Optimization Model

𝑎𝑟𝑔 _𝑚𝑖𝑛 𝑋∗∈ 𝑅𝑛

𝐹(𝑋)=[𝐷𝐶(𝑋), E(𝑋)]𝑇

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑔𝑙(𝑋)

(6-1)

where 𝑋∗ _{∈ 𝑅}𝑛 _{is the optimal decision variables vector, and 𝐹(𝑋) = [𝐷𝐶(𝑋), E(𝑋)]}𝑇 _{is the vector}

of the discomfort (𝐷𝐶) and energy (𝐸) objective functions. 𝑔𝑙(𝑋) is the vector of the constraint

functions that could take into account the regulations and the occupants’ preferences regarding the HVAC and lighting systems’ settings. The inputs to the optimization model are the sensor data as well as the information from the occupancy module. These inputs are the environmental conditions (i.e., the HVAC and lighting systems’ settings), occupants’ preferences and their dynamic profiles. In this study, cooling and heating set-points for each zone along with the illuminance level of each zone are the optimization decision variables. In this study, minimizing discomfort hours (all clo), which is the total discomfort hours when winter or summer clothes are worn (ASHRAE 55, 2010) and minimizing energy consumption are the ultimate goals of the proposed integrated model. The level of occupants’ satisfaction regarding the thermal environmental conditions defines the thermal comfort of occupants. Several mathematical models have been proposed exploring the correlation between thermal comfort variables to predict the thermal satisfaction of occupants. Among these models, the Graphic Comfort Zone method suggested by ASHRAE Standard 55-2010 (ASHRAE 55, 2010) is used in this study to measure the discomfort hours. Based on this method, the total number of discomfort hours is calculated based on whether the humidity ratio and the operative temperature are within the regions provided in ASHRAE Standard 55-2010. These regions are derived from the Predicted Mean Vote (PMV) and Predicted Percent Dissatisfied (PPD) indices developed by Fanger (1972). According to ASHRAE 55, the PMV index between +0.5 and -0.5 can be used as an indication of the thermally comfortable environment when setting the zone cooling and heating temperatures (ASHRAE 55, 2010; Sehar et al., 2017). Hence, the optimal control strategies override the normal heating and cooling set-point temperatures at zone level to maintain the PMV within the comfort range. Therefore, unlike global set-point adjustment, occupants’ thermal requirements can be met at zone level resulting in fewer discomfort hours.

The energy (𝐸) objective function is defined as a combination of the energy consumption of the HVAC system (𝐸_𝐻), which is the summation of cooling (𝐶𝐸_𝑡,𝑧𝑠 _{) and heating (𝐻𝐸}

𝑡,𝑧

𝑠 _{) power}

consumption, and lighting (𝐸_𝐿) system as shown below:

𝐸(𝑋) = 𝑀𝑖𝑛(𝐸𝐻(𝑋) + 𝐸𝐿(𝑋)) (6-2) 𝐸𝐻(𝑋)=∑ ∑ ∑(𝐶𝐸_𝑡,𝑧𝑠 𝑇𝑂𝑇 𝑡=1 𝑍 𝑧=1 𝑆 𝑠=1 + 𝐻𝐸𝑡,𝑧𝑠 ) (6-3) 𝐸_𝐿(𝑋)=∑ ∑ ∑𝐿𝑃_{𝑡 ,𝑧}𝑠 𝑇𝑂𝑇 𝑡=1 𝑍 𝑧=1 𝑆 𝑠=1 (6-4)

The HVAC system heating/cooling load is dependent on the internal heat gains (𝐼𝐻𝐺) including gains from occupants (𝐼𝐻𝐺_𝑜𝑐𝑐), lighting (𝐼𝐻𝐺_𝑙), and equipment (𝐼𝐻𝐺_𝑒𝑞). The cooling energy at time-step t at zone z during season s (𝐶𝐸𝑡,𝑧𝑠 ) is calculated using Equation (6-5) (EnergyPlus, 2015),

where 𝐶𝑂𝑃 is the coefficient of performance of the system. Focusing on occupancy, the summation of the latent (𝑄𝐿𝑜𝑐𝑐) and sensible heat gains (𝑄𝑆𝑜𝑐𝑐) comprises the total occupancy heat gain and

are calculated as follows (Thomas, 2018):

𝐶𝐸_𝑡,𝑧𝑠 =𝑐𝑜𝑜𝑙𝑖𝑛𝑔 𝑙𝑜𝑎𝑑𝑠⁄_𝐶𝑂𝑃 (6-5)

𝑄𝐿𝑜𝑐𝑐= 𝐶𝑜𝑢𝑛𝑡𝑧× 𝐿𝐻𝐺𝑜𝑐𝑐 (6-6)

𝑄𝑆_𝑜𝑐𝑐 = 𝐶𝑜𝑢𝑛𝑡_𝑧× 𝑆𝐻𝐺_𝑜𝑐𝑐× 𝐶𝐿𝐹 (6-7)

Where 𝐶𝑜𝑢𝑛𝑡_𝑧 is the number of occupants at each time-step 𝑡 at zone 𝑧. 𝐿𝐻𝐺_𝑜𝑐𝑐 and 𝑆𝐻𝐺_𝑜𝑐𝑐 are the latent and sensible heat gain per person for the type of activity performed in the zone, respectively, and are derived from standards. Since part of the sensible heat generated by occupants is absorbed by the surroundings and then gradually released into the zone, a cooling load factor (𝐶𝐿𝐹) is considered when calculating 𝑄𝑆𝑜𝑐𝑐 to reflect this time delay. This factor is also obtained

from standards (e.g., ASHRAE). This factor is not needed for 𝐿𝐻𝐺_𝑜𝑐𝑐, which is instantaneously added to the zone (EnergyPlus, 2015). The same concepts are used to calculate the heating energy. The lighting local control strategies are applied in near-real-time using 1-minute occupancy data. Two different resolution levels of 30- and 60-minutes are used to control the HVAC system, as explained in Section 6.3.4. The optimization results are shown as the summation of zone energy

consumption and the number of discomfort hours in a year using Equations (6-3) and (6-4). This is done to have an overall estimation of the effect of optimal local control strategies on building energy performance and comfort.

