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Objective function

5. CASE STUDY RESULTS AND DISCUSSION

5.2 Case Study

5.2.3 Objective function

In the current investigation, the objective function is represented by the global cost per square meter building floor area difference between any design combination that is created through optimization and base case design, added with the sum of all the penalty functions due to constraint violations. Therefore, only the additional cost incurred to achieve a given level of energy savings can be determined and compared while considering design limitations. The objective formula is given in equation 5.1.

β„Ž(π‘₯) = 𝑑𝐺𝐢𝑖

𝐡𝑒𝑖𝑙𝑑𝑖𝑛𝑔 πΉπ‘™π‘œπ‘œπ‘Ÿ π΄π‘Ÿπ‘’π‘Ž+ 𝑃𝐸𝑁 (5.1)

Where

𝑑𝐺𝐢𝑖 : Global cost difference between any design combination and base case building,

𝑃𝐸𝑁 : Sum of all penalty function results.

The reason to divide the GC to the floor area is to reduce the magnitude of the value for better readability.

As explained previously in the methodology section, the purpose of this study is to assist designers in achieving cost-effective high performance building design, therefore the cost function was defined to minimize building energy, water, material, and system related service life costs while maintaining or improving user comfort and reducing building CO2 emission rates. The elements of the main objective function of the case study are therefore as given in equation 5.2:

β„Ž(π‘₯) = 𝑑(βˆ‘ 𝑁𝑃𝑉𝑛1 πΈπ‘›π‘’π‘Ÿπ‘”π‘¦+ βˆ‘ 𝑁𝑃𝑉𝑛1 π‘Šπ‘Žπ‘‘π‘’π‘Ÿ+ βˆ‘ 𝑁𝑃𝑉𝑛1 π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™ +

βˆ‘ 𝑁𝑃𝑉𝑛1 πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘+ βˆ‘ 𝑁𝑃𝑉𝑛1 π‘…π‘’π‘›π‘’π‘€π‘Žπ‘π‘™π‘’)/π΄π‘Ÿπ‘’π‘Ž + βˆ‘ πœ‡π‘˜41 π‘₯π‘ƒπΈπ‘π‘˜

(5.2)

Each cost component is discounted to the present considering the time value of money. The main function is then adapted specifically for the case studies and reformulated.

5.2.3.1 Global cost components

The energy component of the main formula includes net present value of end-use energy consumption due to operation of boiler, chiller, fans (including fan coils and

ventilation fans), circulation pumps, water heating, interior lighting and plugged-in equipment. Moreover, in cases with PV optimization, the net present value of the surplus electricity generated through PV system is subtracted from overall energy cost as a benefit. Equation 5.3 shows end use types and related energy sources. It is assumed that the building maintains its annual energy efficiency performance throughout the long-term calculation period.

βˆ‘ π‘π‘ƒπ‘‰πΈπ‘›π‘’π‘Ÿπ‘”π‘¦ = π‘π‘ƒπ‘‰π‘π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘”π‘Žπ‘ π΅π‘œπ‘–π‘™π‘’π‘Ÿ + π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦πΆβ„Žπ‘–π‘™π‘™π‘’π‘Ÿ 𝑛

1

+ π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦πΆπ‘œπ‘œπ‘™π‘–π‘›π‘” π‘‘π‘œπ‘€π‘’π‘Ÿ

+ π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘¦πΉπ‘Žπ‘›π‘  + π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘¦π‘ƒπ‘’π‘šπ‘π‘  + π‘π‘ƒπ‘‰π‘π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ π‘”π‘Žπ‘ π‘Šπ‘Žπ‘‘π‘’π‘Ÿ β„Žπ‘’π‘Žπ‘‘π‘–π‘›π‘”

+ π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦πΏπ‘–π‘”β„Žπ‘‘π‘–π‘›π‘” + π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦π‘ƒπ‘™π‘’π‘”π‘”π‘’π‘‘ π‘’π‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘βˆ’ (π‘π‘ƒπ‘‰πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘–π‘‘π‘¦π‘ƒπ‘‰ π‘†π‘’π‘Ÿπ‘π‘™π‘’π‘ )

(5.3)

The water component of the main formula includes net present value of water use due to HVAC cooling tower operation and occupancy hot water use, as given in equation 5.4.

