A typical electric power system is related to a number of energy supply, energy conversion and electricity demand activities (shown in Figure 1). The side of energy supply describes the main construction of the power system, including fuelfired power (coal and natural gas), hydro power and wind power. Imported electricity is essential to offset electricity shortage in short term owing to increasing demand. Major energy conversion technologies related to electric power system contains pulverised coal fired technology (PC), integrated gasification combined cycle (IGCC), natural gas combined cycle (NGCC), hydro power conversion, and wind power conversion. Among these five technologies, PC, IGCC and NGCC technologies are the key contributors to CO_{2} emission. Most of generated electricity is distributed to different sectors such as industries, residents, commences, transportations and so on. Planning of such a system is challenged by increasing endusers’ electricity demands, impacts on global climate change induced by CO_{2} emission, and shortage of resources. Besides, many modeling parameters are very inexact and sometimes only be available as intervals, such uncertain information needs to be reflected in an optimization framework. The desired IMINLP model is to tackle a variety of complexities and uncertainties existing in regional electric power systems, and to help decision makers balance electricity supply and demand with minimized total system cost subject to a variety of constraints.
Modeling formulation
The objective function of the IMINLP model consists of costs of energy generation and capacity expansion, costs of applying CCS technologies (i.e. installation of equipments) and corresponding expenditure in operation and periodical maintenance, and costs of imported electricity. The purpose of IMINLP is to minimize the total system costs, and it is supposed to help make decision on (i) planning electricity generation and capacity expansion to meet enduser’s demands, (ii) selecting suitable and affordable CCS technologies to assist mitigation of CO_{2} emission, and (iii) adopting moderate importing measures to keep the balance between supply and demand. Firstly, the objective function without consideration of uncertainties can be formulated as follows:
\begin{array}{l}\text{Min\hspace{0.17em}}f={\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CE{G}_{\mathit{it}}{X}_{\mathit{it}}+{\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CC{E}_{\mathit{it}}{Y}_{\mathit{it}}\to \text{\hspace{0.17em}\hspace{0.17em}(costs of energy generation and capacity expansion)\hspace{0.17em}}\\ +{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{O}_{i}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CI{N}_{ij,t=1}+{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{\sum}_{t=1}^{T}{Y}_{\mathit{it}}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CI{N}_{\mathit{ijt}}\to \text{\hspace{0.17em}(costs of applying CCS technologies)}\\ +{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{\sum}_{t=1}^{T}{X}_{\mathit{it}}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CO{P}_{\mathit{ijt}}\to \text{(costs of operation and maintenance)}\\ +{\sum}_{t=1}^{T}{H}_{t}I{M}_{t}\to \left(\text{costs of imported electricity}\right)\end{array}
(1)
The objective subjects to various technical and environmental constraints, including demand constraints, mass balance constraints, capacity constraints, emission constraints, renewable energy constraints and other technical constraints. The demandrelated activities usually account for the major energy consumption on industrial, residential, commercial and transportational sectors in regional level. In this model, only the total demands for all sectors will be considered. Binary integer variable is used to effectively indicate whether or not a given CCS technology should be employed to capture CO_{2} discharged by fuelfired utilities. All constraints relevant with Equation (1a) are presented as follows:

(i)
constraints for electricity supply and demand balance:
{\sum}_{i=1}^{N}{X}_{\mathit{it}}+I{M}_{t}\ge {D}_{t}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(2)

(ii)
constraints for mass balance:
\left({O}_{i}+{\sum}_{t=1}^{T}{Y}_{\mathit{it}}\right){U}_{\mathit{it}}\ge {X}_{\mathit{it}}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,t
(3)

(iii)
constraints for application of CO_{2} capture technologies:
{Z}_{\mathit{ij}}=\{\begin{array}{c}\hfill 1\text{if technology\hspace{0.17em}}j\text{\hspace{0.17em}is undertaken to facility}i\hfill \\ \hfill 0\text{otherwise}\hfill \end{array}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i\in \left[1,\text{\hspace{0.17em}K}\right]\text{,}\phantom{\rule{0.25em}{0ex}}j
(4)
{\sum}_{j=1}^{J}{Z}_{\mathit{ij}}\le 1\text{,}\phantom{\rule{0.5em}{0ex}}\forall i\in \left[1,\text{K}\right]
(5)

(iv)
constraints for renewable electricity rate:
