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This paper presents a quantitative and computational method to determine the optimal tax rate among generating units. To strike a balance between the reduction of carbon emission and the profit of energy sectors, the proposed bilevel optimization model can be regarded as a Stackelberg game between the government agency and the generation companies. The upperlevel, which represents the government agency, aims to limit total carbon emissions within a certain level by setting optimal tax rates among generators according to their emission performances. The lowerlevel, which represents decision behaviors of the grid operator, tries to minimize the total production cost under the tax rates set by the government. The bilevel optimization model is finally reformulated into a mixed integer linear program (MILP) which can be solved by offtheshelf MILP solvers. Case studies on a 10unit system as well as a provincial power grid in China demonstrate the validity of the proposed method and its capability in practical applications.
Global climate change is mostly attributed to the excessive emission of greenhouse gases, especially carbon dioxide (CO_{2}). According to [
In order to combat global warming issues, many countries have launched their future plans in emission reduction and devoted substantial efforts to achieve their goals through some marketbased policies. Such policies regulate customers' behavior through market forces. Briefly speaking, the emission of CO_{2} would be reduced if the price of emission intensive products increases and consequently the demands shift to other alternative environmentallyfriendly goods. From an economic point of view, marketbased policies effectively achieve the target of emission reduction at the lowest cost to society [
In the C&T program, a government agency specifies a limit or cap on the total amount of carbon emissions. Then the cap is allocated to entities under regulation in the form of carbon credits. Generation companies (GENCOs) whose carbon credits are more than their actual emissions are allowed to trade them [
The carbon tax is a direct charge imposed on carbon emissions. GENCOs are obligated to pay fees proportional to the quantity of their emissions. In the early 1990s, Sweden became the first country attempting to manage the carbon emission with the carbon tax scheme. In the following 20 years, a number of countries have tried to implement carbon tax or energy tax schemes that aim at reducing carbon emissions [
There has been a hot debate between C&T and carbon tax scheme [
Despite the debate about C&T and carbon tax policy, a couple of strategic approaches have been studied to reduce carbon emission from energy and manufacturing industry. An inexact mixinteger twostage program model was proposed in [
Inspired by the work in [
The reminder of this paper is organized as follows: the mathematical formulation is provided in Section 2. The solution method is developed in Section 3. Test results on a 10unit system as well as a provincial power gird in China are given in Section 4. Finally, conclusions are given in Section 5.
In this paper, the problem is established against the following background. A certain number of generators are operated by a control center, say an Independent System Operator (ISO), or a regional dispatch center in China. The carbon emission of each generator is assumed to be proportional to its output. Aiming at limiting the carbon emission within a prespecified level in a target year, the government determines a tax rate for carbon emission on each generator. In response to the tax signal, the ISO executes an ED program by taking into account the carbon tax cost in its objective function and determines the electricity production of each generator.
Different from C&T programs directly setting the quantity of carbon emission, the carbon tax scheme indirectly controls the emission level through prices. Therefore, the most critical problem confronting the tax decision maker is to determine an appropriate tax rate that achieves the target of emission reduction in a desired period. If the carbon tax rate is too high, the GENCOs may suffer from financial burdens, and the economic growth will be hindered. On contrary, if the carbon tax rate is too low, the actual emission cannot be reduced to the desired level. Therefore, it is important for the tax decision maker to strike a proper balance between the environment protection and economy growth.
A distinguished feature of the proposed model is it captures the reactions from the ISO to the carbon tax signal from a lower level optimization problem, so the quantity of emission corresponding to the ISO's dispatch decision can be easily estimated by the tax maker. This feature overcomes the traditional difficulty of the carbon tax policy. In the following subsections, a load model is firstly introduced to simplify the ED problem of ISO, and then the bilevel optimization model of the taxing problem is established.
Similar to [
This manifestation can be regarded as rearranging the yearly time sequence load curve into a quantity descending sequence, and merging periods with similar load levels into a single demand block. This approach is appropriate for a long term policy making problem. However, this modeling usually does not allow accurately representing time coupled operating constraints such as minimum up/down time constraints and ramping constraints of conventional units.