βˆ‘ π‘π‘ƒπ‘‰π‘Šπ‘Žπ‘‘π‘’π‘Ÿ = π‘π‘ƒπ‘‰π‘Šπ‘Žπ‘‘π‘’π‘ŸπΆπ‘œπ‘œπ‘™π‘–π‘›π‘” π‘‘π‘œπ‘€π‘’π‘Ÿ+ π‘π‘ƒπ‘‰π»π‘œπ‘‘ π‘Šπ‘Žπ‘‘π‘’π‘Ÿπ‘‚π‘π‘π‘’π‘π‘Žπ‘›π‘π‘¦

𝑛

1

(5.4)

The material component of the main formula includes net present value of ownership costs of building materials tested through optimization including insulation for external walls and roof, roof cover layer, glazing unit and external wall element, as given in equation 5.5. As the window-to-wall-ratio of external wall varies, the cost of external brick wall also varies as a dependent variable; therefore, its influence is also taken into account. The net present value covers initial, installation, maintenance, replacement and disposal costs of each element.

βˆ‘ π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™ = π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™πΈπ‘₯π‘‘π‘’π‘Ÿπ‘›π‘Žπ‘™ π‘€π‘Žπ‘™π‘™ π‘–π‘›π‘ π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› 𝑛

1

+ π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™π‘…π‘œπ‘œπ‘“ π‘–π‘›π‘ π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›

+ π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™π‘…π‘œπ‘œπ‘“ π‘π‘œπ‘£π‘’π‘Ÿ+ π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™π‘Šπ‘–π‘›π‘‘π‘œπ‘€ + π‘π‘ƒπ‘‰π‘€π‘Žπ‘‘π‘’π‘Ÿπ‘–π‘Žπ‘™πΈπ‘₯π‘‘π‘’π‘Ÿπ‘›π‘Žπ‘™ π‘€π‘Žπ‘™π‘™

(5.5)

The equipment component of the main formula given in equation 5.6 includes net present value of HVAC equipment selected during optimization including boiler and

and fan coils are taken into account as well. The cost of ventilation fans and circulation pumps are ignored for simplification. The formula also includes NPV of water heating equipment and lighting control system. The value covers initial, installation, maintenance, replacement and disposal costs of each element.

βˆ‘ π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘ = π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘π΅π‘œπ‘–π‘™π‘’π‘Ÿ

𝑛

1

+ π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘πΆβ„Žπ‘–π‘™π‘™π‘’π‘Ÿ + π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘πΆπ‘œπ‘œπ‘™π‘–π‘›π‘” π‘‘π‘œπ‘€π‘’π‘Ÿ

+ π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘π‘Šπ‘Žπ‘‘π‘’π‘Ÿ β„Žπ‘’π‘Žπ‘‘π‘’π‘Ÿ+ π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘πΉπ‘Žπ‘› π‘π‘œπ‘–π‘™ + π‘π‘ƒπ‘‰πΈπ‘žπ‘’π‘–π‘π‘šπ‘’π‘›π‘‘πΏπ‘–π‘”β„Žπ‘‘π‘–π‘›π‘” π‘π‘œπ‘›π‘‘π‘œπ‘™

(5.6)

When the base case is integrated with the PV and solar water heating systems, renewable system component given in equation 5.7 is also added to the global cost.

βˆ‘ π‘π‘ƒπ‘‰π‘…π‘’π‘›π‘’π‘€π‘Žπ‘π‘™π‘’ = π‘π‘ƒπ‘‰π‘…π‘’π‘›π‘’π‘€π‘Žπ‘π‘™π‘’π‘ƒπ‘‰

𝑛

1

+ π‘π‘ƒπ‘‰π‘…π‘’π‘›π‘’π‘€π‘Žπ‘π‘™π‘’π‘†π‘Šπ» (5.7)

The renewable system component includes net present value of ownership of selected and sized renewable system equipment, namely photovoltaic and solar thermal collectors, and rest of the supporting equipment required for a successful implementation. The net present value covers initial, installation, maintenance, replacement and disposal costs of each equipment.