{\sum}_{i=K+1}^{N}{X}_{\mathit{it}}\ge {N}_{t}{D}_{t}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(6)

(v)
constraints for CO_{2} emission:
{X}_{\mathit{it}}{\eta}_{i}\left(1{\sum}_{j=1}^{J}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}{\lambda}_{\mathit{ij}}\right)\left(1{\sum}_{j=1}^{J}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}{r}_{\mathit{ij}}\right)\le {G}_{\text{it}}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i\in \left[1,\text{K}\right],t
(7)

(vi)
nonnegativity constraints:
{X}_{\mathit{it}}\ge 0\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,t
(8)
0\le {Y}_{\mathit{it}}\le E{max}_{\mathit{it}}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,t
(9)
0\le I{M}_{t}\le {E}_{t}{D}_{t}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(10)
Dimensions:i, electricity generation facilities, i = 1, 2, …, K, K + 1, …, N (i ≤ K indicate all combustion facilities with CO2 emission; j, CO2 capture technologies, j = 1, 2, …, J; t, time periods, t = 1, 2, …, T.; Xit, electricity generated from facility i during period t (PJ); Yit, scale of capacity expansion needs to be undertaken to the facility i during period t (GW); Zij, binary variables identifying whether or not CO2 capture technology j needs to be undertaken to the facility i; IMt, imported electricity during period t (PJ).; CEGit, cost for electricity generation of facility i during period t ($106/PJ); CCEit, capital cost for capacity expansion of facility i during period t ($106/GW); Oi, existing capacity of facility i (GW); Fij, binary variables indicating if CO2 capture technology j is applicable to facility i (1: applicable, 0: not applicable); CINijt, cost for installing equipments in accordance with CO2 capture technology j to facility i during period t ($106/GW); COPijt, operating cost (including all expenditure in transporting and storing captured CO2) for CO2 capture equipments which are installed to facility i during period t ($106/PJ); Ht, cost of imported electricity during period t ($106/PJ); Dt, total electricity demand during period t (PJ); Uit, units of electricity production generated by per unit of capacity of facility i during period t (PJ/GW); Nt, minimum rate of renewable energy supplied electricity in the total demand during period t; ηi, units of CO2 emitted by per unit of electricity production for fkacility i ∈ [1, K] (106kg/PJ); λij, reduced rate of CO2 emission for facility i ∈ [1, K] after CO2 capture technology j has been applied (106kg/PJ); rij, CO2 capture efficiency of technology j for facility i ∈ [1, K] (0 < rij < 1); Git, allowable upper bounds of CO2 emission for facility i ∈ [1, K] during period t (106kg).; Emaxit, allowable upper bounds of capacity expansion for facility i during period t (GW).; Et, maximum rate of imported electricity in the total demand during period t..
Decision variables:i, electricity generation facilities, i = 1, 2, …, K, K + 1, …, N (i ≤ K indicate all combustion facilities with CO2 emission; j, CO2 capture technologies, j = 1, 2, …, J; t, time periods, t = 1, 2, …, T.; Xit, electricity generated from facility i during period t (PJ); Yit, scale of capacity expansion needs to be undertaken to the facility i during period t (GW); Zij, binary variables identifying whether or not CO2 capture technology j needs to be undertaken to the facility i; IMt, imported electricity during period t (PJ).; CEGit, cost for electricity generation of facility i during period t ($106/PJ); CCEit, capital cost for capacity expansion of facility i during period t ($106/GW); Oi, existing capacity of facility i (GW); Fij, binary variables indicating if CO2 capture technology j is applicable to facility i (1: applicable, 0: not applicable); CINijt, cost for installing equipments in accordance with CO2 capture technology j to facility i during period t ($106/GW); COPijt, operating cost (including all expenditure in transporting and storing captured CO2) for CO2 capture equipments which are installed to facility i during period t ($106/PJ); Ht, cost of imported electricity during period t ($106/PJ); Dt, total electricity demand during period t (PJ); Uit, units of electricity production generated by per unit of capacity of facility i during period t (PJ/GW); Nt, minimum rate of renewable energy supplied electricity in the total demand during period t; ηi, units of CO2 emitted by per unit of electricity production for fkacility i ∈ [1, K] (106kg/PJ); λij, reduced rate of CO2 emission for facility i ∈ [1, K] after CO2 capture technology j has been applied (106kg/PJ); rij, CO2 capture efficiency of technology j for facility i ∈ [1, K] (0 < rij < 1); Git, allowable upper bounds of CO2 emission for facility i ∈ [1, K] during period t (106kg).; Emaxit, allowable upper bounds of capacity expansion for facility i during period t (GW).; Et, maximum rate of imported electricity in the total demand during period t..
