(36).

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3.4.1. Thermo economic performance

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The total annual cost (TAC) of the first year of a specific geothermal system is given by Eq. (36). The TAC function should be minimized due to variation of the formulated design variables by the chosen solving method.

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(36)

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The process variables

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The integer design variable

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database (Appendix A). The integer variable

and

are modeled as non-negative stationary design variables. is representing the chosen specific heat pump provided in a is responsible for a proper amount of wells.

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(37)

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3.4.3. Pareto optimal performance

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As objective function to calculate Pareto optimal solutions were the already shown Eq. (36) and Eq. (28) taken. The division results in Eq. (38).

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Accepted version

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344

(38)

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3.4.2. Environmental performance

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Eq. (28) can be taken as optimization criteria and supplemented as objective function as shown in Eq. (39). 349 350

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4. Solution algorithms

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The new GSHP design method was solved through application of a Generalized-ReducedGradient-2 algorithm (GRG2) and an Evolutionary algorithm (EA). Both solvers are implemented in the basic Microsoft Excel 2010 environment.

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4.1. Generalized reduced gradient 2 (GRG2)

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The Microsoft Excel 2010 Solver employs the GRG2 Algorithm [36] for solving nonlinear problems [37-38]. The method extended first-order reduced gradients with second-order information based on reduced Hessian. This enables solutions also of constrained nonlinear problems. Integer constrained nonlinear problems are solved by a branch and bound algorithm which starts an optimization process by solving the relaxed problem using GRG2 [39]. Iteratively solved sub-problems update the best bounds with the best objectives until a by the is satisfied as shown in Eq. (40). The number of sub-problems may user fixed tolerance

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grow exponentially [40].

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(40)

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4.2. Evolutionary algorithm (EA)

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The applied method follows a nondeterministic approach and is a hybrid, based on the principles of genetic algorithms and evolutionary algorithms. In a genetic algorithm the problem is often encoded in a series of bit strings that are manipulated by the algorithm. In an evolutionary algorithm the decision variables and problem functions are used directly. It generates many trial points and uses “constraint repair” methods to satisfy the integer constraints. The constraint repair methods include classical methods, genetic algorithm methods, and integer heuristics from the local search literature. This approach cannot guarantee the optimality [40].

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5. Application of the new GSHP optimization model

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The provided model evaluation includes a validation case where a comparison between model results and measurements (Fig. 3) under equal conditions is presented. A constitutive comparison of optimal results applying the proposed objectives is shown in Tab. 2. 12

Accepted version

(39)


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Furthermore follows in section 5.3. an extensive analysis of the optimization model with three hypothetical comparable case studies, where every point entails a different optimal GSHP design.

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5.1. Validation case

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Design and measurement data were taken from [41] and compared with results of the crucial variables generated by this method. The origin construction of the associated values was equipped with two ground-coupled heat pumps, each with a max. capacity of approx. 82 kW. This heat pump type is included in the provided data base and referred to as Type 3 (Tab. A.5). For validation purpose was only one single heat pump considered and related results for the electrical power demand can be taken from Fig.3C. A ground-source heat exchanger field with a total length of 2400 m and 16 boreholes was installed on site. Half of the pipes were up to 8 m insulated, whereupon for validation only the separate available data with no influence of the insulation was used. Regarding the distributed annual temperatures contained a freecooling and a heating period. The average fluid inlet temperature became for this period 8.8 °C and the outlet temperature 6.8 °C into the ground. The plant was designed with a average heat flux of 35.6 W m-1. This value could nearly be confirmed with 34 W m-1 measured. In October of the first heating period the maximal heat flux was 39.3 W m-1, what is used as criterion for the calculated design cases shown in Tab. 2. As an annual average was 13.02 W/m calculated. The heat conduction of the submerged borehole filling material was estimated with 1.45 W m-1 K-1 and the PE-pipe leg distance with 0.035 m. The crucial rare available data is tightly arranged in Tab. 1. Further in this section not given values can be taken from and Tab. 6.

