Collaborative robust dispatch of electricity and carbon under carbon allowance trading market

0 Introduction

In July 2021,carbon markets were officially launched in Beijing,Shanghai,and Wuhan.The first execution cycle of the electricity industry in the carbon market officially began in the same year.The electric power industry accounts for approximately 40% of China’s carbon emissions [1] and plays an important role in the carbon market.Conversely,the carbon market influences and alters the economic operation and dispatch modes of power systems.Therefore,research on the impact mechanism of the carbon market on power-system operation has attracted widespread attention.

The introduction of a carbon market has effectively increased the total generation cost of thermal power units.Low-carbon dispatches (LCDs) have become necessary[2-4].LCD modelling depends on carbon-management methods.For example,when the total carbon emissions are limited,a carbon-emissions constraint must be added to the LCD model [5,6].The carbon-emission cost is added to the objective function of the dispatch model to obtain LCD schemes [7,8] when carbon-emission responsibility is considered.Carbon-emission cost models can be further divided into constant [9,10] and step-based price models[11-14].A step-based price model can limit entities with high carbon emissions and create spatiotemporal differences in carbon prices.These differences can encourage flexible power-system resources to participate in carbonreduction operations.For example,resources such as energy storage [15] and demand-side responses [16] can be used as effective measures to reduce carbon emissions.Electrical energy is like a hub that connects and coordinates with multiple types of energy systems,such as heat,transportation,and integrated energy systems,to achieve comprehensive carbon reduction [17-19].Carbon-emission costs influence power-system planning [20,21] in the long term.

Carbon-capture power plants (CCPP) are important for carbon reduction [22].The optimal dispatch of a CCPP involves a trade-off between energy-consumption costs and carbon-reduction benefits.Therefore,the carbon-reduction benefit depends on multiple factors,such as carbon-capture energy consumption and the price of carbon and electricity.Carbon-capture equipment can generate profits if the carbon price is sufficiently high,or if the cost of electricity consumption in the carbon-capture process is sufficiently low.The CCPP can benefit from renewable-energy generation by utilizing the flexibility in power regulation[17,23].It can also provide flexibility to power systems in the ancillary market to improve economic efficiency.The carbon capture,utilization,and storage (CCUS) project is an important link connecting multiple carbon markets and energy systems [24].Ref.[25] proposed a CCUS-powerto-gas (P2G) joint-operation framework and established an optimized schedule model for a comprehensive,electricityhydrogen energy system in the carbon market.In summary,the profit model of the CCPP is a complex issue,particularly under a market mechanism based on carbon allowances in China [26].To address this issue,the economic operating conditions of a CCPP were investigated in this study.

The development of the carbon and green-certificate markets has had different impacts on the cost structure of power generation in various plants.While the carbon market increases the power-generation cost in thermal power units[27],the green-certificate market reduces the renewableenergy generation cost.Changes in the production cost of electricity also affect the composition of marginal electricity prices in power systems [28].However,the mechanism of impact of carbon-market prices on the power industry is complex and depends on the scheduling strategies of the power-system operator and the market strategies of each electricity entity.Individual entities in power systems face decision-making issues that involve simultaneous participation in diverse markets.The strategic behavior of a generation company considering both the electricity and carbon (E&C) markets was modeled in [29].An optimal dispatch model for a virtual power plant in a joint energyreserve-carbon market was proposed in [30].In [31],a peerto-peer trading mechanism was introduced to co-optimize E&C in distribution systems.A bi-level optimization model for network operations and joint E&C trading was established.

Market-equilibrium issues are interdependent.Ref.[32]develops an approach to find an equilibrium between gas and E&C markets.The coupled relationship between the E&C markets was investigated in [33].They found that carbon prices have a negative impact on electricity prices.The settlement cycles of various markets are asynchronous,which increases the difficulty of analyzing the coupling mechanism between various markets.To the best of our knowledge,this issue has not been discussed previously.