Parameters:i, electricity generation facilities, i = 1, 2, …, K, K + 1, …, N (i ≤ K indicate all combustion facilities with CO2 emission; j, CO2 capture technologies, j = 1, 2, …, J; t, time periods, t = 1, 2, …, T.; Xit, electricity generated from facility i during period t (PJ); Yit, scale of capacity expansion needs to be undertaken to the facility i during period t (GW); Zij, binary variables identifying whether or not CO2 capture technology j needs to be undertaken to the facility i; IMt, imported electricity during period t (PJ).; CEGit, cost for electricity generation of facility i during period t ($106/PJ); CCEit, capital cost for capacity expansion of facility i during period t ($106/GW); Oi, existing capacity of facility i (GW); Fij, binary variables indicating if CO2 capture technology j is applicable to facility i (1: applicable, 0: not applicable); CINijt, cost for installing equipments in accordance with CO2 capture technology j to facility i during period t ($106/GW); COPijt, operating cost (including all expenditure in transporting and storing captured CO2) for CO2 capture equipments which are installed to facility i during period t ($106/PJ); Ht, cost of imported electricity during period t ($106/PJ); Dt, total electricity demand during period t (PJ); Uit, units of electricity production generated by per unit of capacity of facility i during period t (PJ/GW); Nt, minimum rate of renewable energy supplied electricity in the total demand during period t; ηi, units of CO2 emitted by per unit of electricity production for fkacility i ∈ [1, K] (106kg/PJ); λij, reduced rate of CO2 emission for facility i ∈ [1, K] after CO2 capture technology j has been applied (106kg/PJ); rij, CO2 capture efficiency of technology j for facility i ∈ [1, K] (0 < rij < 1); Git, allowable upper bounds of CO2 emission for facility i ∈ [1, K] during period t (106kg).; Emaxit, allowable upper bounds of capacity expansion for facility i during period t (GW).; Et, maximum rate of imported electricity in the total demand during period t..
The above mixedinteger nonlinear programming (MINLP) model treats all parameters as deterministic. However, in many realworld problems, quality of information for all parameters may not be good enough to be expressed one fixed value (Huang et al. [1995b]). For example, the total electricity demand D_{
t
} is constantly changing all the times as there are a lot of uncertainties in enduser’s electricity related activities. However, the demand should fluctuate between a base demand {D}_{t}^{} and a peak demand {D}_{t}^{+}, hence the total electricity demand in period t can be expressed as an interval parameter {D}_{t}=\left[{D}_{t}^{},{D}_{t}^{+}\right]. In general, interval approach can be employed to tackle such uncertainties of parameters for LP models (Huang et al. [1992]). Consequently, interval parameters are introduced into Model (1) to facilitate communication of uncertainties into the optimization process, resulting in an IMINLP model for regional electric power system as follows:
\begin{array}{l}\text{Min}{f}^{\pm}={\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CE{G}_{\mathit{it}}^{\pm}{X}_{\mathit{it}}^{\pm}+{\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CC{E}_{\mathit{it}}^{\pm}{Y}_{\mathit{it}}^{\pm}\\ +{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{O}_{i}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CI{N}_{ij,t=1}^{\pm}+{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{\pm}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CI{N}_{\mathit{ijt}}^{\pm}\\ +{\sum}_{i=1}^{K}{\sum}_{j=1}^{J}{\sum}_{t=1}^{T}{X}_{\mathit{it}}^{\pm}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}CO{P}_{\mathit{ijt}}^{\pm}\\ +{\sum}_{t=1}^{T}{H}_{t}^{\pm}I{M}_{t}^{\pm}\end{array}
(11)
subject to:
{\sum}_{i=1}^{N}{X}_{\mathit{it}}^{\pm}+I{M}_{t}^{\pm}\ge {D}_{t}^{\pm}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(12)
\left({O}_{i}+{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{\pm}\right){U}_{\mathit{it}}^{\pm}\ge {X}_{\mathit{it}}^{\pm}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,\phantom{\rule{0.5em}{0ex}}t
(13)
{Z}_{\mathit{ij}}=\{\begin{array}{l}1\phantom{\rule{1em}{0ex}}\text{if technology}\phantom{\rule{0.5em}{0ex}}j\phantom{\rule{0.5em}{0ex}}\text{is undertaken to facility}\phantom{\rule{0.5em}{0ex}}\mathit{i}\hfill \\ 0\phantom{\rule{1em}{0ex}}\text{otherwise}\hfill \end{array}\text{,}\forall i\in \left[1,\text{K}\right],j
(14)
{\sum}_{j=1}^{J}{Z}_{\mathit{ij}}\le 1\text{,}\phantom{\rule{0.5em}{0ex}}\forall i\in \left[1,\text{K}\right]
(15)
{\sum}_{i=K+1}^{N}{X}_{\mathit{it}}^{\pm}\ge {N}_{t}^{\pm}{D}_{t}^{\pm}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(16)
{X}_{\mathit{it}}^{\pm}{\eta}_{i}^{\pm}\left(1{\sum}_{j=1}^{J}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}{\lambda}_{\mathit{ij}}^{\pm}\right)\left(1{\sum}_{j=1}^{J}{F}_{\mathit{ij}}{Z}_{\mathit{ij}}{r}_{\mathit{ij}}^{\pm}\right)\le {G}_{\mathit{it}}^{\pm}\text{,}\phantom{\rule{0.5em}{0ex}}\forall i\in \left[1,\text{K}\right],\phantom{\rule{0.5em}{0ex}}t
(17)
{X}_{\mathit{it}}^{\pm}\ge 0\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,\phantom{\rule{0.5em}{0ex}}t
(18)
{X}_{\mathit{it}}^{\pm}\ge 0\text{,}\phantom{\rule{0.5em}{0ex}}\forall i,\phantom{\rule{0.5em}{0ex}}t
(19)
0\le I{M}_{t}^{\pm}\le {E}_{t}^{\pm}{D}_{t}^{\pm}\text{,}\phantom{\rule{0.5em}{0ex}}\forall t
(20)
where the parameters with superscript “±” are interval numbers. An interval number can be expressed as a^{±} = a^{−}a^{+}, representing this parameter can be any value of the interval with minimum value of a^{−} and maximum one of a^{+} (Huang et al. [1992, 1995b]).
Solution method
In the IMINLP model (2), there are four decision variables X_{
it
}Y_{
it
}Z_{
ij
}IM_{
t
}. The arithmetic products (i.e. X_{
it
}Z_{
ij
} and Y_{
it
}Z_{
ij
}) make this model nonlinear, so the twostep method developed by Huang et al. ([1992]) to solve ILP models is not applicable in this case. Due to the binary integer variable Z_{
ij
} being used to indicate whether CO_{2} capture technology j should be applied to facility i, that means the total number of combinations of technology and facility is always limited in reality. Therefore, the IMINLP model can be converted into a number of ILP models by enumerating all possible values of Z_{
ij
}. Then, Huang’s twostep method can be used to solve each ILP model separately. The final optimal solution must locate in the result set containing output of all ILP models, and it can be obtained according to corresponding criteria. Figure 2 illustrates the process of solving the IMINLP model.