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Table 1 Design conditions and measurements taken from [41]. Parameter Value Unit Maximal heat load 111.2 kW Maximal heat flux 39.3 W m-1 Mean soil conduction 2.4 W m-1 K-1 Full operation hours 4111 h y-1 Average air temperature 8 °C Heating circuit temperature 35 °C Borehole diameter 0.115 m Pipe diameter 0.032 m Mass flow rate 0.16 kg/s

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For one heating season are compared results between measurements and modeled data shown in Fig.3. The comparison emphasizes the mean fluid temperature given in Eq. (15) and the electrical power demand given in Eq. (9) inclusive all related equations. These values represent the main target values of operating conditions. For the calculation of the mean fluid temperature were the measured single borehole depth, mass flow rate, heat conductions, the pipe radius, the borehole radius and the leg distance of PE-pipes, the heat flux and borehole temperatures used as inputs. According to the plant operation were the measured heat flux and borehole temperatures varied, whereupon the other inputs were left constant. Depending values can be taken from Fig.3A-B. The heat pump operated from 300 up to 691 h a month. 13

Accepted version



412 413 414 415 416 417 418 419 420 421 422

Note that each measured monthly heat flux was converted to 740 h to get averaged ground related values for each month. To compare proper the borehole temperatures the measured points were reduced by an average of 0.3 °C, due to slightly differences between the modeled (borehole edge) and measured (inside a borehole) temperature position. To compare the electrical power demand as shown in Fig.3C the measured mean fluid inlet temperature and the heat pump operation hours were time dependent applied, as well as a fixed heat pump specific operating caused conversion factor of 0.85, which was derived from data. The relative error between the measured and modeled data was in these cases under 1% (Fig.1CD). The heat pumps can be assumed to be validated due to publication of empirical based characteristic curves. The R-squared value was for every in this paper applied heat pump above 0.99.

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5.2 Evaluation of the objectives

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Further a constitutive comparison of optimal results applying the proposed objectives and a not optimized base case are shown in Tab. 2. The calculated optimal data were generated by application of the objective functions given in Eqs. (37)-(39) and the constraints given in Eqs. (1)-(14), Eqs. (16)-(28) and Eqs. (30)-(36). The data base of considered heat pumps can be taken form Table A.5. The data base includes also the heat pump type used on site, namely Type 4. For comparability similar model values were inserted, they are given in section 5.1, Table 1 and Table A.6. The calculated optimal results provided by GRG2 and EA can be taken from Table 2. The GRG2 considered a solution accuracy of 10-4, used automatic scaling, forward differences, and zero as lower bound for not constraint variables. The EA considered a tolerance of 10-2, a mutation rate of 0.5, population of 100, automatic scaling and zero as lower bound for not constraint variables. The optimality criterion for integer design variables was in both cases zero.

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Table 2 Base case values and optimal results obtained by the proposed algorithm solved with the GRG2 and EA. The optimal number of heat pumps is aligned with Tab. 5 and arranged

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as follows: Type 1/ Type 2/ Type 3/ Type 4. All given differences are related to the base case.

Units Base case Eq.(37) – GRG2 Diff. [%] Eq.(37) – EA Diff. [%] Eq.(38) – GRG2 Diff. [%] Eq.(38) – EA Diff. [%] Eq.(39) – GRG2 Diff. [%] Eq.(39) – EA Diff. [%]

0/0/2/0 0/0/0/6 1/1/0/0 0/0/0/6 1/1/1/2 0/0/0/9 0/0/0/9

€ 267516 239670 -10.4 254603 -4.8 239670 -10.4 279421 4.5 249001 -6.9 249156 -6.9