In China,the carbon-allowance trading market adopts an annual settlement mechanism [26].Its cycle is different from that of day-ahead,power-system dispatch.This asynchronous coupling mechanism must be clearly understood before an LCD model can be constructed.Current literature treats carbon price as a deterministic parameter [34,35].However,carbon prices change significantly over the course of a year.The annual uncertainty feature is still unclear and obstructs the LCD modelling.Modelling carbon-price uncertainty is an essential pre-requisite for modelling LCDs.

Optimization models are classified as stochastic optimization (SO) and robust optimization (RO) models.RO models depend less on probabilistic information than SO models do,as they solve for worst-case scenarios[36];the current,effective solving method for RO models is the column-and-constraint generation (C&CG)algorithm [37,38].Adjustable RO facilitates avoidance of scenarios in which all uncertain parameters have the worst values simultaneously,to reduce conservativeness [39].Distributionally robust optimization (DRO) was introduced to take advantage of both the SO and RO models.This determines the optimal solution for the worst probability distribution within a known confidence set [19].Although robust LCD models have been constructed previously,they consider only the uncertainty of renewable-energy generation [34] and electricity prices [40].This study adopted the RO model to deal with the dual uncertainties of annual carbon prices and power systems.The contributions of this study are summarized as follows.

i) An asynchronous coupling mechanism between the carbon-allowance trading market and the day-ahead,powersystem dispatch was demonstrated.The reasonability of considering the annual carbon-price uncertainty in the daily dispatch of power systems was elaborated.ii) A data-driven approach was used to model the uncertainty in annual carbon prices.A collaborative low-carbon,robust dispatch model was constructed considering the dual uncertainties(carbon price and renewable-energy generation).iii) An economic operation model of the CCPP under the carbonallowance trading mechanism was established,and its critical profitability condition was derived.

The effectiveness of the proposed,low-carbon,RO method for power-system dispatch was verified through a comparison with the expected-value optimization model and the traditional,economic dispatch (ED) model in case studies.The composition of the marginal generation cost under the carbon-allowance trading mechanism is also discussed in this paper.The impacts of carbon-quota coefficients and conservative levels on the LCD results were analyzed.The conclusions obtained in this study will provide important guidance for practical,carbon-market design and LCD-strategy selection.

1 Carbon-market and power-system-dispatch coupling analysis

1.1 Carbon-trading requirement of thermal power units in carbon market

Currently,the Chinese carbon-allowance trading market consists of a carbon-quota system and a carbon-trading market.Thermal power plants obtain a certain free-carbon quota in accordance with their actual generation.When the actual carbon emissions of a power plant are greater than its free quota,the excess must be purchased from the carbon market -these are termed paid carbon-emissions indicators.When the actual carbon emissions are less than the free quota,the excess carbon quota can be sold in the carbon market for a profit.Carbon-emissions management adopts an annual settlement approach.A power plant or generation enterprise must provide its annual carbon-emission indicators to the ecological-environment regulatory department at the end of a year.

The free-carbon quota,QA,i obtained from a thermal power plant is proportional to its generation [38].The freecarbon quota can be expressed as

where εi is the free carbon-quota coefficient of the power plant i.Considering the differences in thermal units,it is assumed that the free carbon-quota coefficient of coal-fired units is greater than gas-fired units.is the output of a thermal power plant;Δt is the length of a dispatch period.

Based on the coal-consumption characteristics of the thermal power units,their actual carbon emissions can be expressed as a quadratic function of the output:

where ai,bi,and ci are the quadratic-function coefficients of coal consumption for thermal power units;Mi is the coalconsumption quantity; ηi is the carbon-emission coefficient per unit of coal burned by the units.

The carbon-trading volume QB,i of a thermal power plant in the carbon market is the difference between its actual carbon emissions and free quota,i.e.QB,i = QC,i - QA,i.The cost/benefit of carbon trading can be expressed as

where CC refers to the carbon price.

