## 3 Problem Formulation of Total Optimization of Energy Networks in a Smart City
### Objective Function  
The objective function of the smart city problem considers three terms. The function considers minimization of energy costs of whole smart city as the first term. Actual electric power at high load hours are minimized by the second term of the function. The function also considers minimization of CO\({}_{2}\) emission in the whole city as the third term. The functions are shown in the following Equation (1):  
\[min\left\{w_{1}\sum\limits_{n=1}^{NumSec}\sum\limits_{j=1}^{T}\left(pg_{ni} \right)\times GU_{nj}+p_{nj}\times EU_{nj}\right)+w_{2}\sum\limits_{n=1}^{NumSec} \sum\limits_{m=-PS}^{PE}\left(el_{mn}\right)+w_{3}\sum\limits_{n=1}^{NumSec} \sum\limits_{j=1}^{T}\left(pg_{nj}\times GC+p_{nj}\times EC\right)\right\} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sector** & **Decision Variables** \\ \hline Industrial sector & Output of electric power of a gas turbine generator (GTG), Heat output of turbo refrigerators (TRs), Heat output of stream refrigerators (SRs), Charged or discharged electric power of a storage battery (SB) \\ \hline Building sector & Output of electric power of a GTG, Heat output of TRs, Heat output of SRs \\ \hline Residential sector & Heat output of SRs, Output of electric power of a fuel cell, Heat output of a heat pump water heater, Charged or discharged electric power of a SB \\ \hline Railroad sector & The number of passengers/h, Average of journey distance by one passenger/h, The number of operated trains/h, The numbers of passenger cars/set, Average of journey distance by one train/h, Average of speed/h, The number of passengers/car \\ \hline Drinking water & Inflow from river, Inflow of water into a service reservoir, Output of electric power \\ treatment plant sector & output of a co-generator (CoGen), Charged or discharged electric power of a SB \\ \hline Wastewater treatment & Input of Pumped wastewater, Output of electric power of a CoGen, Charged or \\ plant sector & discharged electric power of a SB \\ \hline \hline \end{tabular}
\end{table}
Table 1: Decision variables in various sectors.  
Figure 2: A configuration of an industrial model (© 2018 IEEE [27]).  
where \(NumSec\) is the number of sectors except electric power and natural gas utilities, \(T\) is the number of hours per day (=24), \(pg_{nj}\) is natural gas purchase at hour \(j\) of sector \(n\), \(GU_{nj}\) is a natural gas purchase unit cost at hour \(j\) of sector \(n\), \(pe_{nj}\) is electric power purchase at hour \(j\) of sector \(n\), \(EU_{nj}\) is an electric power purchase unit cost at hour \(j\) of of sector \(n\), \(el_{nm}\) is an actual electric power load at hour \(m\) of sector \(n\), \(PS\) is the start hour of peak load hours of actual electric power loads, \(PE\) the end hour of peak load hours of actual electric power loads, \(GC\) is a coefficient between the purchased natural gas and \(CO_{2}\) emission, \(EC\) is a coefficient between purchased electric power and \(CO_{2}\) emission, and \(w_{1}\), \(w_{2}\), and \(w_{3}\) are weighting coefficients (\(w_{1}+w_{2}+w_{3}=1\)).  
Authors include one of a member of a smart city model committee in IEE of Japan. In the committee, a scope of a smart city has been discussed in order to consider a closed community such as industrial parks and local governments. In such a case, purchase costs of various fuels by electric power and gas companies from resource companies are not out of the scope of the city and we follow the scope in (1). After decision variables are calculated, the amounts of natural gas and electric power purchase are calculated as dependent variables. Hence, when the dependent variables violate limits of the constraints, penalty function values are calculated and they are added to the objective function values.