An interval mixed-integer non-linear programming model to support regional electric power systems planning with CO₂ capture and storage under uncertainty

Modeling formulation

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..

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 non-linear, so the two-step 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 CO₂ 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 two-step 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.

where Q _j indicates the set of subscript i for y _i  = 1, and j ∈ [1, 2 ^N ].

Assume the optimal solutions of $f_j^{+}$ submodel were $x_{\left(j\right)iopt}^{+},f_{jopt}^{+}$. Thus, we have the solution for model (4): $f_{jopt}^{\pm }=\left[f_{jopt}^{-},f_{jopt}^{+}\right],x_{\left(j\right)iopt}^{\pm }=\left[x_{\left(j\right)iopt}^{-},x_{\left(j\right)iopt}^{+}\right],y_{\left(j\right)iopt}=1\left(i\in Q_i\right),y_{\left(j\right)iopt}=0\left(i\in N-Q_i\right)$. Accordingly, the other 2 ^N -1 solutions can be obtained by repeating the above procedure. Define $f_{jopt}^{\pm }¯$ is the median value of interval $f_{jopt}^{\pm }=\left[f_{jopt}^{-},f_{jopt}^{+}\right]$. 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 $\overline{f_{kopt}^{\pm }}=min\left(\overline{f_{1opt}^{\pm }}\text{,}\overline{f_{2opt}^{\pm }}\text{,}\overline{f_{3opt}^{\pm }}\text{,}\dots \text{,}\overline{f_{2^{\text{N}}opt}^{\pm }}\right)$.

Assume the optimal solutions of f⁺ submodel were $X_{itopt}^{+},Y_{itopt}^{+},I{M}_{topt}^{+},f_{\mathit{opt}}^{+}$. Thus, we have the solution for model (9) as follows: $f_{\mathit{opt}}^{\pm }=\left[f_{\mathit{opt}}^{-},f_{\mathit{opt}}^{+}\right],X_{itopt}^{\pm }=\left[X_{itopt}^{-},X_{itopt}^{+}\right],Y_{itopt}^{\pm }=\left[Y_{itopt}^{-},Y_{itopt}^{+}\right],I{M}_{topt}^{\pm }=\left[I{M}_{topt}^{-},I{M}_{topt}^{+}\right]\text{.}$

Overview of the study system

The IMINLP model will be employed to facilitate planning for this regional electric power system. The general solution method is to be used under two scenarios of CO₂ emission limitation (i.e. high and low emission standards) in order to help planners well understand its impacts on the results (shown in Table 4). In reality, choosing suitable CO₂ capture technology for a given electricity facility is not only decided by technical feasibility, but also related to geographical location availability for carbon transportation and storage, as well as its impacts on the social community and economic development. However, such information is usually not available or needs to be further investigated. Therefore, planners’ preferences on CO₂ capture technologies will be helpful and needs to be taken into consideration during the solving process. In this study, we assume that decision makers are only interested in three policies: (i) all facilities with post-combustion capture technologies; (ii) facilities 1, 2 with post-combustion capture, facility 3 with pre-combustion; and (iii) all facilities with oxyfuel combustion technologies.

Electricity generation facilities	High emission scenario${\stackrel{}{\mathit{G}}}_{\mathit{it}}$(106kg)			Low emission scenario${\stackrel{}{\mathit{G}}}_{\mathit{it}}$(106kg)
	t = 1	t = 2	t = 3	t = 1	t = 2	t = 3
PC (i = 1)	[3000, 3150]	[2800,3000]	[2500, 2650]	[1600, 1650]	[1300, 1350]	[1150, 1230]
NGCC (i = 2)	[1800, 1900]	[1700, 1800]	[1600, 1700]	[800, 860]	[760, 800]	[720, 780]
IGCC (i = 3)	[2500, 2600]	[2300, 2500]	[2000, 2150]	[1000, 1200]	[950, 1000]	[900, 980]