16 8 -50 15 -6.3 8 -50 5 -69 7 -56 7 -56

m 150 295 96.7 122 -18.7 295 96.7 277 84.7 303 102 304 103

m 2400 2358 -1.8 1831 -23.7 2358 -1.8 1387 -42.2 2121 -11.6 2122 -11.6

5.32 5.73 7.7 4.52 -15.0 5.73 7.7 4.97 -6.6 5.74 7.9 5.74 7.9

5.51 5.92 7.4 4.65 -15.6 5.92 7.4 5.12 7.1 5.93 7.6 5.93 7.6

14

°C 7.7 9.5 23.4 7.4 -3.9 9.5 23.4 9.3 20.8 9.7 26.0 9.7 26.0

Accepted version



440 441 442 443 444 445 446 447 448 449 450 451 452 453 454

The in Table 2 shown differences between the calculations of the two solving methods and the base case have a causal connection to a change of the single well amount, well length, heat pump capacity and arrangement. The system with the about 10.4% improved costs delivers parallel an improved efficiency of about 7.7%. The best performance of the individual optimal GSHPS provided by the EA and the GRG2 is almost 8% higher than the base case. It may be useful to compare the calculated COP, as the algorithm considers operating points. Looking for a longer term in literature was the annual SPF given with 4.6 of the regarded heating season. Related to this value exists an optimization potential up to the best numerical annual SPF with 5.93 of 28.9%. This could be assessed on the basis of additional optimized control and conversion strategies, which are not subject matter here. Along with the objective related increased optimal performance decreased the TAC about 18.515 € which is equivalent to almost 7%. In accordance to the assumptions the optimal fluid temperature and total well length seemed to be sufficient estimates. As expected the total optimal length decreased along with a lower efficiency and increased along with a higher efficiency. The required optimal start-up energy varied between 419 and 628 kWh/a.

455

5.3. Three case study series

456 457 458 459 460 461 462

In the following sensitivity study three separate hypothetical design series are considered. In these case studies optimal values for three different case series were calculated. In the first case series was the input parameter heat load varied, in the second case the heat flux and in the third the mean outside temperature. All the other input values were left constant. The used values are given in Table 3 and Table A.6. The three heat pumps in the data basis are providing a heating system temperature of 35 °C and the associated parameter values are given in Table A.4.

463

5.3.1. Model set-up

464 465 466 467 468 469 470 471 472

The heat load was varied with an interval of 10 kW between 60 kW and 350 kW. This range can be assigned to the group of new industrial size GSHPSs. In the other two series the heat load was fixed with 100 kW. The range for heat flux values was chosen between 31 W m -1 and 59 W m-1 with an interval of 1 W m-1. For the third series the average outside temperature varies between 5.5 °C and 14.5 °C with an interval of 0.5 °C. Further input values were taken for the ground properties, hydraulic properties, operating conditions and economical properties and can be taken from Table 6. The related results are shown in Fig.4-6, for heat flux variations they are indicated with A, for heat load variations they are indicated with B and for temperature they are indicated with C.

473 474

Table 3 Parameter values and -ranges used as input variation. Heat load 60-350 100 Heat flux 50 31-59 Average annual temperature 10 10

100 50 5.5-14.5

kW W m-1 °C y-1

475 476 477

15

Accepted version



478

5.3.2. Results and discussion

479 480

Optimization programming for the sensitivity study was solved with the GRG2 algorithm and calibrated as shown in section 5.2. Additional used values are presented in Tab. 6.

481

5.3.2.1. Optimal investment and operational costs

482 483 484 485 486 487 488 489 490

The optimal values for investment and operational costs are shown in Fig. 4. With an increasing heat flux are the investment costs falling, but the operating costs remain almost the same. The integer caused variations indicate the dependent relationship between the investment and operational costs. This seems to be caused by a connected change in the optimal heat pump configuration. Even though discrete variables exist both cost components are with an increasing heat demand almost parallel rising. Surprisingly investment and operational costs related to a variation of heat load is almost smooth and linear. The rising average outside temperature tends to result in lower costs. The impact of the discretized solution domain shows compared to the other ones the main significance.