In order to clearly address the general solution method, Model (2) can be rewritten as follows:
\text{Min}f={\sum}_{i=1}^{N}{e}_{i}^{\pm}{x}_{i}^{\pm}+{\sum}_{i=1}^{N}{g}_{i}^{\pm}{x}_{i}^{\pm}{y}_{i}
(21)
subject to:
\{\begin{array}{c}\hfill {\sum}_{i=1}^{N}{a}_{i}^{\pm}{x}_{i}^{\pm}\ge {b}_{i}^{\pm}\phantom{\rule{0.75em}{0ex}}\hfill \\ \hfill {\sum}_{i=1}^{N}{c}_{i}^{\pm}{x}_{i}^{\pm}{y}_{i}\ge {d}_{i}^{\pm}\hfill \\ \hfill {y}_{i}=0\text{or}1\text{,\hspace{0.17em}\hspace{0.17em}}\forall i\phantom{\rule{0.25em}{0ex}}\hfill \\ \hfill {x}_{i}^{\pm}\ge 0\text{,}\forall i\phantom{\rule{1.5em}{0ex}}\hfill \end{array}
(22)
Define one combination of binary integer variable y as (y_{
1
}, y_{
2
}, …, y_{
N
}), then the total number of combinations for y is 2^{N}. Therefore model (3) can be disassembled into 2^{N} ILP models, and the j_{
th
} ILP model can be expressed as:
\text{Min}{f}_{j}^{\pm}={\sum}_{i=1}^{N}{e}_{i}^{\pm}{x}_{i}^{\pm}+\sum _{i\in {Q}_{j}}{g}_{i}^{\pm}{x}_{i}^{\pm}
(23)
subject to:
\{\begin{array}{c}\hfill {\sum}_{i=1}^{N}{a}_{i}^{\pm}{x}_{i}^{\pm}\ge {b}_{i}^{\pm}\phantom{\rule{3em}{0ex}}\hfill \\ \hfill \sum _{i\in {Q}_{j}}{c}_{i}^{\pm}{x}_{i}^{\pm}\ge {d}_{i}^{\pm}\phantom{\rule{3em}{0ex}}\hfill \\ \hfill {y}_{i}=1\text{,}i\in {Q}_{j}\phantom{\rule{2.25em}{0ex}}\hfill \\ \hfill {y}_{i}=0\text{,}i\in N{Q}_{j}\phantom{\rule{0.5em}{0ex}}\hfill \\ \hfill {x}_{i}^{\pm}\ge 0\forall i\phantom{\rule{3.5em}{0ex}}\hfill \end{array}
(24)
where Q_{
j
} indicates the set of subscript i for y_{
i
} = 1, and j ∈ [1, 2^{N}].
Obviously, such an ILP as model (4) can be tackled by being divided into two LP submodels \left({f}_{j}^{}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}{f}_{j}^{+}\right) according to Huang’s twostep method (Huang et al. [1992, 1995a, b]; Cao and Huang [2011]; Huang and Cao [2011]; Fan and Huang [2012]). The objective of this model is to minimize the cost, so {f}_{j}^{} submodel should be firstly considered. It can be formulated as follows:
\text{Min}{f}_{j}^{}={\sum}_{i=1}^{N}{e}_{i}^{}{x}_{i}^{}+\sum _{i\in {Q}_{j}}{g}_{i}^{}{x}_{i}^{}
(25)
subject to:
\{\begin{array}{c}\hfill {\sum}_{i=1}^{N}{a}_{i}^{}{x}_{i}^{}\ge {b}_{i}^{}\phantom{\rule{3em}{0ex}}\hfill \\ \hfill \sum _{i\in {Q}_{j}}{c}_{i}^{}{x}_{i}^{}\ge {d}_{i}^{}\phantom{\rule{3em}{0ex}}\hfill \\ \hfill {y}_{i}=1\text{,}\phantom{\rule{0.75em}{0ex}}i\in {Q}_{j}\phantom{\rule{2.25em}{0ex}}\hfill \\ \hfill {y}_{i}=0\text{,}\phantom{\rule{0.75em}{0ex}}i\in N{Q}_{j}\phantom{\rule{0.5em}{0ex}}\hfill \\ \hfill {x}_{i}^{}\ge 0\text{,}\forall i\phantom{\rule{3.5em}{0ex}}\hfill \end{array}
(26)
Let {x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{},{y}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}\text{,}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}{f}_{j\phantom{\rule{0.25em}{0ex}}opt}^{}, and be the optimal solutions of {f}_{j}^{} submodel. Then the {f}_{j}^{+} submodel can be formulated as:
\text{Min}{f}_{j}^{+}={\sum}_{i=1}^{N}{e}_{i}^{+}{x}_{i}^{+}+{\sum}_{i=1}^{N}{g}_{i}^{+}{x}_{i}^{+}{y}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}
(27)
Subject to:
\{\begin{array}{c}\hfill {\sum}_{i=1}^{N}{a}_{i}^{+}{x}_{i}^{+}\ge {b}_{i}^{+}\phantom{\rule{1.5em}{0ex}}\hfill \\ \hfill {\sum}_{i=1}^{N}{c}_{i}^{+}{x}_{i}^{+}{y}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}\ge {d}_{i}^{+}\hfill \\ \hfill {x}_{i}^{+}\ge {x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{}\text{,}\forall i\phantom{\rule{1.25em}{0ex}}\hfill \end{array}
(28)
Assume the optimal solutions of {f}_{j}^{+} submodel were {x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{+},{f}_{j\phantom{\rule{0.25em}{0ex}}opt}^{+}. Thus, we have the solution for model (4): {f}_{j\phantom{\rule{0.25em}{0ex}}opt}^{\pm}=\left[{f}_{j\phantom{\rule{0.25em}{0ex}}opt}^{},{f}_{j\phantom{\rule{0.25em}{0ex}}opt}^{+}\right],{x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{\pm}=\left[{x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{},{x}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}^{+}\right],{y}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}=1\left(i\in {Q}_{i}\right),{y}_{\left(j\right)i\phantom{\rule{0.25em}{0ex}}opt}=0\left(i\in N{Q}_{i}\right). Accordingly, the other 2^{N}1 solutions can be obtained by repeating the above procedure. Define {f}_{j\phantom{\rule{0.5em}{0ex}}opt}^{\pm}\xaf is the median value of interval {f}_{j\phantom{\rule{0.5em}{0ex}}opt}^{\pm}=\left[{f}_{j\phantom{\rule{0.5em}{0ex}}opt}^{},{f}_{j\phantom{\rule{0.5em}{0ex}}opt}^{+}\right]. Since the objective of model (3) is to find the minimum value of f, the screening rule for the optimal solution from result set can be summarized as that k_{
th
} solution is the best solution if and only if \overline{{f}_{k\phantom{\rule{0.25em}{0ex}}opt}^{\pm}}=min\left(\overline{{f}_{1\phantom{\rule{0.25em}{0ex}}opt}^{\pm}}\text{,}\phantom{\rule{0.25em}{0ex}}\overline{{f}_{2\phantom{\rule{0.25em}{0ex}}opt}^{\pm}}\text{,}\phantom{\rule{0.25em}{0ex}}\overline{{f}_{3\phantom{\rule{0.25em}{0ex}}opt}^{\pm}}\text{,}\dots \text{,}\phantom{\rule{0.25em}{0ex}}\overline{{f}_{{2}^{\text{N}}\phantom{\rule{0.25em}{0ex}}opt}^{\pm}}\right).
As for the specific case of IMINLP expressed as model (2), there would be (J + 1)^{K} ILP models. In reality, CO_{2} capture technologies mainly include postcombustion, precombustion and oxyfuel combustion (Damen et al. [2006]). That means J equals to 3, thus the total number of ILP models is 4^{K}. The value of K is also countable in a real regional electric power system. Hence the solution method discussed above is feasible in practice. Furthermore, if there is enough information helpful for decision makers to eliminate impossible combinations of Z_{
ij
}, or the decision makers only prefer several combinations rather than all of them, the number of ILP models to be considered will decrease significantly. In other words, to solve such IMINLP model effectively, it is very important to screen the essential scenarios beforehand based on decision makers’ concerns. For example, if only the scenario that all facilities are employed postcombustion capture technology to reduce CO_{2} emission needs to be considered, thus we have the corresponding combination of Z_{
