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Multisource coordinated lowcarbon optimal dispatching for interconnected power systems considering carbon capture
Energy Informatics volume 7, Article number: 61 (2024)
Abstract
The overall electricity consumption of electrolytic aluminum and ferroalloy loads is significant, and some of these loads have dispatch potential that can be used to locally absorb wind power while reducing dependence on conventional thermal power. To characterize the uncertainty of wind power, a fuzzy set of wind power forecasting error probability distribution based on the Wasserstein distance was first established, and the approximate radius of the fuzzy set was corrected under extreme scenarios. By introducing joint chance constraints, the inequalities of uncertain variables were established at the lowest confidence level to improve the reliability of the model. Next, a twostage distributed robust optimal scheduling model for sourceload coordination was developed. In the first stage, wind power forecasting information was fully utilized to schedule the electrolytic aluminum load and optimize unit commitment. In the second stage, the uncertainty of wind power output was considered to schedule the ferroalloy load and optimize unit output. The model was approximately transformed into a mixedinteger linear programming problem and solved using a sequential algorithm. The IEEE 24bus system was used for case validation. The validation results show that the model can effectively improve wind power absorption capacity, reduce overall operating costs, and achieve a balance between low carbon emissions and robustness.
Introduction
In recent years, with the increasing penetration rate of wind power, due to the limited regulation capacity of conventional power supply, the grid peak regulation and frequency regulation capacity is insufficient, resulting in a large number of wind abandonment phenomenon (Ran et al. 2020; Ye et al. 2021; Li et al. 2021). Pi et al. (2020) improves wind power consumption capacity by tapping the flexible regulation role of high energycarrying loads. High energycarrying loads are characterized by large capacity and good adjustability, which can solve the problem of obstruction of wind power consumption by tapping the loadside adjustability to consume wind power in the vicinity. High energycarrying loads have the characteristics of discrete and continuous regulation, and can participate in wind power consumption from both the electricity and power aspects, the most representative of which are electrolytic aluminum loads and ferroalloy loads (Liu and Lin 2021). Under the background of high wind power penetration, traditional scheduling methods are difficult to cope with the various challenges brought by wind power grid integration, so the study of sourceload coordinated optimal scheduling method considering the uncertainty of wind power is of practical research significance.
Zhang et al. (2024) introduces a hybrid MLPBiLSTMTCN model for highly accurate shortterm wind power prediction by combining the strengths of multiple networks and optimizing weights via linear programming. However, it mainly focuses on deterministic forecasts, with limited work on probabilistic intervals. Ren and Bao (2023) presents a grid planning strategy for active distribution networks using an improved kmeans algorithm, addressing voltage and load fluctuation issues from distributed photovoltaics. The approach focuses on shape similarity, but may require further consideration of grid dynamics and operational constraints.
Yan et al. (2023) presents a mixedinteger linear programming method for optimizing hybrid AC/DC distribution networks, enhancing efficiency and ensuring optimal solutions through structured hierarchies and linearization. However, it lacks discussion on practical challenges and limitations in largescale implementations. Xu et al. (2023) introduces an economic dispatch model for combined heat and power systems, optimizing carbon emissions with mixedinteger linear programming and considering both networks' constraints. The model's effectiveness is demonstrated through case studies, but its robustness and scalability under varying conditions need more analysis. Zhang and Zhou (2023) develops a MILPRL algorithm improving the economic performance and adaptability of microgrid dispatch in response to renewable energy fluctuations and load changes. The scalability and practicality of this algorithm in larger or more complex grids, however, are not thoroughly examined. Ma et al. (2023) presents an optimization model for emergency resource allocation in power systems, utilizing dynamic programming and an enhanced particle swarm algorithm to effectively handle sudden electrical incidents and meet dynamic resource needs. However, the study focuses solely on economic goals and lacks a multiobjective approach, suggesting future research should include more decisionmaking factors.
Xu et al. (2021) introduces a system for restructuring smart distribution networks using dynamic programming, enhancing voltage quality and reducing losses with notable sensitivity and reliability. Yet, the paper does not adequately verify the system’s performance in largescale networks or discuss the dynamic programming algorithm's efficiency and scalability for complex network topologies. Elghitani and Zhuang (2018) introduces an AGC optimization dispatch method for virtual power plants using an enhanced quantum genetic algorithm, boosting frequency regulation profits and renewable energy integration. However, the paper mainly examines optimization results and does not sufficiently analyze the algorithm's applicability and stability across various grid complexities. Qi et al. (2021) proposes a comprehensive energy optimization method using an improved genetic algorithm that adaptively adjusts crossover and mutation rates, enhancing global search capabilities and robustness. Nevertheless, the discussion on the algorithm’s practicality and adaptability in diverse or larger integrated energy systems is limited.
The Distributed Robust Optimization (DRO) method has become a better way to deal with power system uncertainty because it combines the advantages of SO (Stochastic Optimization) and RO (RobusOptimization) methods (Yu et al. 2018; Liu et al. 2020; Lin and Tian 2017). Compared with other DRO methods, the Wasserstein distancebased DRO method has better risk resistance by constructing an initial empirical probability distribution from historical data instead of weighting the probability distribution (Lin and Tian 2017; Nadeem et al. 2019; Yan et al. 2019; Sun et al. 2021). The selection of the radius of the Wasserstein sphere plays a crucial role in the performance of the Wasserstein distancebased DRO model. Sun et al. (2019), Zhang et al. (2012) utilized an approximate computational method to determine the Wasserstein ball radius, but did not consider the effect of differences in the ability of specific models to cope with uncertainty on the selection of the Wasserstein ball radius.
This paper employs the Distributionally Robust Optimization (DRO) approach, which presents four key advantages:

(1)
Enhanced Model Adaptability: By using fuzzy uncertainty sets that allow error values to vary within a specified range rather than fixing them at a single point, the predictive model can adapt to fluctuations in actual wind power output. This increases the model’s adaptability and accuracy under different operational conditions.

(2)
Improved Economic Performance of Dispatch Decisions: The application of fuzzy set methods in electrogas coupled systems optimizes resource allocation while meeting system operational constraints, reducing operational costs, and enhancing economic efficiency.

(3)
Handling of Forecast Error Distribution Information: Fuzzy set methods do not rely on specific distribution assumptions but use fuzzy logic to manage uncertainty. This makes the model more versatile and applicable to various realworld scenarios.

(4)
Reduced Computational Complexity: In largescale systems, fuzzy set methods can lower computation requirements. By employing approximation techniques and heuristic algorithms, they enhance solution speed, allowing the model to quickly respond to system changes.
Along these lines, a twostage distributed robust optimization scheduling model with sourceload coordination based on joint chance constraints is developed, and the conservatism of the DRO model results is further reduced by optimizing the Wasserstein sphere radius and introducing joint chance constraints. In the first stage, thermal power units and aluminum loads are scheduled with the objective of minimizing the total system operating cost considering wind abandonment cost. In the second stage, the uncertainty of wind power prediction error is considered, the fuzzy uncertainty set of probability distribution based on Wasserstein distance is established, and the adjustability of DRO algorithm is optimized by introducing the joint opportunity constraints, and then the twophase DRO model is transformed into a mixedinteger linear programming problem, which is solved by sequential algorithms, and the worst probabilitydistributed thermal unit output correction and ferroalloy load dispatching quantities are obtained. Ferroalloy load dispatch, and finally get the optimal solution of the model under the worst probability distribution. This approach increases the utilization of wind power, which is fully allocated to the production of electrolytic aluminum and ferroalloys. Simultaneously, it indirectly achieves the implicit goal of reducing carbon emissions. Finally, the effectiveness of the twostage DRO model proposed in this paper is verified through examples.
Coordinated sourceload scheduling model
The sourceload coordinated dispatch model containing wind farms, thermal power units, aluminum electrolysis and ferroalloy loads is shown in Fig. 1. In this paper, we consider the incentivebased demand response model, so that the dispatch center signs compensation agreements with the electrolytic aluminum and ferroalloy enterprises participating in the dispatch. The alumina load as a discrete adjustable load participates in the phase 1 dispatch to optimize the unit mix. The ferroalloy load participates in phase 2 dispatch as a continuously adjustable load, which can provide spare capacity for the units when the spinning reserve of the units is insufficient, so as to maximize the consumption of wind power.
Considering the distinct energy characteristics of manufacturing aluminum alloys and iron:

(1)
Aluminum has a melting point of approximately 660.3 °C (1220.5°F), and the operational temperature of devices is generally set above 700 °C. Iron has a higher melting point of about 1538 °C (2800°F), with devices typically operating at temperatures above 1600 °C.

(2)
The smelting time for aluminum is relatively short, with a process cycle ranging from 0.5 to 1.0 h. In contrast, iron smelting takes significantly longer, often spanning several hours per cycle.