491

5.3.2.2. Optimal heat pump configuration

492 493 494 495 496 497 498 499 500 501 502 503 504

Information about the optimal type and number of heat pumps for each design case provided by solutions of the optimization model can be taken from Fig. 5. A huge range of optimal points related to a change of the heat flux remain almost constant with a tendency to an increased total number of heat pumps along with a higher capability of the ground. The second case study series shown in Fig. 5B with solutions of optimal heat pumps created through a varying heat load corresponds to the progress of investment and operational costs. With a higher heat load the total amount of heat pumps increases almost linear and stepwise. The heat pump type tendency gradually switches with higher heat loads to a higher number of heat pump types, in our case study number three, as shown in Table A.4, this is the major heat pump in the provided database. The left heat load gap is filled with a combination of the other fitting and cheaper heat pumps. The third case study series shown in Fig. 5C shows a step at 10 °C and this is significant correlated with the costs. A change in the average outside temperature causes an only slightly lower number of necessary heat pumps.

505

5.3.2.3. Optimal well configuration

506 507 508 509 510

The optimal borehole configuration can be taken from Fig. 6. The almost equal number of heat pumps shown in Fig. 5 can be reached due to a decreasing amount of wells within a developing higher heat flux. For higher heat loads there is a need of a higher number of wells, whereupon the total well length linearly rises. A higher average outside temperature causes only a slightly higher stepwise demand of a total well length.

511

5.3.3. Optimal COP

512 513 514 515 516

The optimal COP value is for a change of the heat flux for the investigated data range 7.31 with a standard deviation of approx. 0.061. For a change of the heat load amounts the value to 7.26 with and standard deviation of 0.083. And for a change of the average ground surface temperature is the average COP value 7.04 and the standard deviation 0.581. For the variation of the heat load is the COP increasing with a higher heat load (Fig. 7). This integer constraint

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Accepted version



517 518 519 520 521 522 523 524 525 526 527 528 529 530 531

solution is comparative shown with calculated relaxation solutions and can be taken from Fig. 7. A bound was applied as shown in section 3.2.1.1 for a maximal single well depth and created three different algorithm cases with the values 300 m, 900 m and 1500 m. For each one were optimal design points related to a varying heat load calculated. The average COP shown in Fig. 7A is related to Fig. 4B, 5B and 6B. There were 300 m applied, in Fig.7B and Fig.7C 600 m and respectively 900 m applied. The GRG2 found best relaxation based solutions near the bound of 300 m with increasing heat loads. For this reason also the COP remains at a value of approx. 7.37 with a bound at 300 m. With a change of the upper bound to a value of 900 m, the COP increases significant. The solver uses the higher solution domain and finds solutions which vary close to an almost constant linear distribution around a COP of 8.75. Changing the maximal single length bound to a “virtual” value of 1500 m, the effect of a higher variation in the solutions increases only slightly with a variation of heat load. The optimal COP results for the heat load variation problem with integer constraints, without integer constraints (due to relaxation) and with a variation of a single bound can be taken from Fig. 7.

532

6. Conclusion and future work

533 534 535

A new method for design optimization of GSHPSs is presented. The proposed method combines financial and thermodynamic aspects with the selection problem of real components and delivers an optimal designed system.

536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558

It became obvious that a huge range of starting points within the design space led to converge to a local minimum instead of the global minimum. However, the GRG2 found often sufficient solutions in a time span under 1s. The EA found often solutions in reasonable time below 360s when used as hybrid with the GRG2 applied in advance. Note that it might be convenient to adjust narrow bounds for each design variable using the EA. In general requires the GRG2 less computational effort than the EA. A comparison of different objectives showed that different systems might be optimal, where in the studied case the straight minimization of costs seemed to be as well ecological sufficient. The sensitivity of the costs, well and heat pump arrangement related to a variation of three input parameter values (heat load, heat flux and the average temperature) were analyzed. Results show that the investment costs drop with increased thermal properties. Heat pump variations depend on the size and quality of pumps included in the data base and the algorithm prefers for an optimal design of industrial size problems lower units of heat pumps with a higher capacity. The heat load has a strong impact especially on the configuration of heat pumps and wells. Comparing this influence with the influence of the variation of the average ground temperature and heat flux these variables have a less impact on the selection of heat pumps and wells. The peak-load method seems to provide sufficient design solutions. All optimal thermo-economic configurations had beneficial environmental balance. The heat load dependency related to investment costs, operational costs, total well length and number of total heat pumps is even under the constraints of integer values almost linear. Optimal results have shown that required heat pump start-up energy can be a crucial design criterion. This can lead to different optimal heat pump arrangements. Due to a change of local average ground surface temperatures it is more suitable to change the conditions of the well arrangement in the ground than the

