# Period Cycle Optimization of Integrated Energy Systems with Long-Term Scheduling Consideration
## 2 Materials and Methods
### System's Optimization Algorithms
#### 2.1.2 Constraint Conditions  
In this system, the load ratios of various components are constrained by the balance of different resources, power generation, energy storage, and supply. For solar photovoltaic panels, their electrical output is nonlinearly correlated with solar radiation, which can be described by the following calculation formula:  
\[pv_{elec}=pv_{capa}\times lr=pv_{capa}\times f(si) \tag{9}\]  
where \(pv\), \(hm\), \(br\), \(cu\), \(gb\), \(pg\), \(ng\), \(wt\), and \(hp\) are the photovoltaics, refrigeration unit, lithium bromide absorption chiller, CHP, gas boiler, power grid, natural gas station, wind turbine, and ground source heat pump, \(elec\), \(heat\), \(cold\), and \(gas\) represent electricity, heat, cool, and gas loads, \(capa\) is the capacity of the energy component, \(si\) is solar radiation in kW, and \(sp\) is wind speed in m/s. For the wind turbine, its electrical output is nonlinearly related to wind speed. \[wt_{elec}=wt_{capa}\times lr=wt_{capa}\times f(sp) \tag{10}\]  
For the ground source heat pump, its electrical input is nonlinearly related to its load rate. \[hp_{elec}=\left(hp_{capa}\times lr\right)/f(lr) \tag{11}\]  
For airborne main engines and double-working condition main engines, their electrical input is nonlinearly related to their load rate. \[hm_{elec}=\left(hm_{capa}\times lr\right)/f(lr) \tag{12}\]  
For lithium bromide refrigeration machines, their heat input is nonlinearly related to their load rate. \[br_{heat}=\left(br_{capa}\times lr\right)/f(lr) \tag{13}\]  
For CHP and gas boilers, their gas flow rate is nonlinearly related to their load rate. \[cu_{gas}=\left(cu_{capa}\times lr\right)/f(lr) \tag{14}\]  
\[gb_{gas}=\left(gb_{capa}\times lr\right)/f(lr) \tag{15}\]  
The power grid can be regarded as a flexible generator with constant efficiency, and its output can be calculated with Equation (22), with the output power not exceeding its maximum carrying capacity. \[pg_{out}\leq max(pg_{capa},pg_{out}) \tag{16}\]  
where \(pg_{out}\) is the electrical output power from the grid in kWh. The CHP and gas boilers are dispatchable variables; the output power of the natural gas station is calculated with Equation (25). \[ng_{out}\leq max(ng_{capa},ng_{out}) \tag{17}\]  
where \(pg_{out}\) represents the natural gas flow rate, and the gas flow rate (m\({}^{3}\)/h) is converted into power (kW) by using the calorific value of natural gas (kWh/m\({}^{3}\)). The energy storage and output power within a certain period can be expressed as shown in Equation (18), with the maximum energy storage and power transmission conditions subjected to the constraints of Equations (20) and (21). \[st_{i}=\frac{st_{i}^{c}}{\eta^{c}}\delta_{c}\Big{(}st_{i}^{total}\Big{)}-st_{i }^{d}\eta^{d}\delta_{d}\Big{(}st_{i}^{total}\Big{)} \tag{18}\]  
\[wt_{elec}=wt_{capa}\times lr=wt_{capa}\times f(sp) \tag{10}\]0  
\[wt_{elec}=wt_{capa}\times lr=wt_{capa}\times f(sp) \tag{10}\]1\[wt_{elec}=wt_{capa}\times lr=wt_{capa}\times f(sp) \tag{10}\]2  
where \(st_{i}^{total}\) is the total stored energy of each piece of energy storage equipment, \(st_{i}\) is the energy storage power within a certain time period, \(\eta^{c}\) is the energy storage efficiency of the energy storage component, \(\eta^{d}\) is the energy transfer efficiency of the energy storage component, \(t\) is the duration of energy storage and transfer for the energy storage component, \(t\) is the number of components, and \(\delta_{c}\) and \(\delta_{d}\) denote the efficient storage and output energy related to the total energy storage, respectively. In the integrated energy systems, electricity is the most commonly used energy source, and its energy structure is also the most complex. The electricity balance equation for it is as follows:  
\[wt_{elec}=wt_{capa}\times lr=wt_{capa}\times f(sp) \tag{10}\]3  
where \(ld\) is the power consumed by the user, which refers to the power inputted into the user load. Its unit is kW.