(3)
The average carbon emissions for producing one ton of electrolytic aluminum include 10.7 tons from electricity, whereas producing one ton of iron results in an average of 3.07 tons of carbon emissions from electricity.
Given the quick adjustment capabilities in aluminum smelting and its higher carbon emissions, it is prudent to consider the unique characteristics of both aluminum and steel production when adopting a twostage optimization scheduling approach.
Probability of wind power forecast error based on Wasserstein distance distribution fuzzy uncertainty set
Wind power prediction error set
Historical data of wind power prediction error using extracted wind power group M denotes the set obeyed by the wind power prediction error:
Which Z = \({[{I}_{\zeta },{I}_{\zeta }]}^{T}\), z = \({[{1\mu }_{\zeta },{\mu }_{\zeta }]}^{T}\), \({I}_{\zeta }\) and \({\mu }_{\zeta }\) are the variance and mean, respectively, of the normalized historical wind power forecast error.
Wasserstein distance for probability distributions
Considering the following advantages of Wasserstein distance:
(1) Compared to other distance metrics such as KL divergence or JS divergence, Wasserstein distance is more robust to outliers or minor inconsistencies in distributions. This makes it particularly useful in practical applications, especially when data contains noise or outliers.
(2) In certain models, methods based on Wasserstein distance often achieve better stability and convergence.
The real probability distribution that the wind power prediction error ξ obeys is denoted as P. The limited wind power historical data ξ cannot accurately characterize P, but the empirical probability distribution \({P}_{M}\) can be obtained, and then the fuzzy uncertainty set that contains P as much as possible is constructed with \({P}_{M}\) as the center. In this paper, we construct a fuzzy uncertainty set based on the Wasserstein distance of the probability distribution, and the Wasserstein distance between the empirical probability distribution and the real probability distribution is defined as:
Which: Π is the joint probability distribution of \({P}_{M}\) and P.
Constructing a fuzzy uncertainty set for the probability distribution of wind power prediction errors
According to the literature (Elghitani and Zhuang 2018), the radius \(\rho (M)\) at a confidence level β can be roughly determined by the following equation roughly determined:
where: D is the coefficient, which can be determined by Eq. (4).
where: μ is the mean value of the sample; the minimum value of the function on α can be obtained by the bisection search method; k is the group number of wind power prediction error historical data. The minimum value of the function on α can be obtained by the bisection search method; k is the group number of the historical data of wind power prediction error.
From Eq. (3), the radius \(\rho (M)\) will gradually decrease as the number of extracted samples M increases, and when M tends to infinity, the radius \(\rho (M)\) will be infinitely close to 0, which will make the robustness gradually weaker. In order to ensure the robustness of the model, this paper incorporates the extreme scene into the extracted scene, the extreme scene is the situation where the uncertain variable obtains the extreme value, and the extreme scene set in this paper is a onedimensional scene set containing the uncertain variable of wind power, and the two endpoints of the onedimensional data need to be selected as the extreme scene. If the extreme value of wind power historical data is \({w}_{min}\) and \({w}_{max}\). These two extreme sample scenarios are involved as extracted samples in the semidiameter \(\rho (M)\) calculation, thus providing an effective support for the robustness of the model.
The fuzzy uncertainty of wind power prediction error based on Wasserstein distance is expressed as follows set expression is as follows:
where: M (Ξ) denotes the full probability distribution on Ξ.
Twostage distributed robust optimal scheduling model for sourceload coordination
Considering the uncertainty of wind power, by appropriately adjusting the scheduling of electrolytic aluminum loads, a portion of the wind power can be absorbed. This adjustment can appropriately reduce the operational costs of the system, achieving the objective of the first phase of optimization. Building on this, by further adjusting the scheduling of ferroalloy loads, the aim is to maximize the absorption of wind power. The general form of the twostage distributed robust optimization model is as follows:
where X and \({Y}_{0}\) are their corresponding feasible domains, \({E}_{P}[g(y,\zeta )]\) is the minimum of the expected value of the phase 2 scheduling cost under the worstcase probability distribution.
Phase 1 deterministic optimal scheduling model
The scheduling model for Phase 1 is based on wind power forecast information and has the minimum integrated system cost as the objective function as follows:
where: \({C}_{11}\) represents the generation cost of the unit;\({C}_{12}\) represents the startup and shutdown cost of the unit; \({C}_{2}\) represents the cost of wind abandonment; \({C}_{3}\) represents the cost of aluminum load dispatch.\({C}_{2}\) represents the cost of wind abandonment, and \({C}_{3}\) represents the cost of aluminum load dispatch.
where: T is the total number of time slots; \({N}_{g}\) is the total number of thermal power units; \({P}_{i,t}\) is the power generation of the ith unit at the ttime; \({U}_{i,t}\) is the unit start/stop flag bit, which is 1 for 1. \({P}_{i,t}\) is the power generation capacity of the ith unit at time t; \({U}_{i,t}\) is the flag bit of the unit's on/off status; 1 represents the on state and 0 represents the off state; \({a}_{i}, {b}_{i}\),\({c}_{i}\) are the flag bits of the thermal power unit. \({U}_{i,t}\) is the on/off flag bit of the unit, a 1 represents the on state and a 0 represents the off state; \({a}_{i},{b}_{i}\),\({c}_{i}\) are the cost coefficients of the thermal power unit; \({H}_{Ti}\) and \({c}_{i}\) are the cost coefficients of the thermal power unit. \({a}_{i},{b}_{i}\),\({c}_{i}\) are the cost coefficients of thermal power units; \({H}_{Ti}\) and \({H}_{Di}\) are the startup and shutdown costs of thermal power units, respectively; \({u}_{i,t}^{on}\) are the startup and shutdown costs of thermal power units; \({u}_{i,t}^{on}\) represent the startup variables of thermal power units; 1 means startup, and 0 means no startup action;\({u}_{i,t}^{off}\) represents the unit shutdown variable, a value of 1 means shutdown, and a value of 0 means no shutdown action is performed. \({u}_{i,t}^{off}\) represents the unit shutdown variable, a value of 1 means shutdown, a value of 0 means no shutdown action; \({N}_{w}\) is the number of wind farms; φ is the wind abandonment penalty \({N}_{w}\) is the number of wind farms; φ is the wind abandonment penalty coefficient; \({P}_{i,t}^{wp}\) is the predicted wind power output; \({P}_{i,t}^{wr,wp}\) is the predicted wind power output; \({P}_{i,t}^{wr}\) is the planned wind power output; \({a}_{d,i}\) is the dispatching cost of the alumina load; and \({a}_{d,i}\) is the dispatch cost of the aluminum load; \({\Delta P}_{ad,i}^{D}\) is the dispatch cost of the aluminum load; \({\Delta P}_{i,t}^{D}\) is the dispatch quantity of the aluminum load.