	Facility	High emission scenario			Low emission scenario
		t = 1	t = 2	t = 3	t = 1	t = 2	t = 3
IM t (PJ)		[68.47, 68.47]	[39.85, 39.85]	[40.18, 40.18]	[74.40, 74.40]	[103.50, 103.50]	[133.00, 133.00]
X it (PJ)	i = 1	[495.00, 507.66]	[636.50, 636.50]	[650.00, 657.68]	[434.78, 471.05]	[353.26, 353.26]	[312.50, 329.18]
	i = 2	[200.00, 220.00]	[212.50, 212.50]	[315.00, 315.00]	[229.32, 252.25]	[280.50, 280.50]	[309.68, 309.68]
	i = 3	[73.50, 83.87]	[67.60, 80.65]	[58.80, 69.35]	[47.50, 47.50]	[219.24, 219.24]	[290.89, 290.89]
	i = 4	[89.00, 115.20]	[187.50, 223.70]	[216.00, 263.50]	[140.00, 150.00]	[187.50, 236.70]	[216.00, 263.50]
	i = 5	[4.00, 4.80]	[6.00, 6.80]	[50.00, 54.29]	[4.00, 4.80]	[6.00, 6.80]	[67.94, 73.76]
Y it (GW)	i = 1	[0.00, 0.00]	[1.20, 1.20]	[1.00, 1.00]	[0.00, 0.00]	[0.00, 0.00]	[0.00, 0.00]
	i = 2	[0.00, 0.00]	[0.00, 0.00]	[1.00, 1.00]	[0.37, 0.37]	[0.80, 0.80]	[0.94, 0.94]
	i = 3	[0.27, 0.34]	[0.18, 0.25]	[0.06, 0.13]	[0.00, 0.00]	[1.69, 1.69]	[2.27, 2.27]
	i = 4	[0.78, 1.04]	[2.00, 2.30]	[2.20, 2.60]	[1.50, 1.50]	[2.00, 2.46]	[2.20, 2.60]
	i = 5	[0.00, 0.00]	[0.00, 0.00]	[1.23, 1.23]	[0.00, 0.00]	[0.00, 0.00]	[1.74, 1.74]
f ($106)		[53714.63, 69399.39]	[65326.26, 81953.30]

Result analysis

1. (1)

   Optimization solutionsFirstly, the results of planning without consideration of decision makers’ interests in choosing CO₂ capture technologies are discussed. That means we need to cover all combination of technologies when disassembling the IMINLP model into (J + 1) ^K ILP models. The optimal solutions for two CO₂ emission scenarios concerning imported electricity, generation of local facilities and their corresponding capacity expansion in periods 1, 2, and 3 are listed in Table 5. The total cost for high emission scenario is [53714.63, 69399.39]$10⁶, which is obviously lower than the cost at low emission scenario about [65326.26, 81953.3]$10⁶. In order to make better understanding of the results, some comparisons based on these two scenarios are further conducted. Figure 3 shows the comparison of electricity supply schemes between high and low CO₂ emission scenarios. There are apparent differences in the trends of energy supply from import for facility 1 and 3. In the high emission scenario, contributions of imported sector and facility 3 are decreasing within the range below 100 PJ, while electricity generated by facility 1 is increasing from approximate 500 to 650 PJ. By contrast, the low emission scenario indicates another situation in the opposite way regarding the electricity supplied by imports for facility 1 and 3. The electricity contribution of imports and facility 3 are always growing within the whole planning horizon, especially, generation of facility 3 has jumped from about 50 to 300 PJ. Meanwhile, facility 1 shows a descending way from 400 to 300 PJ. This comparison reveals that facilities 1 and 3 are playing important role in the total CO₂ emission of the study power system as there are significant difference between high and low emission scenarios. As for the other three facilities (2, 4, and 5), there are no obvious differences in the comparison. In other words, it can be seen that the contributions of these facilities in some extent have relative smaller or no impacts on the CO₂ emission. In fact, the facilities 4 and 5 indicate hydro and wind power plants, and facility 2 means natural gas-fired power plant. Hydro and wind power indeed have no CO2 emission except the natural gas power, however, its impact stands less than that of coal-fired plants (i.e. facilities 1 and 3). The electricity is then imported to cover the shortage while capacity expansion can not meet the increasing demands under restricted CO₂ emission standards. The comparison of capacity expansion for two scenarios is presented in Figure 4. The expansions for hydro and wind power are almost keeping a stable level. But for the facilities 1, 2, and 3, the capacity expansions are very different. In particular, the expanding scale of facility 1 stands at the highest among these three facilities in the high emission scenario; however, its expansion is not suggested at all in the low emission scenario. The reason is obviously related to its important contribution to the total CO₂ emission of the entire electric power system.Secondly, the rates of imported electricity and renewable energy in the two emission scenarios are compared to further assess the security of power system structure. The results are shown in Figure 5. The rate of imported electricity is increasing to about 8% in the low emission scenario; the reason is that capacity expansion of local facilities is restricted by the low emission standards. Therefore, electricity needs to be imported to meet the growing demands. The declining trend of imported electricity in high emission scenario also demonstrates its interaction in the opposite way. Obviously, the higher the rate of imported electricity, the more insecurity or instability the power supply structure will be. In turn, the lower the rate, the more CO₂ will be emitted. Therefore, there is a tradeoff between the safety of power supply framework and lower CO₂ emission. As shown in Figure 5, there is no significant change in the rate of renewable energy in two scenarios. Such relative stability is mainly limited by the corresponding constraints in the IMINLP.