The lighting power is calculated using Equation (6-8) (EnergyPlus, 2015):

𝐿𝑃_{𝑡 ,𝑧}𝑠 ₌ 𝐿𝐸𝑧× 𝐼𝑡,𝑧

𝑠 _{× 𝐴}

𝑧

100 (6-8)

where:

𝐿𝑃𝑡 ,𝑧𝑠 : Lighting power at time-step t at zone z during season s (W)

𝐿𝐸_𝑧: Lighting energy at zone z (W/m2_{/100 lux)}

𝐼_𝑡,𝑧𝑠 : Zone illuminance level at time-step t at zone z during season s (lux) 𝐴𝑧: Zone floor area (m2)

The discomfort (𝐷𝐶) objective function is defined as the normalized summation of all discomfort time at all zones:

𝐷𝐶(𝑋) =∑ (𝐴𝑧∑ ∑ 𝐷𝑇𝑡,𝑧 𝑠 𝑇𝑂𝑇 𝑡=1 𝑆 𝑠=1 ) 𝑍 𝑧=1 ∑𝑍 𝐴_𝑧 𝑧=𝑖 (6-9)

where 𝐷𝑇_𝑡,𝑧𝑠 _{is the discomfort time according to ASHRAE55 at time-step t in zone z during}

season 𝑠.

6.3.1.1 Selection of Optimization Algorithm

Optimization problems can be mainly categorized as single objective or multi-objective optimization problems, where the former have only one objective function and the latter have more than one objective function. These objective functions are usually in conflict with each other in real-world engineering optimization problems so that the improvement of one of them leads to worsening the others. Therefore, multi-objective optimization offers the near-optimal set of solutions, which are called Pareto points or Pareto front, rather than a single near-optimal solution. In this set, there is not any answer that dominants the others (Deb, 2005).

Different analytics and heuristic optimization methods have been used by researchers in order to solve the optimization problems related to energy management objectives, such as improving

energy efficiency, reducing energy cost, and increasing the occupants’ comfort. The approximations (or heuristics) algorithms are used when finding exact optimal solutions is not applicable. Although heuristic optimization algorithms find approximate feasible solutions within a reasonable time frame, there is no guarantee of optimality. To overcome this shortcoming, meta- heuristic methods were developed that employ heuristics techniques with guidance through the search space to obtain near-optimal solutions (Mellouk et al., 2015).

Among different meta-heuristic optimization algorithms, the Genetic Algorithm (GA) followed by Particle Swarm Optimization (PSO) are the most used ones in the energy management field (Shaikh et al., 2016). This is due to the capability of GA in solving complex multi-objective optimization problems while maintaining the simplicity of its computational steps. GAs mimic the process of natural selection in order to find proper solutions to optimization problems based on the ideas of the evolutionary theory (Holland, 1975). PSO algorithm is an evolutionary computation technique, which is motivated by the behavior of bird flocks. Similar to GA, the PSO algorithm generates a population of random solutions called particles. However, unlike GA, each particle is also associated with a randomized velocity. Thus, particles fly around a multi-dimensional search space to find out optimal solutions (Shi, 2001; Sun et al., 2004). Based on the literature, while both PSO and GA obtain high-quality solutions, the number of computational steps for GA is lower than that of PSO, which is due to the communication between the particles after each generation (Panda and Padhy, 2008). The difference in computational effort between PSO and the GA is problem-dependent. PSO, in general, outperforms GA for unconstrained nonlinear problems with continuous design variables. However, when applied to highly nonlinear, constrained optimization problems, that are typical for complex energy management problems, GA is more efficient and requires less computational time (Hassan et al., 2005). Therefore, choosing an optimization algorithm with less computational steps, such as GA, would result in producing near-optimal solutions while reducing the complexity of the problem.

Among various multi-objective evolutionary algorithms (MOEAs), the Non-Dominated Sorting Genetic Algorithm (NSGA) was one of the first methods to create Pareto-optimal solutions (Srinivas and Deb, 1994). However, in order to alleviate some of the problems associated with NSGA, a better and faster algorithm, called NSGA-, was introduced a few years later. Computational complexity, lack of elitism, and the need for sharing parameters were some of those

problems (Deb et al., 2002). The new algorithm performs better and faster to find the non- dominated solutions by providing a better distribution of the population. According to (Wang, 2016) the NSGA- is a mature multi-objective optimization algorithm at present. The main advantages of NSGA- includes the flexibility to be applied to a wide range of optimization problems of significant complexity (McCall, 2005; Deb et al., 2002), the simplicity of its computational steps, especially when it is integrated with simulation models, and its ability to effectively solve multi-objective optimization problems. Therefore, NSGA- is selected as the optimization engine in this research.