ij
} as below:
{Z}_{\mathit{ij}}=\{\begin{array}{c}\hfill \phantom{\rule{0.25em}{0ex}}1\phantom{\rule{0.75em}{0ex}}j=1\phantom{\rule{1.25em}{0ex}}\hfill \\ \hfill \phantom{\rule{0.5em}{0ex}}0\phantom{\rule{0.75em}{0ex}}j=2\text{,}\phantom{\rule{0.25em}{0ex}}3\phantom{\rule{0.5em}{0ex}}\hfill \end{array}\text{,}\forall i\in \left[1,\text{K}\right]
(29)
where, j = 1 indicates postcombustion technology, and j = 2,3 mean precombustion and oxyfuel combustion capture technologies, respectively. Correspondingly, the model (2) can be expressed as:
\begin{array}{l}\text{Min}{f}^{\pm}={\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CE{G}_{\mathit{it}}^{\pm}{X}_{\mathit{it}}^{\pm}+{\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CC{E}_{\mathit{it}}^{\pm}{Y}_{\mathit{it}}^{\pm}\\ +{\sum}_{i=1}^{K}{O}_{i}{F}_{i,j=1}CI{N}_{i,j=1,t=1}^{\pm}+{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{\pm}{F}_{i,j=1}CI{N}_{i,j=1,t}^{\pm}\\ +{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{X}_{\mathit{it}}^{\pm}{F}_{i,j=1}CO{P}_{i,j=1,t}^{\pm}\\ +{\sum}_{t=1}^{T}{H}_{t}^{\pm}I{M}_{t}^{\pm}\end{array}
(30)
subject to:
{\sum}_{i=1}^{N}{X}_{\mathit{it}}^{\pm}+I{M}_{t}^{\pm}\ge {D}_{t}^{\pm}\text{,}\forall t
(31)
\left({O}_{i}+{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{\pm}\right){U}_{\mathit{it}}^{\pm}\ge {X}_{\mathit{it}}^{\pm}\text{,}\forall i,t
(32)
{\sum}_{i=K+1}^{N}{X}_{\mathit{it}}^{\pm}\ge {N}_{t}^{\pm}{D}_{t}^{\pm}\text{,}\forall t
(33)
{X}_{\mathit{it}}^{\pm}{\eta}_{i}^{\pm}\left(1{F}_{i,j=1}{\lambda}_{i,j=1}^{\pm}\right)\left(1{F}_{i,j=1}{r}_{i,j=1}^{\pm}\right)\le {G}_{\mathit{it}}^{\pm}\text{,}\forall i\in \left[1,\text{K}\right],t
(34)
{X}_{\mathit{it}}^{\pm}\ge 0\text{,}\forall i,t
(35)
0\le {Y}_{\mathit{it}}^{\pm}\le E{max}_{\mathit{it}}^{\pm}\text{,}\forall i,t
(36)
0\le I{M}_{t}^{\pm}\le {E}_{t}^{\pm}{D}_{t}^{\pm}\text{,}\forall t
(37)
This ILP model apparently can be solved through twostep method. The objective is to minimize system costs, therefore {f}_{j}^{} submodel will be firstly considered. It can be formulated as:
\begin{array}{l}\text{Min\hspace{0.17em}}{f}^{}={\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CE{G}_{\mathit{it}}^{}{X}_{\mathit{it}}^{}+{\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CC{E}_{\mathit{it}}^{}{Y}_{\mathit{it}}^{}\\ +{\sum}_{i=1}^{K}{O}_{i}{F}_{i,j=1}CI{N}_{i,j=1,t=1}^{}+{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{}{F}_{i,j=1}CI{N}_{i,j=1,t}^{}\\ +{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{X}_{\mathit{it}}^{}{F}_{i,j=1}CO{P}_{i,j=1,t}^{}\\ +{\sum}_{t=1}^{T}{H}_{t}^{}I{M}_{t}^{}\end{array}
(38)
subject to:
{\sum}_{i=1}^{N}{X}_{\mathit{it}}^{}+I{M}_{t}^{}\ge {D}_{t}^{}\text{,}\forall t
(39)