1.2 Mechanism of carbon-market impact on the thermal power-generation costs

The current settlement cycle of carbon-emission indicators in the Chinese carbon market is one year.This means that carbon-emitting entities can settle their carbonemission indicators before the end of a year.Carbonemission indicators generated by thermal power plants on a certain day can be purchased from the carbon market on any day of the year -even in advance,to balance their potential carbon-emission-indicator requirements.This means that the carbon-emission costs of a power plant are related to the annual carbon prices,which vary over the course of a year.Therefore,the LCD model,proposed in this study,takes the annual carbon-price uncertainty of into account.

Based on historical data of carbon prices from the national carbon-trading market,a nonparametric probability density function (PDF) of the annual carbon price was obtained using the kernel function method.

where Ns represents the number of historical carbon-price samples;Ci is the historical carbon price;σ is the shape parameter of kernel function that represents the standard deviation.

Currently,the uncertainty in the annual carbon price is unclear,and there is no universal,parameterized,probability-modelling method.A data-driven approach,namely,the kernel-density estimation method,was adopted in this study to obtain the nonparametric probabilitydistribution characteristics of carbon prices.Based on historical,carbon-price data in a real carbon market,a PDF is generated,as shown in Fig.1.The PDF has a unimodal asymmetric shape,indicating that the distribution of carbon prices is relatively concentrated.For comparison,a frequency-distribution histogram of the carbon price is also presented.This indicates the frequency at which the historical sample values of carbon prices fall within the corresponding range.

Fig.1 PDF of annual carbon price

The cost for coal-fired thermal power plants is:

where,CM,the coal price,is generally constant.

The total generation cost for a thermal power plant can be expressed as the sum of the coal-fired generation costand the uncertain,carbon-price-based carbon-emission trading cost

2 CCPP Operation model in the carbon market

A CCPP is a renovated,traditional,thermal power unit that adds a carbon-capture system at the inlet of an absorption tower to control the amount of flue gas and carbon-emissions.Carbon-capture systems require a certain amount of energy,thereby changing the net output of the thermal power units.The energy consumptionof the carbon-capture system is proportional to its CO2-capture capacity [8]:

where ωC is the energy-consumption coefficient for carbon capture.is the quantity of captured CO2.

During operation,the quantity of CO2 captured by the carbon-capture system can be adjusted as follows:

The free-carbon quota for a CCPP is allocated according to the net output fed into the power system.represents the coal-fired cost andrepresents the carbon-trading cost,which can be expressed as

whererepresents the original CCPP carbon emissions;is the carbon-capture quantity;is the free-carbon quota for the CCPP;εcc is the carbon-quota coefficient.

3 Low-carbon,robust dispatch modelling

In the LCD of power systems,both coal-fired and carbon-trading costs of thermal power plants are considered.Carbon-trading costs are closely affected by carbon prices,which remain uncertain for more than a year.In this section,a collaborative,RO model is established considering the uncertainty in annual carbon price and renewable-energy generation.The collaborative,RO model was compared with the expected-value optimization model.The PDF of the carbon price CC is represented by f c(CC),its expected value is represented by

3.1 RO model considering multiple uncertainties

RO aims to determine the optimal solution for the worstcase scenario.In this case,carbon price is an uncertain parameter.The compact form of the low-carbon,robust dispatch model can be expressed as

Carbon price,as an uncertain parameter,appears only in the objective function as a coefficient of carbon-emission variables.It can be proven that the worst-case scenario under an uncertain carbon price corresponds to the upper limit.Therefore,the objective function can be re-expressed as follows:

Constraints of the robust dispatch model are listed as follows.

System power balance constraints:

whereis the system marginal generation cost,which is composed of the marginal coal-fired cost and marginal carbon-trading cost.

System upward/downward rotating reserve constraint:

Output constraints of conventional thermal units:

Constraints for the minimum start-up/shut-down duration of conventional thermal power units are as follows:

where,represent the startup and shutdown state variables of the unit,respectively,and Ui,Di represent the minimum startup and shutdown durations of the unit,respectively.

