Model formula

This section summarizes the theoretical formulation of AIM/Technology, which was developed based on AIM/Enduse framework [1] [2]. Here, capital characters refer to endogenous variables, while lowercase characters indicate exogenous parameters. The model determines the technology installation and operating conditions to minimize total energy system costs, which is the sum of initial investment, operating and management costs and emission costs.

List of set elements, parameters, and variables

Sets and indices

Sets and indices

Symbol	Description
\(i \in I\)	sector
\(j \in J\)	service type
\(k \in K\)	energy type
\(l \in L\)	device
\(m \in M\)	gas type
\(r \in R\)	region
\(y \in Y \subset H\)	set of year based on the simulation time horizon \(Y\) as a subset of all years \(H\) including past years. \(y^{*} (\in Y)\) denotes the simulation year.
\(ME_{r,i}\)	set of regions for energy constraint
\(MR_{r,i}\)	set of regions for emission constraint
\(i \in G_m\)	set of gas types for emission constraint
\(N_{j,k}\)	combination of internal service \(j\) and energy \(k\)
\(I_{i,N}\)	combination of region \(i\) and service \(N\) for internal service balance
\(l_m\)	technology group
\(ML_{l_m,l}\)	technology group combination
\(k_m\)	energy group
\(MK_{k_m,k}\)	energy group combination

Endogenous variables

Endogenous variables

Symbol	Description
\(VD_{r,i,j}\)	service output of service type \(j\)
\(VE_{r,i,k}\)	energy demand for energy type \(k\)
\(VQ_{r,i,m}\)	emission of gas type \(m\)
\(VX_{r,i,l}\)	operating quantity of device \(l\)
\(VS_{r,i,l}\)	stock quantity of device \(l\)
\(VR_{r,i,l}\)	new installation of device \(l\)
\(VC_{r,i,l_1,l_2,h}\)	device retrofit from \(l_1\) to \(l_2\) installed in year \(h\). \(\left( l_1,l_2 \right) \in L\)
\(VSC_{r,i,l,h}\)	stock quantity of device \(l\) accounting technology retrofit
\(VP_{r,i,l}\)	replacement of device \(l\)
\({DVPG}_{r,i,l,j,dum2}\)	dummy variable for stabilizing hourly changes of power generation and storage
\(VTC\)	total energy system cost
\(\delta^{serv}_{r,i,j}\)	slack variable for service demand balance
\(\delta^{end}_{I_{i,N},N_{j,k}}\)	slack variable for intermediate carrier input/output balance
\(\delta^{occ}_{r,i,l}\)	slack variable for technology operation rate balance

Exogenous parameters

Exogenous parameters

Symbol	Description
\(a_{r,i,l,j}\)	service output coefficient of device \(l\) for service type \(j\)
\(e_{r,i,l,k}\)	energy input coefficient of device \(l\) for energy type \(k\)
\(d_{r,i,j}\)	exogenous service demand for service type \(j\)
\(b^n_{r,i,l}\)	initial cost of new installation
\(c^n_{r,i,l}\)	annualized initial cost of new installation
\(b^r_{r,i,l,l_1}\)	initial cost of retrofit
\(c^r_{r,i,l,l_1,h}\)	annualized initial cost of retrofit
\(b^p_{r,i,l}\)	initial cost of replacement
\(c^p_{r,i,l}\)	annualized initial cost of replacement
\(g^e_{r,i,l}\)	variable operation and management cost
\(g^o_{r,i,l}\)	fixed operation and management cost
\(\gamma_{r,i,l}\)	maximum capacity factor for device \(l\)
\(\epsilon_{r,i,k,m}\)	emission factor of energy type \(k\) for gas type \(m\)
\(\theta_{r,i,l,j}\)	maximum/minimum share of device \(l\) for provision of service \(j\)
\(\phi_{r,i,l,j}\)	service efficiency change for device \(l\) for service \(j\)
\(\xi_{r,i,l,k}\)	energy efficiency change for device \(l\) for energy type \(k\)
\(\zeta_{r,i,m}\)	emission tax for gas \(m\)
\({ss}_{r,i,l,h}\)	existing stock of device \(l\) installed in year \(h\)
\({sr}_{r,i,l}\)	retired stock of device \(l\) due to technology lifetime
\(t_{r,i,l}\)	lifetime
\(t^r_{r,i,l,h}\)	remaining lifetime of existing device installed in year \(h\) for retrofit cost calculation
\(sc^n_{r,i,l}\)	subsidy rate of new installation
\(sc^r_{r,i,l,l_1}\)	subsidy rate of retrofit
\(sc^p_{r,i,l}\)	subsidy rate of replacement
\(\alpha_{r,i,l}\)	discount rate
\(\hat{q}_{r,i,m}\)	maximum emission
\(\hat{e}_{ME_{r,i},k_m}\)	maximum/minimum energy supply of energy group \(k_m\)
\(\tau_{r,i,l_m}\)	maximum/minimum new installations of device group \(l_m\)
\(\rho_{r,i,l}\)	maximum/minimum stock quantity of device \(l\)