(1)
Power balance constraints
$$\sum^{\underset{{N}_{g}}{i=1}}{P}_{i,\iota }+\sum^{\underset{{N}_{w}}{i=1}}{P}_{i,\iota }^{wp}={P}_{\iota }^{L}+\sum^{\underset{{N}_{d}}{i=1}}\Delta {P}_{i,\iota }^{D}$$(9)where: \({P}_{t}^{L}\) is the total load in addition to the electrolytic aluminum load at time t; \({N}_{d}\) is the number of electrolytic aluminum loads; \({\Delta P}_{i,t}^{D}\) is the scheduling amount of active power consumed by the ith electrolytic aluminum load at time t.

(2)
Thermal power unit operation constraints
$$\left\{\begin{array}{c}{U}_{i,t}{P}_{i,min}\le {P}_{i,t}\le {U}_{i,t}{P}_{i,max}\\ {P}_{i,t}{P}_{i,t1}\le {r}_{i,t}^{+}\\ {P}_{i,t1}{P}_{i,t}\le {r}_{i,t}^{}\end{array}\right.$$(10)where: \({P}_{i,min}\) and \({P}_{i,max}\) represent the minimum and maximum output power of the ith unit, respectively; \({U}_{i,t}\) represents the startstop state, with 1 for the startup state and 0 for the shutdown state; \({r}_{i,t}^{+}\) and \({r}_{i,t}^{}\) are the limit values of the upward and downward climbing power, respectively.

(3)
Currents constraints
The specific form of the tidal model constraints used as follows:
$${\overline{P} }_{line}\le {Q}^{g}{P}_{t}^{g}+{Q}^{w}{P}_{t}^{w}{Q}^{l}{P}_{t}^{l}{Q}^{d}{\Delta P}_{t}^{d}\le {\overline{P} }_{line}$$(11)where: \({Q}^{g},{Q}^{w}\),\({Q}^{l }\text{ and }{Q}^{d}\) are the power transfer allocation coefficient matrices, respectively; \({P}_{t}^{g}\) is the diagonal matrix of thermal unit output; \({P}_{t}^{w}\) is the diagonal matrix of wid power output. \({P}_{t}^{g}\) is the diagonal matrix of thermal unit output; \({P}_{t}^{w}\) is the diagonal matrix of wind power output; \({P}_{t}^{l}\) is the load demand matrix; \({\Delta P}_{t}^{d}\) is the load demand matrix; \({\Delta P}_{t}^{d}\) is the aluminum load dispatch quantity matrix. \({\overline{P} }_{line}\) is the upper limit of branch circuit power.