5.2.3.2 Penalty function components

There are four penalty functions, which are thermal comfort, CO2 emission rate, equipment capacity and payback period of renewables, used to restrict the design space to a user-defined eligible region.

Equipment capacity

The methodology requires the HVAC equipment to be sized first through a sizing calculation then, the optimization attempts to select a suitable equipment from the equipment database that can satisfy the autosized capacities while performing well at on and off-reference conditions. Sizing factors are applied to determine an allowable capacity range.

In the case study, for boiler equipment, the capacity lower limit factor is set as 0.99, and the capacity upper limit is chosen as 1.25.

Thus, the capacity penalty equation for boiler takes the form in equation 5.8.

π‘ƒπΈπ‘πΆπ‘Žπ‘π‘Žπ‘π‘–π‘‘π‘¦π΅π‘œπ‘–π‘™π‘’π‘Ÿ

= πœ‡π‘π‘™π‘šπ‘Žπ‘₯(π‘šπ‘Žπ‘₯(0, (π΅πΏπΆπ‘Žπ‘π‘‘π‘’π‘Žπ‘™βˆ’ π΅πΏπΆπ‘Žπ‘’π‘‘π‘œπ‘ π‘–π‘§π‘’βˆ— 1.25 )))π‘ž + πœ‡π‘π‘™π‘šπ‘–π‘›(π‘šπ‘Žπ‘₯(0, (π΅πΏπΆπ‘Žπ‘’π‘‘π‘œπ‘ π‘–π‘§π‘’βˆ— 0.99 βˆ’ π΅πΏπΆπ‘Žπ‘π‘‘π‘’π‘Žπ‘™)))q

(5.8)

Similarly, the capacity lower limit factor is set as 0.99, and the capacity upper limit is chosen as 1.15 for chiller equipment. Therefore, the capacity penalty equation for chiller is expressed as in equation 5.9.

π‘ƒπΈπ‘πΆπ‘Žπ‘π‘Žπ‘π‘–π‘‘π‘¦πΆβ„Žπ‘–π‘™π‘™π‘’π‘Ÿ

= πœ‡π‘π‘™π‘šπ‘Žπ‘₯(π‘šπ‘Žπ‘₯(0, (πΆπΏπΆπ‘Žπ‘π‘‘π‘’π‘Žπ‘™βˆ’ πΆπΏπΆπ‘Žπ‘’π‘‘π‘œπ‘ π‘–π‘§π‘’βˆ— 1.15 )))q + πœ‡π‘π‘™π‘šπ‘–π‘›(π‘šπ‘Žπ‘₯(0, (πΆπΏπΆπ‘Žπ‘’π‘‘π‘œπ‘ π‘–π‘§π‘’βˆ— 0.99 βˆ’ πΆπΏπΆπ‘Žπ‘π‘‘π‘’π‘Žπ‘™)))π‘ž

(5.9)

Penalty parameters πœ‡π‘π‘™π‘šπ‘Žπ‘₯, πœ‡π‘π‘™π‘šπ‘–π‘›, πœ‡π‘π‘™π‘šπ‘Žπ‘₯, πœ‡π‘π‘™π‘šπ‘–π‘› and penalty power factor q are determined in the pre-optimization phase based on design of experiments.

The application of capacity constraints makes sure that optimization selects right-sized equipment.

Thermal Comfort

In thermal comfort penalty function, the target thermal comfort metric is chosen according to European standard EN 15251. The standard indicates four categories of state of comfort for mechanically heated and cooled buildings through PMV and PPD metrics, as shown in Table 5.12.

Table 5.12 : Recommended categories for design of mechanically heated and cooled buildings according to EN 15251.

Category PPD % PMV

I (high level of expectation), < 6 -0.2 < PMV < + 0.2 II (normal level of expectation), <10 -0.5 < PMV < + 0.5 III (moderate level of expectation), <15 -0.7 < PMV < + 0.7 IV (acceptable only for a limited

part of the year). >15 PMV < - 0.7;

or + 0.7 < PMV

β€œCategory II: normal level of expectation” is taken to define the boundaries of the comfort zone, therefore the target PPD index is determined as 10 per cent. Equation 5.10 introduces the penalty formula used for the case study application.

