Penalty functions

In the current study, the constraints regarding the HVAC equipment sizing, greenhouse gas emissions, indoor comfort levels and renewable system payback period are imposed in the form of penalty functions.

The equation 4.13 shows the elements of main penalty function:

∑ 𝑃𝐸𝑁 = 𝜇_𝑒𝑐𝑃𝐸𝑁_{𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦}+ 𝜇_𝑒𝑚𝑃𝐸𝑁_{𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛} + 𝜇_𝑐𝑓𝑃𝐸𝑁_{𝐶𝑜𝑚𝑓𝑜𝑟𝑡}

+ μ_pbPEN_PayBack (4.13)

Equipment capacity

In the current study, ideal primary side HVAC equipment is aimed to be selected via optimization with rest of the design variables from a database, simultaneously. As explained previously, the optimization algorithm firstly combines design variables, runs a design day simulation, and determines the required equipment loads. Then, it tries to assess the annual performance at the same instance. In the equipment library, there exists a wide range of equipment with varying capacities and dynamic performances. Therefore, to prevent a capacity mismatch between the recommended equipment’s actual capacity and the required capacity occurs due to new combination of design variables, a penalty is added to the main objective every time an equipment violates sizing rules set by the designer. The calculation steps of the equipment capacity penalty function are depicted in Figure 4.9.

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Figure 4.9 : Equipment capacity penalty value calculation algorithm.

The penalty calculation formula is based on equation 4.14.

𝐸𝐶_{𝑎𝑢𝑡𝑜𝑠𝑖𝑧𝑒}∗ 𝑆𝐹_{𝐿𝑜𝑤𝑒𝑟} ≤ 𝐸𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙} ≤ 𝐸𝐶_{𝑎𝑢𝑡𝑜𝑠𝑖𝑧𝑒}∗ 𝑆𝐹_{𝑈𝑝𝑝𝑒𝑟} (4.14) Where,

𝐸𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙} : Capacity of the actual equipment in database,

𝐸𝐶_{𝑎𝑢𝑡𝑜𝑠𝑖𝑧𝑒} : Required equipment capacity determined via autosizing calculation, 𝑆𝐹_{𝐿𝑜𝑤𝑒𝑟} : User-defined sizing factor to determine undersizing limit,

𝑆𝐹_{𝑈𝑝𝑝𝑒𝑟} : User-defined sizing factor to determine oversizing limit.

Therefore, the penalty function for equipment capacity becomes as following:

𝑃𝐸𝑁_{𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦} = 𝜇_{𝑚𝑎𝑥𝑐𝑎𝑝}(𝑚𝑎𝑥 (0, (𝐸𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙}− 𝐸𝐶_{𝑎𝑢𝑡𝑜𝑠𝑖𝑧𝑒}∗ 𝑆𝐹_{𝑈𝑝𝑝𝑒𝑟} )))^𝑞 + 𝜇_{𝑚𝑖𝑛𝑐𝑎𝑝}(𝑚𝑎𝑥(0, (𝐸𝐶_{𝑎𝑢𝑡𝑜𝑠𝑖𝑧𝑒}∗ 𝑆𝐹_{𝐿𝑜𝑤𝑒𝑟}− 𝐸𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙})))^𝑞

(4.15)

Where,

𝑃𝐸𝑁_{𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦} : Calculated penalty for being above or below user-set capacity limits, μ_maxcap : User-assigned maximum equipment capacity penalty parameter, μ_mincap : User-assigned minimum equipment capacity penalty parameter, 𝑞 : Nonnegative constant as penalty power factor.

CO2 emission

A good environmental performance of a building is aimed to be assured by setting a minimum-achievable performance target in the form of penalty. Therefore, while optimization searches for the optimum combination of design variables in terms of economic viability, it also makes sure the proposed building emits less than a target level during operational phase.

There are several different types of greenhouse gases with varying levels of global warming potential. The major ones are carbon dioxide, water vapour, methane, and nitrous oxide; however, in this study the target emission is restricted only to CO2

because CO2 remains in the atmosphere longer than the other major heat-trapping gasses and is the dominant source of global warming.

The metric used in the penalty function equation is described as the overall annual amount of carbon dioxide equivalence emitted by the building in kg due to the operational energy consumption from different energy sources. When the emitted overall CO2 emission exceeds the target, a penalty which is calculated according to steps illustrated in Figure 4.10 is added to the main objective function.

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Figure 4.10 : CO₂ emission penalty value calculation algorithm.