Equations

Objective function

The objective function is expressed as the minimization of total energy system costs, including initial, operation and management, and emission costs. New installation of technologies \({VR}_{r,i,l}\) is determined based on the following equation, which minimizes total energy system costs.

\[VTC = \sum_{r,i,l}{\left[VR_{r,i,l}\cdot c^n_{r,i,l}+VP_{r,i,l}\cdot c^p_{r,i,l}+VX_{r,i,l}\cdot g^o_{r,i,l}\right]}+\sum_{r,i,l,l_1,h}{\left[VC_{r,i,l,l_1,h}\cdot c^r_{r,i,l,l_1,h}\right]} +\sum_{r,i,k}{\left[VE_{r,i,k}\cdot g^e_{r,i,k}\right]}+\sum_{r,i,m}{\left[VQ_{r,i,m}\cdot\zeta_{r,i,m}\right]} +\sum_{r,i,l,j,dum2}{\left[DVPG_{r,i,l,j,dum2}\right]} \rightarrow min\]

Service output balance

Total service output is estimated by multiplying the operating quantity of a device with its service output coefficient.

\[VD_{r,i,j} = \sum_{l}{\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right]}\]

Total service output must meet service demand requirement, which is provided exogenously.

\[VD_{r,i,j} = d_{r,i,j}+\delta^{serv}_{r,i,j},\quad \delta^{serv}_{r,i,j}\ge 0\]

For specific services, service share constraint for each device is imposed.

\[ \begin{align}\begin{aligned}\theta^{max}_{r,i,l,j}\cdot VD_{r,i,j}\ge (1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\\\theta^{min}_{r,i,l,j}\cdot VD_{r,i,j}\le (1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\end{aligned}\end{align} \]

The following equations are used to control the service shares of the service and technology groups, respectively.

\[ \begin{align}\begin{aligned}\omega^{max}_{r,i,n,j}\cdot VD_{r,i,j}\ge \sum_{l\in MN_{r,l,n,j}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right]\\\omega^{min}_{r,i,n,j}\cdot VD_{r,i,j}\le \sum_{l\in MN_{r,l,n,j}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right]\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\chi^{max}_{r,i,l,o}\cdot \sum_l \sum_{j\in MO_{r,l,j,o}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right] \ge \sum_{j\in MO_{r,l,j,o}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right]\\\chi^{min}_{r,i,l,o}\cdot \sum_l \sum_{j\in MO_{r,l,j,o}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right] \le \sum_{j\in MO_{r,l,j,o}}\left[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}\cdot VX_{r,i,l}\right]\end{aligned}\end{align} \]

Energy input balance

Energy consumption is estimated by multiplying the operating quantity of a device with its energy input coefficient.

\[VE_{r,i,k} = \sum_l\left[{(1+\xi_{r,i,l,k})\cdot e_{r,i,l,k}\cdot VX_{r,i,l}}\right]\]

Service output must exceed the quantity of the corresponding energy input. For example, total electricity output from the power sector should be greater than total electricity demand in end-use sectors.

\[\sum_{r,i\in I_{i,N}}\sum_{j\in N_{j,k}}VD_{r,i,j} = \sum_{r,i\in I_{i,N}}\sum_{k\in N_{j,k}}VE_{r,i,k}+\delta^{end}_{I_{i,N},N_{j,k}},\quad \delta^{end}_{I_{i,N},N_{j,k}} \ge 0\]

The maximum and minimum constraints on energy consumption can be imposed as below.

\[ \begin{align}\begin{aligned}\hat{e}^{max}_{ME_{r,i},k_m} \ge \sum_{(r,i)\in ME_{r,i}}\sum_{k\in MK_{k_m,k}}VE_{r,i,k}\\\hat{e}^{min}_{ME_{r,i},k_m} \le \sum_{(r,i)\in ME_{r,i}}\sum_{k\in MK_{k_m,k}}VE_{r,i,k}\end{aligned}\end{align} \]

Technology stock balance and operation conditions

The operating quantity of devices must be less than the maximum quantity, which is calculated by multiplying the stock quantity of the device with its capacity factor.

\[VX_{r,i,l} = (1+\gamma_{r,i,l})\cdot VS_{r,i,l}-\delta^{occ}_{r,i,l},\quad \delta^{occ}_{r,i,l} \ge 0\]

The stock capacity of a device is determined by existing stock, newly installed capacity, replacement and retrofit.

\[VS_{r,i,l} = VR_{r,i,l}+VP_{r,i,l}+\sum_y VSC_{r,i,l,y},\quad \left(\forall y\in \{y\mid y+t_{r,i,l}\le y^* \}\subset H \right)\]

\[VSC_{r,i,l,h} = {ss}_{r,i,l,h}+\sum_{l_1} VC_{r,i,l_1,l,h}-\sum_{l_1} VC_{r,i,l,l_1,h}\]

\[{sr}_{r,i,l} \ge VP_{r,i,l}\]