The operation,reserve,and start-up/shut-down constraints of a CCPP are similar to those of conventional,thermal power units.The net output is shown in (9).The constraint for the carbon-capture quantity in each period is given by (8).The relationship between the energy consumption of the carbon-capture process and the quantity of carbon captured is represented by (6) and (7).Renewable energy generation constraints are as follows:

where is the predicted day-ahead output of the renewable-energy generation.When the dispatch result pt re is less than the predicted value,power curtailment is planned.

The RO model can be solved using the C&CG algorithm[37,38].The details are presented in Appendix A1.As the uncertainty of renewable-energy generation has been widely researched previously,this study mainly focuses on carbonprice uncertainty.

To reduce the conservative level of the RO,an adjustable robust parameter Γ is introduced.It assumes that all the uncertain variables are unlikely to reach their bounds simultaneously.Consequently,adjustable RO partially compromises the uncertainty interval to improve the economy of the dispatch results [41,42].

Conventional ED does not consider carbon-trading costs.Its objective function is expressed as (28),with no carbon-emission-related constraints.

3.2 Expected-value optimization dispatch model

Because the carbon-market price only appears in the objective function,the carbon-emission cost can be calculated using the expected carbon price in the expectedvalue optimization model;the carbon price takes the value of its expected valueThe renewable-energy generation output is also represented by its expected value,which is higher in the lower-bound scenario.The objective function is expressed as follows:

The constraints are the same as those of the RO dispatch model of P2.The expected-value model is a single-layer optimization problem and belongs to a risk-neutral dispatch.

4 Case studies

A three-unit power system is adopted for the case studies.It consists of carbon-capture thermal power,coal-fired thermal power,gas-fired thermal power,and wind power units.The coal and gas consumption characteristics of the three thermal power units are listed in Table 1.Their carbonquota coefficients are 0.42,0.42,and 0.36 ton/MWh for generation,respectively.The maximum power of the carboncapture system in G1 was 10 MW,which accounted for 5%of the maximum generation output.The energy-consumption coefficient of carbon capture was set to 0.27 MWh/ton referring to [8,43].The expected wind power output and boundary of its uncertainty set are shown in Fig.2.

Table 1 Generator Parameters

Fig.2 Wind power and its uncertainty set

Carbon-price data for the national carbon-trading market were collected between January and December 2022.The data-driven probabilistic distribution of carbon prices is shown in Fig.1.The expected value was 58.08 RMB/ton.The maximum and minimum values were 61.6 RMB/ton and 50.51 RMB/ton,respectively.The carbon prices at different quantile levels are listed in Table 2.

Table 2 Carbon prices at different quantile levels

4.1 Results analysis of robust optimization dispatch

Because the uncertain parameter of the carbon price appears only in the objective function of the robust dispatch model,the worst-case scenario corresponds to the highest carbon price.The dispatch results corresponding to the worst-case scenario are shown in Fig.3.As a gas-fired unit,G3 was turned on during peak-load periods.When considering carbon costs,the outputs of the gas-fired unit increase,and those of the coal-fired unit decrease as verified through a comparison with the ED model in the following section.

Fig.3 Dispatch results under robust optimization

Dispatch results with different adjustable robust parameter Γ are shown in Fig.4.With the reduction of Γ,namely the decrease of conservative level in the carbon market,the generation of gas-fired unit decreases,and the total carbon emission of the system gradually increases.

Fig.4 Dispatch results under different adjustable robust parameters

When the value of Γ reduced to 0.5,the carbon price in objective function takes the expected value.This indicates that the generation of G3 under full RO increased from 634 to 660 MWh compared to the scenario with Γ=0.5.Meanwhile,the total fuel cost of the system increases slightly,indicating that a rise in the carbon price or an increase in the conservative level of carbon-price uncertainty leads to a decrease in carbon emissions and an increase in fuel costs.