The parameters and variables used in the model are defined as follows:
Parameters
Δ
Variables
The tax policy making problem can be formulated as a bilevel optimization, and regarded as a Stackelberg game, in which the government is the leader, whose strategy is the tax rate, and ISO is the follower, whose strategy is the output of generators. The upperlevel (UL, the leader's problem) represents the tax rate decision process of the government agency with the target of minimizing the total levied tax subject to restricting the emission within a certain level. The lowerlevel (LL, the follower's problem) represents the ISO's ED problem subject to operating constraints with the tax rate fixed. The bilevel optimization model is provided as follows:
The UL problem represents the total tax minimization for the government, constrained by: (i) tax rate bound (
It should be pointed out that:
In the UL problem, the tax rate may vary among different units. This setting provides more flexibility and economic superiority than using an equal tax rate on all generators. To see this, adding the constraint
The value of
Because the time scale involved in the ED problem is one year, so system upgrading is not directly considered in the current formulation. To pay the tax is the only choice of generator owners. However, if long term decisions, such as the generation expansion planning decisions, are incorporated, our modeling framework is able to model long term resorts to reduce emissions under the taxation policy in the time scale of several decades rather than paying the tax, such as upgrading the generation equipment, investment on new technologies and renewable energies can be incorporated. This will lead to a different formulation of the decision problem, but the solution method directly applies. It is a very interesting topic that we are still working on.
Direct Current (DC) power flow constraints also can be imposed in the LL ED problem to prevent transmission congestions.
As for the objective function
It's important to restrict the cost coefficient (
As analyzed before, the UL problem (1) and the collection of LL problems (2) are interacted with each other, they should be solved simultaneously. To use offtheshelf solvers, it's necessary to convert the bilevel model (1)–(2) into a single level optimization problem by replacing the LL problem with its first order optimality condition. Because the LL problem appears to be a linear program (LP) when the tax rate is fixed, two options are available for this task: the KKT formulation and the primaldual formulation [
Note that for fixed tax rates, the LL problems (2) are LPs and decoupled with respect to block
However, problem (4) is instinctively hard to solve because the MangasarianFromovitz constraint qualification fails at every feasible point [
The single level optimization problem (4) includes two kinds of nonlinearities:
the complementarity and slackness conditions in
the bilinearity in the objective function
They are subsequently linearized in the following two subsections.
The complementarity and slackness conditions in
Variables
As a result:
The same procedure can be applied to drive the bound of
The value of
To sum up,
The bilinearity in the objective function
Thus:
Substituting
The right hand side of
In model (4), replacing the objective function
To validate the effectiveness of the proposed model and algorithm, numeric experiments on a 10unit system and a provincial power grid in China are carried out. YALMIP is used to formulate the model. CPLEX 12.2 is used to solve related MILP problems.
The parameters of generators and demand blocks of the 10unit system are given in
In the 10unit system, the parameters of generators are collected from typical generators in China. Each generator has a coal consumption rate
In this case, the parameter
Then the optimal emission problem (14) is solved. The optimal value is
The value of
The optimal tax rate under different permitted emission level α is computed from MILP (12) and shown in
From
With the optimal tax rates fixed, the LL ED problem is solved. The production cost (∑
From
The realistic Guangdong power grid of China is studied in this case. One hundred and seventy four (174) generating units with a total 58,744 MW capacity are available in this system. The topology of the 500 kV main transmission network is shown in
In this system, it's found that some generators in the same power plant have identical parameters. To simplify the bilevel model (1)–(2) and reduce binary variables in MILP (12), generators with the same cost coefficient
It's also found that the equivalent production costs of most newbuilt generators with CO_{2}/SO_{2} capture facilities are usually higher than the old ones without advanced environmental protection equipment. Therefore, a traditional costminimized energy scheduling scheme will result in higher emissions. If carbon tax is imposed on Guangdong power grid, bilevel model (1)–(2) is used to study how to determine an appropriate tax rate among generating units. The results are shown through
According to our modeling method, each aggregated unit has a corresponding optimal tax rate for a fixed permitted emission level. To illustrate them clearly, the 52 aggregated generators are divided into 6 groups. Each group contains a number of aggregated generators whose coal consumption rate is within a certain range. The coal consumption interval of each group is given in
From
The permitted amount of carbon emission
Finally,
This paper presents a bilevel optimization model for the carbon emission taxing problem that can strike a proper balance between the target of emission reduction and the profit of emission sectors. The model is finally converted into a tractable MILP without any approximation and can be efficiently solved by commercial solvers. Our model possesses two distinguished features: (1) The Stackelberg competition between the tax maker and emission sectors under regulation is explicitly modeled; (2) The actual emission corresponding to the optimal ED solution is no more than the prespecified value.