\left({O}_{i}+{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{}\right){U}_{\mathit{it}}^{}\ge {X}_{\mathit{it}}^{}\text{,}\forall i,t
(40)
{\sum}_{i=K+1}^{N}{X}_{\mathit{it}}^{}\ge {N}_{t}^{}{D}_{t}^{}\text{,}\forall t
(41)
{X}_{\mathit{it}}^{}{\eta}_{i}^{+}\left(1{F}_{i,j=1}{\lambda}_{i,j=1}^{}\right)\left(1{F}_{i,j=1}{r}_{i,j=1}^{}\right)\le {G}_{\mathit{it}}^{}\text{,}\forall i\in \left[1,\text{K}\right],t
(42)
{X}_{\mathit{it}}^{}\ge 0\text{,}\forall i,t
(43)
0\le {Y}_{\mathit{it}}^{}\le E{max}_{\mathit{it}}^{}\text{,}\forall i,t
(44)
0\le I{M}_{t}^{}\le {E}_{t}^{}{D}_{t}^{}\text{,}\forall t
(45)
Let {X}_{it\phantom{\rule{0.5em}{0ex}}opt}^{},{Y}_{it\phantom{\rule{0.5em}{0ex}}opt}^{},I{M}_{t\phantom{\rule{0.5em}{0ex}}opt}^{},{f}_{\mathit{opt}}^{} be the optimal solutions of f^{−} submodel. Then the f^{+} submodel can be formulated as:
\begin{array}{l}\text{Min\hspace{0.17em}}{f}^{+}={\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CE{G}_{\mathit{it}}^{+}{X}_{\mathit{it}}^{+}+{\sum}_{i=1}^{N}{\sum}_{t=1}^{T}CC{E}_{\mathit{it}}^{+}{Y}_{\mathit{it}}^{+}\\ +{\sum}_{i=1}^{K}{O}_{i}{F}_{i,j=1}CI{N}_{i,j=1,t=1}^{+}+{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{+}{F}_{i,j=1}CI{N}_{i,j=1,t}^{+}\\ +{\sum}_{i=1}^{K}{\sum}_{t=1}^{T}{X}_{\mathit{it}}^{+}{F}_{i,j=1}CO{P}_{i,j=1,t}^{+}\\ +{\sum}_{t=1}^{T}{H}_{t}^{+}I{M}_{t}^{+}\end{array}
(46)
subject to:
{\sum}_{i=1}^{N}{X}_{\mathit{it}}^{+}+I{M}_{t}^{+}\ge {D}_{t}^{+}\text{,}\forall t
(47)
\left({O}_{i}+{\sum}_{t=1}^{T}{Y}_{\mathit{it}}^{+}\right){U}_{\mathit{it}}^{+}\ge {X}_{\mathit{it}}^{+}\text{,}\forall i,t
(48)
{\sum}_{i=K+1}^{N}{X}_{\mathit{it}}^{+}\ge {N}_{t}^{+}{D}_{t}^{+}\text{,}\forall t
(49)
{X}_{\mathit{it}}^{+}{\eta}_{i}^{}\left(1{F}_{i,j=1}{\lambda}_{i,j=1}^{+}\right)\left(1{F}_{i,j=1}{r}_{i,j=1}^{+}\right)\le {G}_{\mathit{it}}^{+}\text{,}\forall i\in \left[1,\text{\hspace{0.17em}K}\right],t
(50)
{X}_{\mathit{it}}^{+}\ge {X}_{it\phantom{\rule{0.25em}{0ex}}opt}^{}\text{,}\forall i,t
(51)
{Y}_{it\phantom{\rule{0.25em}{0ex}}opt}^{}\le {Y}_{\mathit{it}}^{+}\le E{max}_{\mathit{it}}^{+}\text{,}\forall i,t
(52)
I{M}_{t\phantom{\rule{0.25em}{0ex}}opt}^{}\le I{M}_{t}^{+}\le {E}_{t}^{+}{D}_{t}^{+}\text{,}\forall t
(53)
Assume the optimal solutions of f^{+} submodel were {X}_{it\phantom{\rule{0.5em}{0ex}}opt}^{+},{Y}_{it\phantom{\rule{0.5em}{0ex}}opt}^{+},I{M}_{t\phantom{\rule{0.5em}{0ex}}opt}^{+},{f}_{\mathit{opt}}^{+}. Thus, we have the solution for model (9) as follows: {f}_{\mathit{opt}}^{\pm}=\left[{f}_{\mathit{opt}}^{},{f}_{\mathit{opt}}^{+}\right],{X}_{it\phantom{\rule{0.25em}{0ex}}opt}^{\pm}=\left[{X}_{it\phantom{\rule{0.25em}{0ex}}opt}^{},{X}_{it\phantom{\rule{0.25em}{0ex}}opt}^{+}\right],{Y}_{it\phantom{\rule{0.25em}{0ex}}opt}^{\pm}=\left[{Y}_{it\phantom{\rule{0.25em}{0ex}}opt}^{},{Y}_{it\phantom{\rule{0.25em}{0ex}}opt}^{+}\right],I{M}_{t\phantom{\rule{0.25em}{0ex}}opt}^{\pm}=\left[I{M}_{t\phantom{\rule{0.25em}{0ex}}opt}^{},I{M}_{t\phantom{\rule{0.25em}{0ex}}opt}^{+}\right]\text{.}