The equation 4.16 describes mathematically the penalty formulation.

𝑃𝐸𝑁_{𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛} = 𝜇_𝑒𝑚(𝑚𝑎𝑥 (0, (𝐶𝑂2_{𝑎𝑐𝑡𝑢𝑎𝑙}− 𝐶𝑂2_{𝑡𝑎𝑟𝑔𝑒𝑡})))^𝑞 (4.16)

Where,

PEN_Emission : Penalty value due to violation of CO₂ emission criteria, 𝐶𝑂2_{𝑎𝑐𝑡𝑢𝑎𝑙} : Proposed building overall CO2 emission amount, 𝐶𝑂2_{𝑡𝑎𝑟𝑔𝑒𝑡} : User set overall CO₂ emission target,

𝜇em : User-assigned CO2 emission penalty parameter, q : Nonnegative constant as penalty power factor.

The overall building CO2 emission amount, for either actual case or base case, is a summation of CO2 emission due to different energy sources used in the building and is calculated according to the equation 4.17:

𝐶𝑂2_{𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛} = ∑ 𝐶𝐼_𝑖𝐸𝑛_𝑖

𝑖

1

(4.17)

Where,

𝐶𝑂2_{𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛} : Overall building CO₂ emission amount,

𝐶𝐼_𝑖 : Carbon dioxide equivalent intensity index in kg.EqCO2/kWh for each available energy source,

𝐸𝑛_𝑖 : Energy consumptions in different fuel forms.

The carbon dioxide equivalent intensity indexes are determined by public bodies according to the nature of the national energy market.

User thermal comfort

When performing a building design optimization, it is also crucial to maintain thermal comfort in the building. For instance, if the thermal comfort is not included in the calculations, it is very likely that the design that turns up as cost-effective, could lead to overheating or underheating problems. Therefore, in the current study, thermal comfort is added to the objective function as a penalty to make sure that design alternatives, which violate a user-set thermal comfort criterion is eliminated from design alternatives and the solution region is restricted to a comfort zone.

The penalty function for thermal comfort is defined mathematically as following:

𝑃𝐸𝑁_{𝐶𝑜𝑚𝑓𝑜𝑟𝑡} = 𝜇_𝑐𝑓(𝑚𝑎𝑥 (0, (𝑇𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙}−𝑇𝐶_{𝑡𝑎𝑟𝑔𝑒𝑡})))^𝑞 (4.18)

Where,

𝑃𝐸𝑁_{𝐶𝑜𝑚𝑓𝑜𝑟𝑡} : Penalty value due to violation of comfort criteria, 𝑇𝐶_{𝑡𝑎𝑟𝑔𝑒𝑡} : Target thermal comfort metric set by designer,

𝑇𝐶_{𝑎𝑐𝑡𝑢𝑎𝑙} : Calculated thermal comfort metric for proposed building,

𝜇_𝑐𝑓 : User-assigned weighting factor for thermal comfort penalty function, 𝑞 : Nonnegative constant.

Thermal comfort can be defined as ‘that condition of mind which expresses satisfaction with the thermal environment’ (EN ISO 7330, 2006). The determination

of thermal comfort level is not straight forward since it results from a combination of environmental factors and personal factors including air and radiant temperature, humidity, air velocity, activity level of occupant and clothing insulation. There are many techniques available for estimating likely thermal comfort. In this study, however, Predicted Mean Vote (PMV) and Percentage People Dissatisfied (PPD) is adapted as suggested by EN ISO 7730 (2006), EN ISO 15251 (2007) and ASHRAE 55 (2004) standards. PPD is a quantitative measure of the thermal comfort of a group of people at a particular thermal environment and described as the percentage of occupants that are dissatisfied with the given thermal conditions. PPD is calculated according to equation 4.19 given in EN ISO 7330.

𝑃𝑃𝐷 = 100 − 95𝑒−(0.03353𝑃𝑀𝑉⁴+0.2179𝑃𝑀𝑉²) (4.19) The PPD can be deduced from the Predicted Mean Vote (PMV) as suggested in EN ISO 7730 given in equation 4.20:

𝑃𝑀𝑉 = (0.303𝑒^{−0.036𝑀𝑒𝑡}+ 0.28)(𝐻 − 𝐿) (4.20) Where,

𝑀𝑒𝑡 : Metabolic rate,

𝐻 : Internal heat production rate of an occupant per unit area, 𝐿 : All the modes of energy loss from body.