    
2. (2)

   Policy analysisDecision makers’ preference plays an important role in the selection and penetration of CO₂ capture and storage technologies; furthermore, it could affect the structure of electricity supply and capacity expansion planning in the regional electric power system. Therefore, three scenarios are conducted to demonstrate the influences of different policies for CO₂ sequestration. Policy on all facilities being applied post-combustion capture technologies is considered in scenario A; facilities 1, 2 with post-combustion capture, facility 3 with pre-combustion is considered in scenario B; and all facilities with oxyfuel combustion technologies is processed in scenario C. The total system costs for three scenarios are [58232.08, 74128.19] $10⁶, [57777.78, 73682.44] $10⁶, and [56265.91, 73380.01] $10⁶ respectively. The electricity supplies under different scenarios during the planning period are shown in Figure 6. There is no difference in the structure of electricity supply between scenarios A and B. The results for capacity expansion are also the same. The reason should be the only difference in choosing CO₂ sequestration technologies for facility 3. However, the system costs for scenario A and B are entirely different. This indicates the oxyfuel combustion is a cheaper way for facility 3 compared with post-combustion technology. Under scenario C, the electricity supply changes a lot by enhancing coal-fired power during the whole planning period, while scenarios A and B are both showing decreasing trends. Another apparent difference lies on the facility 3 with NGCC conversion technology, which plays an important role on the electricity supply under scenarios A and B in period 3. In contrast, its contribution in scenario C shows considerable decline, and electricity supplied by facility 1 is correspondingly increased to meet the end users’ demands. Meanwhile, the results for capacity expansion of facility 1 and 3 under three scenarios are changing according to their proportions in the total electricity supply. For example, there is no need to expand the capacity of facility 1 for both scenario A and B during the planning horizon, but under scenario C, its capacity can not satisfy the necessary supply any more. Consequently, the expansion options of [1.2, 1.2] GW and [1.0, 1.0] GW should be taken in the period 2 and 3 for facility 1 at scenario C. There is no apparent discrepancy in electricity supply of facility 2, 4 and 5 for three scenarios, so is the capacity expansion. As for the imported electricity, it holds a noticeable position in the whole electricity supply in period 1 and 2 for all scenarios; however, it decreases to zero in period 3, which means the shortage of electricity can be handled through capacity expansion.The above analysis could generate alternative decision bases for planners regarding CO₂ sequestration technologies. For example, scenario C with the least cost may be preferred in recessionary period; however, this cost-efficient strategy should be based on sufficient coal supply. If there are more oil and gas reserved in this region, scenarios A and B should be considered. Although these two scenarios generate the same schemes for both electricity supply and capacity expansion, scenario B is more efficient in the total system cost than scenario C. Therefore, scenario B would be preferred.