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Accepted version



559 560 561 562 563 564

arrangement of heat pumps. Altogether delivers the proposed optimization model good estimates for quick design investigations of real components to design an optimal vertical GSHPS. Already Dickinson [42] pointed out that of course the optimization of GSHPSs remains only part of the required methodology to reduce economic and CO2 cost of using energy in buildings. Additionally should the building designer weight up the most cost effective solution to result in the desired benefits.

565 566 567 568 569 570

Further investigations could extend the model with a heating circuit, the latest heat pump technology or study the impact of a huge heat pump data base in detail. With an extended time depended heat flux one could investigate possible limits of the related operational time. As one next step the author will investigate and provide a quantitative global sensitivity analysis to get a deep insight of the new optimization model and the related processes and potentials.

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Accepted version




573

Acknowledgements

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The authors wish to acknowledge support from the Wirtschaftsförderung Bremen GmbH (WFB Bremen – Economic Development) and the EUROPEAN UNION: Investing in your future – European Regional Development Fund.

Accepted version

577

19


578

Appendix A

579 580 581

For completeness are the used mathematical coefficients and some further parameter values characterizing investigated heat pumps provided. Also the model set-up parameter values used for specific calculations are presented.

582 583 584 585 586

A.1. Heat pump data base functions and coefficients

587 588 589 590 591 592

The provided coefficients and parameter values given in Table A.4 and A.5 were determined from manufacturer´s data sheets with the creation of linear regression lines for each specific heat pump. The correlation coefficient was for each function above 0.99. The parameter used in Eq. (8) was for all cases 0.8. The minimal flow rate in the soil circuit could be estimated with 20% of the nominal flow rate. Table 4 Properties for heat pumps 1-3 with a close spread capacity selection, valid for a mean heating circuit inlet temperature of 35 °C. Unit heat pump 1 heat pump 2 heat pump 3

593 594 595 596

7.900 10.907 13.914

-0.0013 0.0000 -0.0025

1.63 2.20 2.80

14.8 19.8 26.8

5658 6212 6693

A 17.00 20.00 23.00

m³ h-1 0.38 0.52 0.68

Table 5 Properties for heat pumps 1-4 with a wide spread capacity selection, valid for a mean heating circuit inlet temperature of 35 °C. Unit heat pump 1 heat pump 2 heat pump 3 heat pump 4

597 598 599 600 601 602 603 604

0.261 0.344 0.489

1.83 3.15 1.74 0.48

63.45 108.8 55.7 16.3

0.03 0.100 0.002 -0.002

19.27 26.00 12.96 3.65

88.00 160.00 82.00 24.50

25359 39451 26785 5537

A 63.00 80.00 64.00 30.00

m³ h-1 0.78 1.36 0.70 0.76

A.2. Model Set-up parameters Further values for case studies shown in section 5.3. and for the cases 5.1 and 5.2 which are not in the sections or Tab. 1 given are shown here in Table A.6. Table 6 Input conditions. Ground properties Mean soil conductivity Borehole diameter Grout thermal conductivity Geothermal gradient Hydraulic properties Specific heat capacity

2.0 0.15 1.0 0.06

W m-1 K-1 m W m-1 K-1 W m-2

3800

J kg-1 K-1 20

Accepted version


PE pipe diameter Thickness PE-pipe material Operating conditions Full operation hours Seasonal operation hours HP work cycle Electrical to heat efficiency Economic properties Energy price Borehole drilling cost Additional connection drilling costs

0.032 0.0037

m m

2300 4500 3 0.85

h y-1 h y-1 Unit h-1 -

0.25 60 30

€ kWh-1 € m-1 % of IC


605 606 607 608 609 610 611 612

A.3. Fluid density The fluid density depends strongly on the average fluid temperature as shown in Eq. (41). The density for a binary water-propylenglycol mixture of 25% propylene glycol and 75% water is approximated from manufacturer´s data sheets and can be calculated with Eq. (42) taken from [43].