PMV is representative of what a large population would think of a thermal environment using a seven-point thermal sensation scale. It is derived from the physics of heat transfer and empirical correlations.

Accordingly, when the thermal comfort criterion is taken as PDD index, the penalty function takes the following mathematical form:

𝑃𝐸𝑁_{𝐶𝑜𝑚𝑓𝑜𝑟𝑡}= 𝜇_𝑐𝑓(𝑚𝑎𝑥 (0, (𝑃𝑃𝐷_{𝑎𝑐𝑡𝑢𝑎𝑙}−𝑃𝑃𝐷_{𝑡𝑎𝑟𝑔𝑒𝑡})))^𝑞 (4.21)

Where,

𝑃𝐸𝑁_{𝐶𝑜𝑚𝑓𝑜𝑟𝑡} : Penalty value due to violation of comfort criteria, 𝑃𝑃𝐷_{𝑎𝑐𝑡𝑢𝑎𝑙} : calculated PPD index for proposed building,

𝑃𝑃𝐷_{𝑡𝑎𝑟𝑔𝑒𝑡} : Target PPD index set by designer, q : nonnegative constant.

The PMV-PPD indices are included in the national and international thermal comfort standards. Therefore, the designer can select the target PPD metric according to recommended values and can define the boundaries of the comfort zone.

The PDD index of actual building is however computed through building simulation at each optimization step. For multi-zone buildings, PDD is calculated for each zone during occupied times and then each PPD can be used as an individual comfort penalty otherwise an average PPD of all zones representing the whole building can be adopted.

Figure 4.11 represents calculations steps for comfort penalty through an average PPD index approach.

Figure 4.11 : User thermal comfort penalty value calculation algorithm.

Setting up a thermal comfort metric requires taking into account a range of environmental and personal factors however in the current study, it is assumed that all environmental factors other than air temperature and radiant temperature are constant. Commonly, control strategies are implemented in building simulation to maintain air temperatures within standard-defined comfort limits. However, in the optimization study HVAC plant equipment is selected from the equipment library based on a capacity calculation. Therefore, a capacity mismatch can be prevented through comfort criteria check, too. Moreover, radiant temperature is influenced a great deal by the change in building envelope design variables and thermal comfort can be improved based on radiant temperature.

Payback period for renewable systems

The payback period is the time in which the initial cash outflow of an investment is expected to be recovered from the cash inflows generated by the investment.

Therefore, payback period measures the time required to recover initial investment costs. The payback period of a given investment is an important measure of whether or not to undertake the investment, since longer payback periods are typically not desirable for investors.

In the current study, a penalty is added to the main objective to set a limit on the payback period of a considered renewable system based on designer’s expectancy.

The simple payback method is used to calculate payback period as explained in

Figure 4.12 : Renewable payback period penalty value calculation algorithm.

The calculation algorithm is based on the equation 4.22.

𝑃𝐸𝑁_{𝑝𝑎𝑦𝑏𝑎𝑐𝑘} = 𝜇_𝑝𝑏(𝑚𝑎𝑥 (0, (𝑆𝑃𝐵_{𝑎𝑐𝑡𝑢𝑎𝑙}− 𝑆𝑃𝐵_{𝑡𝑎𝑟𝑔𝑒𝑡})))^𝑞 (4.22)

Where,

𝑃𝐸𝑁_{𝑝𝑎𝑦𝑏𝑎𝑐𝑘} : Penalty value due to violation of payback time criteria, 𝑆𝑃𝐵_{𝑎𝑐𝑡𝑢𝑎𝑙} : Calculated simple payback index for proposed building, 𝑆𝑃𝐵

𝜇_𝑝𝑏 : Payback period penalty parameter, 𝑞 : Nonnegative constant.

The simple payback (SPB) is formulated as in equation 4.23 for renewable system investments in the study (Fuller and Petersen, 1995):

𝑆𝑃𝐵 = 𝑑𝐼₀

[𝑑𝐸₀+ 𝑑𝑀₀] (4.23)

Where,

𝑑𝐼₀ : Additional investment cost, 𝑑𝐸₀ : Savings in energy cost in year t,

𝑑𝑀₀ : Difference in maintenance cost in year t.

SBP is a practical method and it does not use discounted cash flows in the payback calculation. For instance, dE and dM are assumed to be the same every year, which means price escalation is not taken into account. Moreover, non-annually recurring additional costs such as replacements costs are ignored in SPB, too.