It's found in the case studies that: (1) according to the design parameters and operating data of typical generators in China, generators with higher emission coefficients will generally have higher tax rates; (2) the unit cost of carbon emission increases with the decreasing permitted emissions, but the total tax is still within a reasonable range when α varies from 0.2 to 0.6. These results provide practical references for relevant government agencies.
Further work will be focused on extending the proposed formulation to a generation expansion planning framework, thus upgrading the generation equipment, investment on new technologies and renewable energies could be modeled. If the uncertainty resulting from long term decisions and renewable generation is considered, the problem will be more challenging and worthy of studying.
This work is supported in part by the National Natural Science Foundation of China (No. 51007041), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51321005) and the Special grand from EPRI of China (XTB51201303968).
The authors declare no conflict of interest.
Loadblock representation of the yearly load curve.
500 kV main transmission network of the Guangdong power grid.
Weighted average tax rate of each group under different emission level.
Electricity production of each group under different permitted emission level.
Tax revenue under different permitted emission level.
Parameters of generators.
G_{1}  1000  600  554  1004.7 
G_{2}  840  350  536  1034.0 
G_{3}  700  300  518  1063.3 
G_{4}  660  200  540  1096.3 
G_{5}  600  200  445  1114.7 
G_{6}  500  150  400  1129.3 
G_{7}  450  150  372  1169.7 
G_{8}  450  150  330  1140.3 
G_{9}  300  100  346  1191.7 
G_{10}  100  50  320  1257.7 
Parameters of load blocks.
1  5000  1000 
2  4500  3000 
3  4000  3000 
4  3500  1000 
5  3000  760 
Optimal cost and emission.

 

1.6352 × 10^{10}  3.9941 × 10^{10}  1.8149 × 10^{10}  3.8775 × 10^{10} 
Optimal tax rate under different permitted emission level.
G_{1}  0  0  0  0  0 
G_{2}  0  0  0  0  0.0674 
G_{3}  0  0  0.1310  0  0.1310 
G_{4}  0.0502  0.0502  0.0502  0.0502  0.0569 
G_{5}  0  0.3651  0  0.3651  0.3717 
G_{6}  0  0  0  0.5097  0.5162 
G_{7}  0  0  0.5799  0.5831  0.5925 
G_{8}  0  0  0  0  0.7395 
G_{9}  0.6462  0.6492  0.6523  0.6554  0.6646 
G_{10}  0  0  0.6968  0  0.7085 
Production cost and emission under different permitted emission level.
 

α = 0.2  166.39  2.1114  3.9706  3.9692 
α = 0.4  169.31  4.1249  3.9472  3.9450 
α = 0.6  171.98  6.9378  3.9241  3.9233 
α = 0.8  176.64  10.242  3.9006  3.8991 
α = 1.0  181.49  20.370  3.8775  3.8775 
Electricity generated by each unit under different permitted emission level.
G_{1}  5656  6736  7486  8066  8760 
G_{2}  4906  6076  6696  6986  7252 
G_{3}  5278  5678  2978  6132  5788 
G_{4}  1812  2212  3142  3922  4972 
G_{5}  4802  1752  5028  2602  3352 
G_{6}  4190  4190  4380  1564  1964 
G_{7}  3942  3942  1314  1314  1314 
G_{8}  3942  3942  3942  3942  1564 
G_{9}  876  876  876  876  876 
G_{10}  876  876  438  876  438 
Load blocks of the Guangdong power grid.
1  55,805  1,000 
2  49,931  3,000 
3  44,056  3,000 
4  38,182  1,000 
5  35,245  760 
Coal consumption interval of each group (kg/MWh).
0  250  280  300  320  350  
250  280  300  320  350  398 
Comparison of the permitted emission and expected emission.
 

α = 0.2  3.8491  3.8491 
α = 0.4  3.8074  3.8071 
α = 0.6  3.7656  3.7654 
α = 0.8  3.7239  3.7238 
α = 1.0  3.6822  3.6822 