## 4 Problem Formulation
### Objective Function
#### 4.1.2 Cooling Power  
The cooling device power consumption contributes to the maximum electricity consumption of the data center. Dynamic tuning of the cooling load based on the current workload may help reduce power consumption in data centers, which will have a direct impact on the PUE. The cooling power cannot be ignored, as it prevents service disruption caused by the heat generated by servers [49]. To analyze the power consumption of a cooling device, standard computer room air conditioning (CRAC) units are considered in this work. The power consumption of the chiller does not change much with regard to the outside air temperature or IT load [50]. The coefficient of performance (CoP) is the measure used to compute the efficiency of the cooling unit to determine its cooling load. The CoP is the ratio (\(d\)/\(w\)) of heat removed (for server load \(d\)) to the quantity of work (\(w\)) needed to remove the heat. A larger CoP indicates better efficiency, meaning less work is required to remove a greater amount of heat. The CoP of the CRAC unit is a changeable value that increases in proportion to the increase of the supply air temperature in the CRAC unit [51].  
The total carbon footprint (TCF) generated at time \(t\), including overhead power, is formulated as:  
\[TCF(t)=\sum_{d=1}^{tdc}(PUE_{d}\times\sum_{c=1}^{tc}(CFR_{d,c}\times PS_{d,c})) \tag{7}\]  
where \(tdc\), \(tc\), \(M\) and \(N\) represent the numbers of data centers, clusters, machines, and requests, respectively. The overall energy consumption of all servers in a cluster (PSc) within the interval \([0,T]\), partitioned as a sequence of reservation intervals (\(ri\)) in the form of (\(t_{ri}\), \(t_{ri+1}\)) (\(ri\) \(\in\) {0,1,... _ri-1_}), is formulated as:  
\[PS_{c}=\sum_{j=1}^{M}\sum_{\alpha=0}^{ri-1}\sum_{l=0}^{k}\times P_{j,l}(t_{ \alpha+1})\times(t_{\alpha+1}-t_{\alpha}) \tag{8}\]  
The CoP for the CRAC unit can be modeled as in [52]:  
\[CoP=\left\{0.0068T_{sup}^{2}+0.0008T_{sup}+0.458\right\} \tag{9}\]  
where \(T_{\text{sup}}=\) (current_temperature - safe_temperature)  
The PUE of the data center can be calculated as:  
\[PUE_{d}=\left(\frac{Total\ facility\ Power}{IT\ power}\right) \tag{10}\]  
\[\text{Total facility Power}_{d}=\left(\sum_{c=1}^{lc}PS_{c}+\sum_{c=1}^{lc} OP_{c}\right) \tag{11}\]  
\[IT\ Power_{d}=\sum_{c=1}^{lc}PS_{c} \tag{12}\]  
The total overhead power (\(OP\)) of a cluster (\(c\)) is calculated as:  
\[OP_{c}=\left(\frac{PS_{c}}{CoP}\right) \tag{13}\]  
The objective function \(TCF(t)\) is subject several limitations. The total number of VMs allocated to a machine should not exceed the servers computing (\(U\)) and memory capacity (\(mem\)), as follows:  
\[\sum_{i=1}^{N}R_{j,i}{}^{U}\leq PM_{j}^{mem,max} \tag{14}\]  
\[\sum_{i=1}^{N}R_{j,i}^{mem}\leq PM_{j}^{mem,max} \tag{15}\]  
The relation \(R\) between VMs and PMs is many-to-one, meaning \(R\subseteq N\times M\) if:  
\[\forall\ i\ e\ N\ \&\forall\ j,k\ e\ M:(i,j)eR\wedge(i,k)eR\Rightarrow j=k \tag{16}\]  
The total energy (\(eng\)) consumed is supposed to be within the limit of the available brown energy (\(B\)) at the data center, as follows:  
\[\sum_{i=1}^{N}R_{j,i}^{eng}\leq Total\ available\ B \tag{17}\]  
The total brown energy consumed is supposed to be within the limits of the cloud provider's agreed upon grid electricity consumption (\(G\)):  
\[Total\ available\ B\leq Total\ assigned\ G \tag{18}\]