(4)
Wind power output constraints
$$0\le {P}_{i,t}^{wr}\le {P}_{i,t}^{wp}$$(12) 
(5)
Electrolytic Aluminum Load Constraints
$$\left\{\begin{array}{l}\begin{array}{c}{P}_{i,t}^{\text{D}}={P}_{0}^{\text{D}}+\Delta {P}_{i,t}^{\text{D}}\\ {P}_{i\text{min}}^{\text{D}}\le {P}_{i,t}^{\text{D}}\le {P}_{i\text{max}}^{\text{D}}\\ \sum_{k=t}^{t+{T}_{\text{D}}^{\text{min}}} {\text{flag}}_{\text{D}}^{k}=0\end{array}\\ \left.{\text{flag}}_{\text{D}}^{\iota }=\left\{\begin{array}{c}0,{P}_{i,t}^{\text{D}}{P}_{i,t1}^{\text{D}}=0\\ 1,{P}_{i,t}^{\text{D}}{P}_{i,t1}^{\text{D}}\ne 0\end{array}\right..\right\}\\ \sum_{t=1}^{T} {\text{flag}}_{\text{D}}^{t}\le {\Upsilon }_{\text{D}}^{\text{max}}\end{array}\right.$$(13)where: \({P}_{i,t}^{D}\) is the power of the ith electrolytic aluminum load after regulation at time t, \({P}_{0}^{D}\) is the rated power of the electrolytic aluminum load, \({\Delta P}_{i,t}^{D}\) is the amount of power regulation, \({P}_{imin}^{D}\) and \({P}_{imax}^{D}\) are the lower and upper limits of power, respectively; \({T}_{D}^{min}\) is the minimum stable operation duration of the aluminum load; \({flag}^{k}\) is the minimum stable operation duration of the aluminum load. \({T}_{D}^{min}\) is the minimum stable operation duration of the aluminum electrolysis load; \({flag}_{D}^{k}\) is the power regulation flag of the aluminum electrolysis load; \({\Upsilon}_{D}^{max}\) is the maximum power regulation flag of the aluminum electrolysis load in a day. max D is the maximum number of regulation times of the aluminum load in a day.
Phase 2 distributed robust optimal scheduling under joint chance constraints model
On the basis of the phase 1 scheduling plan, the phase 2 scheduling model takes into account the uncertainty caused by the wind power prediction error ξ, corrects the output of thermal power units and gas units, and schedules the ferroalloy loads to jointly consume the wind power.
where: \({C}_{1}^{*}\) represents the thermal unit output adjustment cost; \({C}_{2}^{*}\) represents the dispatch cost of ferroalloy loads; and \({C}_{3}^{*}\) represents the wind abandonment cost in phase 2. The specific expression is
where: \({c}^{g}\),\({c}^{s}\),\({c}^{w}\) are the dispatch cost coefficients of the corresponding variables; \({\Delta P}_{i,t}\) denotes the correction of thermal and gas unit output;\({\Delta P}_{i,t}^{S}\) denotes the dispatch of ferroalloy loads; \({\Delta P}_{i,t}^{w}\) denotes the prediction error of wind power; and \({\Delta P}_{i,t}^{wf}\) denotes the actual wind power output.
1) Power balance constraints
where:\({\Delta P}_{i,t}^{S}\) is the ferroalloy load dispatch quantity.
2) Thermal power unit output constraints
3) Trend constraints
4) Ferroalloy load constraints
where: \({P}_{i,t}^{S}\) is the operating power of the ith ferroalloy load at time t; \({P}_{i,t}^{{S}_{0}}\) is the rated power of the ferroalloy load; \({\Delta P}_{i,t}^{S}\) is the load regulation amount; \({P}_{imin}^{S}\) and \({P}_{imax}^{S}\) are the minimum and maximum operating power of the load, respectively.
5) Actual wind power output constraints
Model solving
The deterministic model in phase 1 is relatively easy to solve, while the distributed robust optimal scheduling model with joint chance constraints in phase 2 is difficult to solve of the problem, as it constitutes an infinite dimensional optimization problem. In order to solve the above problem, according to the literature (Nadeem et al. 2019), the stage 2 decision variable function y (ξ) can be approximated and replaced by Y(ξ), which is the functional correlation matrix of y and ξ. g(y, ξ) in Eq. (6) is approximated and replaced by \({c}_{2}^{T}\) Yξ +\({c}_{3}^{T}\) ξ, which transforms the original problem into a linear decision making problem, and Eq. (21) is reexpressed in the following matrix form