The marginal fuel cost,marginal carbon cost,and total marginal generation cost under LCD is shown in Fig.5.It was also compared with the marginal fuel cost under ED.The total marginal generation cost consists of the marginal fuel cost and marginal carbon cost,which are obviously higher than the marginal fuel cost under ED.

Fig.5 Various marginal cots under LCD and ED

During the peak-load period,the total marginal cost under LCD drops slightly because G3 is turned on and all three units are marginal units.In this period,the marginal fuel cost under the LCD was lower than that under the ED because the outputs of G1 and G2 were lower than those under the ED.The marginal fuel costs of G1 and G2 increase the functions of their outputs.In summary,when considering the carbon cost,the total fuel costs increase,but the changes in the marginal fuel cost in each time slot are bidirectional because the unit commitment results may sometimes change.

4.2 Sensitivity analysis of carbon-quota coefficients

As a measure of macroeconomic regulation,the carbonquota coefficient changes with the development of the carbon market.The impact of the carbon-quota coefficient on the system carbon emissions under the LCD model was analyzed and is shown in Table 3.The decrease in the carbon-quota coefficient indicates an increase in the carbonemission cost of the responsible entities.The optimization results tended to dispatch generation units with low carbonemission intensities.When the carbon-quota coefficient drops to 50% of the initial value,the total carbon emissions decrease from 6,560 to 6,545 ton,and the total fuel cost increases from 2,895,808 to 2,896,671 RMB.This indicates that the carbon-quota coefficient affects LCD results.This affects the unit commitment results of the power system when it decreases to a certain level.

Table 3 Results under different carbon-quota coefficients

The carbon-quota coefficient also influenced the marginal carbon cost of the power system.The marginal carbon costs of generation for different carbon-quota coefficients are shown in Fig.6.A decrease in the carbonquota coefficient indicates that thermal units can obtain a lower free-carbon quota per MWh of generation and need to buy more in the carbon market.Subsequently,the marginal carbon costs increase.In practical applications,carbonmarket managers can encourage power systems to reduce carbon emissions by adjusting the carbon-quota coefficient,which also changes the composition of the power-generation costs.Eventually,this also influences the strategy for generating entities in the carbon market.

Fig.6 Marginal carbon costs under different carbon-quota coefficients

4.3 Analysis of the impact of carbon market on the operation of carbon-capture power plants

The carbon-capture facility of G1 was not operational in this case.This was because the carbon-reduction benefit of carbon capture was lower than the energy-consumption cost.The energy-consumption coefficient ωC of carbon capture was 0.27 MWh/ton.The lowest generation cost was 390 RMB/MWh.The corresponding carbon-capture cost was 113.1 RMB/ton,which is much higher than the current carbon price of 50.5-60.6 RMB/ton.The critical,profitable conditions for the CCPP depend on the energy-consumption coefficient of carbon capture,the marginal generation cost,and the carbon price.In this case,the carbon-capture system profits only when the energy-consumption coefficient decreases to 0.155 MWh/ton.Otherwise,the CCPP is profitable when the marginal generation cost is sufficiently low,such as by utilizing abandoned wind power.

4.4 Comparative analysis of different dispatch models

Table 5 compares the dispatch results of the different dispatch models.Table 4 compares the models.Low-carbon robust dispatch and economic robust dispatch are two-layer optimization models under uncertainty sets of uncertainty parameters.The low-carbon expected-value dispatch is a single-layer optimization model under the expected values of all uncertainty parameters.

Table 4 Model comparison between different dispatch methods

Table 5 Result comparison of different dispatch models

εi: Initial carbon-quota coefficient;Γ: adjustable robust parameter.

As a gas-fired unit,G3 has a high fuel cost and low carbon emissions.It is sensitive to the carbon prices and dispatch strategies.The outputs of the different models are shown in Fig.7.Compared to the ED model (Model E),all LCD models can effectively reduce carbon emissions.The robust,low-carbon Model A reduced carbon emissions by 2.67%.When the adjustable robust parameter decreased to 50% (Model B),the total carbon emissions were 6576 ton,which was slightly higher than that of the fully robust model.