π‘ƒπΈπ‘πΆπ‘œπ‘šπ‘“π‘œπ‘Ÿπ‘‘= πœ‡π‘π‘“(π‘šπ‘Žπ‘₯(0, (π‘ƒπ‘ƒπ·π‘Žπ‘π‘‘π‘’π‘Žπ‘™βˆ’ 10)))π‘ž (5.10)

The PDD index of actual building is computed for each hour of the occupancy work schedule through the year for each thermal zone. For simplification, hourly PPD indices are averaged for the whole year. Then, an area-weighted average PPD of all zones is calculated to represent the comfort conditions in the entire building as given in equation 5.11.

π‘ƒπ‘ƒπ·π΄π‘π‘‘π‘’π‘Žπ‘™ =βˆ‘9𝑛=1𝑃𝑃𝐷 π‘›π΄π‘£π‘”βˆ— π‘π‘œπ‘›π‘’π΄π‘Ÿπ‘’π‘Ž 𝑛

βˆ‘ π‘π‘œπ‘›π‘’π΄π‘Ÿπ‘’π‘Ž91 𝑛 (5.11)

Penalty parameter πœ‡π‘π‘“ and power factor q are determined in the pre-optimization phase based on design of experiments.

CO2 emission rate

In CO2 emission penalty function, a penalty is applied to force the optimum solution into the target zone when the emitted overall building CO2 emission rate exceeds a user-set target.

In the case study, 10 per cent reduction in building annual CO2 emission is aimed to be achieved therefore the final formula becomes as in equation 5.12.

π‘ƒπΈπ‘πΈπ‘šπ‘–π‘ π‘ π‘–π‘œπ‘› = πœ‡π‘’π‘š(π‘šπ‘Žπ‘₯(0, (𝐢𝑂2π‘Žπ‘π‘‘π‘’π‘Žπ‘™βˆ’ 𝐢𝑂2π‘π‘Žπ‘ π‘’βˆ— 0.9)))q (5.12)

The actual amount of CO2 released from base case building due to energy consumption is obtained through application of appropriate carbon dioxide equivalent intensity indexes for each energy carrier. In this study, CO2 emission factor is set at 0.234 kg.eqCO2/kWh for natural gas and 0.617 kg.eqCO2/kWh for electricity in compliance with published national data by Ministry of Environment and Urbanization of Turkey.

Therefore, the actual amount of CO2 emission rate is formulated as in equation 5.13:

𝐢𝑂2π΄π‘π‘‘π‘’π‘Žπ‘™ = 0.617 βˆ— βˆ‘ πΈπ‘™π‘’π‘π‘‘π‘Ÿπ‘–π‘π‘¦ + 0.234 βˆ— βˆ‘ π‘π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™ πΊπ‘Žπ‘  (5.13)

Penalty parameter ΞΌem and power factor q are determined in the pre-optimization phase based on design of experiments.

Payback period

The payback period of a renewable system investment is limited according to a user set time-period. In the case study, 25 years period is chosen as maximum payback limit both for photovoltaic and solar thermal system applications and the related formula is expressed as in equation 5.14. The reason to choose a long period as payback time is to have an opportunity to observe the payback behaviour of renewable systems under different climatic conditions within building life-cycle and to compare climate responsive performances without eliminating systems unless they are not beneficial within the building life.

π‘ƒπΈπ‘π‘π‘Žπ‘¦π‘π‘Žπ‘π‘˜ 𝑃𝑉/𝑆𝐢 = πœ‡π‘π‘π‘ƒπ‘‰/𝑆𝐢(π‘šπ‘Žπ‘₯(0, (π‘†π‘ƒπ΅π‘Žπ‘π‘‘π‘’π‘Žπ‘™βˆ’ 25)))q (5.14)

Penalty parameter πœ‡π‘π‘ and power factor q are determined in the pre-optimization phase based on design of experiments.

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