613

(41)

614 615

(42)

616

21

Accepted version




617

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[4]

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[20] de Paly M, Hecht-Méndez J, Beck M, Blum P, Zell A, Bayer P. Optimization of energy extraction for closed shallow geothermal systems using linear programming. Geothermics; 43: 57-65. [21] Beck M, Bayer P, de Paly M, Hecht-Méndez J, Zell A. Geometric arrangement and operation mode adjustment in low-enthalpy geometrical borehole fields for heating. Energy 2013; 49: 434-443. [22] Michopoulos A, Kyriakis N. Predicting the fluid temperature at the exit of the vertical heat exchangers. Appl Energy 2009; 86: 2065-2070. [23] Hepbasli A, Akdemir O. Energy and exergy analysis of a ground source (geothermal) heat pump syste. Energy & Conversion Manage 2004; 45: 737-753. [24] Hepbasli A. A key review on exergetic analysis and assesment of renewable energy resources for a sustainable future. Renewable and sustainable energy reviews 2008; 12: 593-661. [25] Bi Y, Wang X, Liu Y, Zhang H, Chen L. Comprehensive exergy analysis of a ground/source heat pump system for both building heating and cooling modes. Appl Energy 2009; 86:2560-2565. [26] Torío H, Angelotti A, Schmidt D. Exergy analysis of renewable energy-based climatisation systems for buildings: A critical review. Energy and Buildings 2009; 41: 248-271. [27] Shonder JD, Martin M, Hughes P, Thornton J. Geothermal Heat Pumps in K-12 Schools – A case study of Lincoln, Nebraska, Schools. ORNL/TM-2000/80, Oak Ridge National Laboratory; 2000. [28] Lee CK. Effects of multiple ground layers on thermal response test analysis and ground/source heat pump simulation. Appl Energy 2011; 88: 4405-4410. [29] Florides GA, Christodoulides P, Pouloupatis P. An analyis of heat flow through a borehole heat exchanger valideted model. Appl Energy 2012; 92: 523-533. [30] Lamarche L, Kajl S, Beauchamp B. A review of methods to evaluate borehole thermal resistances in geothermal heat-pump systems. Geothermics 2010; 39: 187-200. [31] Bennet J, Claesson J, Hellstrom G. Multipole Method to Compute the Conductive Heat Transfer to and between Pipes in a Composite Cylinder. Notes on Heat Transfer 31987. Department of Building Physics, Lund Institute of Technology, Lund, Sweden 1987: 1-42. [32] Liu X, Hellstrom G. Enhancements of an integrated simulation tool for ground-source heat pump system design and energy analysis. Proceedings of Ecostock 2006, the 10th International Conference on thermal energy storage, the Richard Stockton College of New Jersey. [33] Hellstrom G, Sanner B. EED – Earth Energy Designer, Version 1.0, User’s Manual. Prof. Dr. Knoblich & Partner GmbH, Wetzlar, Germany, 1997: 43. [34] Spitler JD. GLHEPRO – A design tool for commercial building ground loop heat exchanger. In: Proceedings of the 4th International Conference on Heat Pumps in Cold Climates, 17–18 August, Aylmer, QC, Canada; 2000: 1–16. [35] VDI 4640 Blatt 2 - Verein Deutscher Ingenieure (VDI) [Hrsg.]. Thermische Nutzung des Untergrundes – Erdgekoppelte Wärmepumpenanlagen. Düsseldorf, 2001: 12. [36] Lasdon L, Waren A. GRG2 User's Guide. Department of Management Science and Information Systems, The University of Texas at Austin. 2 October 1997: 1-50. [37] Bernard O, Alata O, Francaux M. On the modelling of breath-by-breath oxygen uptake kinetics at the onset of high-intensity exercises: simulated anneling vs. GRG2 method, J Appl Physiol 2005; 100: 1049-1058. [38] Hyde KM, Maier HR. Distance-based and stochastic uncertainty analysis for multicriteria decision analysis in Excel using Visual Basic for Applications, Environ Modell & Soft 2006; 21: 1695-1710. 23