Fig.7 G3 Dispatch results using different models

The total carbon emissions,total generation costs,and the generation of G3 in each robust dispatch model were higher than those in the expected-value dispatch model(Model D).This is because the solution of RO corresponds to the worst-case scenario (generally,the lower bound of wind power),and the expected-value model corresponds to the expected value of wind power,which is obviously higher than that in the worst-case scenario.

When the carbon-quota coefficient decreases to 50%of the initial quota (Model C),the output of G3 further increases (as shown in Fig.7),and the total carbon emissions of the system are reduced by 2.89% compared with the ED model.The results of the different dispatch models demonstrate that low-carbon units have more generation opportunities with an increase in carbon price and a decrease in the carbon-quota coefficient.

To verify the proposed RO dispatch,different conservative levels of wind uncertainty were simulated.RO involves determining the optimal solution for a worstcase scenario.The wind power in the worst scenario under different adjustable robust parameters (Γ decreasing from 1 to 0.8) is shown in Fig.8.Under the full RO,namely Γ=1,the worst scenario corresponds to the lower bound of wind power uncertainty set.

Fig.8 Wind power in the worst scenario under different adjustable robust parameters

Adjustable RO assumes that not all uncertain variables reach the bounds simultaneously.Consequently,when the adjustable robust parameters decrease,the wind power in the worst-case scenario is high at the lower bound in certain time slots,as shown in Fig.8.As Γ decreases from 1 to 0.8,the total carbon emission decreases from 6,560 to 6,364 ton.It should be noted that this result represents a possible carbon emission in the worst-case scenario,rather than the actual carbon emissions.

5 Conclusions

The asynchronous coupling mechanism between the annual carbon market and day-ahead,power-system dispatch was analyzed in this study.A collaborative,lowcarbon robust dispatch model was proposed considering the uncertainties in the annual carbon price and renewableenergy generation.LCD has an obvious influence on the dispatch schemes and system marginal generation cost.The conclusions of this study are as follows.

1) Under an RO dispatch framework,the uncertainty of the carbon price only changes the structure of the objective function.As a conservative risk-averse model,the RO model prefers dispatch units with low carbon-emission intensity.Compared to the ED model,which does not consider carbon emissions,the robust low-carbon model can reduce carbon emissions by 2.67%.

2) When the conservative level of carbon-price uncertainty decreases,the total generation costs decrease and the total carbon emissions increase.

3) Under the carbon-allowance trading mechanism,a reduction in the carbon-quota coefficient means that carbon-emission entities must buy more carbon indicators,decreasing the carbon in the power system.

4) The energy consumption and carbon trading cost models of carbon capture power plant (CCPP) is established under the current carbon allowance trading mechanism in China.It is found that the critical profitability condition of CCPP depends on the energy consumption coefficient of carbon capture system (CCS),electricity price,and carbon price.Under the current generation cost and the carbon price level in China,the CCS can make profit only when the energy efficiency of carbon capture is lower than 0.155 MWh/ton.

When considering carbon cost,the total generation cost consists of fuel cost and carbon emission cost.The principles of locational marginal electricity price and real-time electricity price will be further investigated in future studies.The dispatch strategy will in turn influence the carbon trading strategy of generation entities.Conducting joint optimization decision-making of low-carbon dispatch and carbon trading at the annual level will have important practical significance.

Appendix A

Column-and-Constraint generation (C&CG) algorism C&CG algorism decomposes the original two-stage optimization model into a master problem (MP) model and a slave problem (SP) model,that are solved iteratively.The MP and SP models of the proposed low carbon dispatch model is presented in the following in the compact form.

Acknowledgements

This work was supported by the Science and Technology Project of State Grid Liaoning Electric Power Co.,Ltd.(No.2023YF-82).

Declaration of Competing Interest

We declare that we have no conflict of interest.