The following equations provide maximum and minimum constraints on stock and the new installation capacity of devices, technology retrofit, respectively.

\[ \begin{align}\begin{aligned}\tau^{max}_{r,i,l_m} \ge \sum_{l\in ML_{l_m,l}} VR_{r,i,l} ,\quad \tau^{min}_{r,i,l_m} \le \sum_{l\in ML_{l_m,l}} VR_{r,i,l}\\\rho^{max}_{r,i,l} \ge VS_{r,i,l} ,\quad \rho^{min}_{r,i,l} \le VS_{r,i,l}\\\kappa^{max}_{r,i,l_1,l_2} \ge \sum_h VC_{r,i,l_1,l_2,h} ,\quad \kappa^{min}_{r,i,l_1,l_2} \le \sum_h VC_{r,i,l_1,l_2,h}\end{aligned}\end{align} \]

Emissions

Greenhouse gas emissions are estimated as energy consumption multiplied by emission factors.

\[VQ_{r,i,m} = \sum_k VE_{r,i,k}\cdot \epsilon_{r,i,k,m}\]

The emission constraint is imposed as below.

\[\hat{q}^{max}_{R_{r,i},G_i} = \sum_{r,i\in MR_{r,i}}\sum_{m\in G_m}VQ_{r,i,m}\]

Other functions

Hourly changes in power generation and storage volume are stabilized by introducing variable DVPG which is to be minimized in the objective function.

\[(1+\phi_{r,i,l,j})\cdot a_{r,i,l,j}*VX_{r,i,l}+DVPG_{r,i,l,j,'1'} = (1+\phi_{r,i,l+1,j+1})\cdot a_{r,i,l+1,j+1}*VX_{r,i,l+1}+DVPG_{r,i,l+1,j+1,'2'}\]

\[DVPG_{r,i,l,j,dum2} \ge 0\]

Technology costs

The initial cost of a device was annualized based on a discount rate and a technology-specific lifespan. The subsidy rate was also considered when applicable. The initial investments are annualized under the assumption of a 10% discount rate, and include 20% for carbon capture and storage technologies, given their uncertainties.

\[c^n_{r,i,l} = b^n_{r,i,l} (1-sc^n_{r,i,l})\cdot \frac{\alpha_{r,i,l}\cdot (1+\alpha_{r,i,l})^{t_{r,i,l}}}{(1+\alpha_{r,i,l})^{t_{r,i,l}}-1}\]

\[c^p_{r,i,l} = b^p_{r,i,l} (1-sc^n_{r,i,l})\cdot \frac{\alpha_{r,i,l}\cdot (1+\alpha_{r,i,l})^{t_{r,i,l}}}{(1+\alpha_{r,i,l})^{t_{r,i,l}}-1}\]

\[c^r_{r,i,l,l_1,h} = b^r_{r,i,l,l_1} (1-sc^r_{r,i,l,l_1})\cdot \frac{\alpha_{r,i,l_1}\cdot (1+\alpha_{r,i,l_1})^{t^r_{r,i,l,h}}}{(1+\alpha_{r,i,l_1})^{t^r_{r,i,l,h}}-1}\]

Dynamic stock changes

Some parameters, such as technology vintage information and the availability of depletable resources, are associated with simulation results for previous years. The stock quantity in year \(y+1\) is updated based on new installations in year \(y\).

\[{ss}_{r,i,l,t}^{y+1} = VR_{r,i,l}^y\]

\[{sr}_{r,i,l} = \sum_y {ss}_{r,i,l,y},\quad \left(\forall y\in \{y\mid y+t_{r,i,l}\ge y^* \}\subset H \right)\]

Energy constraints on depletable resources, including crude oil, coal and natural gas, in year \(y+1\) are updated based on energy consumption in year \(y\).

\[\hat{e}_{ME_{r,i},k_m}^{max,y+1} = \hat{e}_{ME_{r,i},k_m}^{max,y}-\sum_{(r,i)\in ME_{r,i}}\sum_{k\in MK_{k_m,k}}VE_{r,i,k}\]

References

[1]	Kainuma, M., Matsuoka, Y., Morita, T. (2003). Climate policy assessment: Asia-Pacific integrated modeling. Tokyo: Springer. https://doi.org/10.1007/978-4-431-53985-8

[2]

Akashi, O., Ashina, S., Ehara, T., Fujino, J., Fujiwara, T., Hanaoka, T., Hanasaki, N., Harasawa, H., Hibino, G., Hijioka, Y., Kainuma, M., Kawase, R., Masui, T., Matsuoka, Y., Miyashita, M., Shimada, K., Shukla, P. R., Takahashi, K. (2007). Aligning Climate Change and Sustainability -Scenarios, modeling and policy analysis-. https://www.cger.nies.go.jp/publications/report/i072/I072.pdf