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[39] Fylstra D, Lasdon L, Watson J. Waren, A., Design and use of the Microsoft Excel Solver. Interfaces, Vol. 28, No 5, Sept-Oct. 1998: 29-55. [40] Frontline Systems, Frontline Solvers – User Guide. Version 11.5. For use with Excel 2003-2010. Frontline Systems, Inc., Incline Village, Nevada, 2011. [41] Eberhard M. Erdwärmesondenfeld Aarau. Heizen und Kühlen eines grossen Bürogebäudes mit teilweise wärmeisolierten Erdwärmesonden. Bundesamt für Energie. Schlussbericht Oktober 2005. Aarau, 2005. [42] Dickinson J, Jackson T, Matthews M, Cripps A. The economic and environmental optimisation of integrating source energy system into buildings. Energy 2009; 34: 2215-2222. [43] Glück B. Simulationsmodell Erdwärmesonden zur wärmetechnischen Beurteilung von Wärmequellen, Wärmesenken und Wärme-/Kältespeichern. Bericht der RUD. OTTO MEYER – Umwelt – Stiftung, Hamburg, 2007. [44] Eskilson P, Cleasson J, Blomberg T, Sanner B. Earth Energy Designer. Version 2.0 (Dec 19,1999). 620 621 622 623

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Accepted version




624 625 626 627 629 630

Tair

soil rotary pump

631 632

GSHE cycle

633

Tf2,i heat pump cycle

evaporator

634 635

Qhp.i

Qsc,2

Tgs

binary mixture

636

Tf1,i

637 638 639

Qsc,1

ground-scource heat exchanger

Tbi

expansion valve

Li

640 641

Qhc.load

Qsc, 3

642 643 644 645 646 647 648

Fig.1. GSHP system flow with indicated modeled action.

649 650 651 652 653 654 655 656 657 658 659 660 661 25

Accepted version

compressor

628



662 663 664 665 666

Tb R1

668

Tup

669

R2

xc

Tdn R12

rb

rp

Accepted version

667

670 671 672

Fig.2. Thermal resistance circuit with one U-tube (left) and Parameters (right) related to Eqs. (21)-(23).

673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699

26

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Accepted version

Fig. 3. Measured versus modeled mean fluid temperature Tf (D) and total electrical energy demand Ptot (C) considered under equal input conditions (A-B). Relative error is in both cases (C-D) less than 1%.

Retkowski W, Thöming J (2014) Applied Energy (114), 492-503

700

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731

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738 739 740 741

Fig. 4. Investment and operating costs of different GSHP design points due to variations of heat flux, heat load and the average annual temperature.

743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775

28

Accepted version

742



776 777 778 779

782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813

29

Accepted version

781

Fig. 5. Optimal heat pump configurations of different GSHP design points due to variations of heat flux, heat load and the average annual temperature.

780



814 815 816 817 818

821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851

30

Accepted version

820

Fig. 6. Optimal well configurations of different GSHP design points due to variations of heat flux, heat load and the average annual temperature.

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31

Accepted version

Fig. 7. Optimal COP design points of varied heat load conditions with integer constraints and without integer constraints (relaxation). A: Single well length was bounded with max. 300 m. B: Single well length was bounded with max. 900 m. C: Single well length was bounded with 1500 m.

Retkowski W, Thöming J (2014) Applied Energy (114), 492-503

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