The twostage distributed robust optimization model under joint chance constraints is reexpressed in a general form and modeled as follows:
where: Θ is the feasible domain of the decision variables; \({C}_{x}^{T}, {C}_{1}^{T}, {C}_{2}^{T}, {C}_{3}^{T}\) are the cost coefficient matrices. In order to facilitate the solution, the equation constraints involving x and y have been transformed into corresponding inequality constraints by substituting inequality constraints. Bx ≤ h is the inequality constraint of decision variable x, D \({y}_{0}\) ≤ n is the inequality constraint of decision variable\({y}_{0}\), and E \({y}_{0}{E}^{T}\)+ GY ≤ 1 is the coupling inequality constraint of \({y}_{0}\) and Y.
Approximate approximation of joint chance constraints
The distributed robust optimization model under joint chances is difficult to solve directly, and according to the improved CVa \({R}_{\epsilon }\) constraint approximation method proposed in the literature (Nadeem et al. 2019) Eq. (22) can be approximated as Eq. (24).
where: \({\delta }_{k}\) represents the proportion of the kth inequality; K is the total number of rows of the \({A}^{j}\)(Y) matrix.
Transformation processing of the model
References (Xu et al. 2023) and (Qi et al. 2021) can linearize \(\underset{y\in {Y}_{0}}{min}\underset{P\in \Omega }{max}{E}_{P}\left[{c}_{2}^{T}Y\xi +{c}_{3}^{T}\xi \right]\) and Eq. (24), respectively, and finally transform Eq. (23) into solving Eq. (25):
Sequential iterative algorithm
Equation (25) is still a difficult problem to solve because the distributed robust optimization model obtained earlier becomes a nonconvex problem if \({\delta }_{k}^{j}\) is used as an additional decision variable, so it is necessary to solve (x, \({y}_{0}\),Y) and \({\delta }_{k}^{j}\) separately and iteratively.In order to ensure that the solution of the model is converged, the slack variable q is introduced in the objective function and joint chance constraints of Eq. (25) as follows:
where: ω is a large constant; q_jis the value of the slack variable corresponding to the jth chance constraint \(\left({x}^{1},{y}_{0}^{1},{Y}^{1}\right)\).
Firstly, let \({\delta }_{k}^{j}=1/{K}^{j}\), the solution \(\left({x}^{1},{y}_{0}^{1},{Y}^{1}\right)\) and the minimum value U_{1} of the objective function of the 1st iteration are obtained by solving Eq. (26); then the solution \(\left({x}^{1},{y}_{0}^{1},{Y}^{1}\right)\) is substituted into Eq. (27) to solve the new \({\delta }_{k}^{j}\); then the new \({\delta }_{k}^{j}\) is substituted into Eq. (26) to solve the problem until the result of the νth and ν + 1st iteration is less than the error tolerance, and the obtained solution is the optimal solution of the model shown in Eq. (25).
Example analysis
Simulation results
This example is validated by the improved IEEE 24node system, which consists of 10 thermal power units with a total of 4,676 MW, wind farms with an installed capacity of 1,800 MW, six 300 MW wind farms connected to nodes 1, 2, 11, 12, 12, and 16 in that order, two 500 MW aluminum loads, and one 500 MW ferroalloy load, with a wind abandonment cost of 600 RMB/MWh, the dispatch cost of the electrolytic aluminum load is 30 RMB/MWh, and the dispatch cost of the ferroalloy load is 50 RMB/MWh. The information of each unit is referred to the literature (Yu et al. 2018; Liu et al. 2020). The Gurobi solver is invoked in Matlab through the Yalmip toolbox.
The following scenarios are set up for comparative analysis:
Scenario 1: Do not consider the participation of electrolytic aluminum load and ferroalloy load in scheduling, and only consider the participation of thermal power units in wind power consumption.
Scenario 2: Aluminum, ferroalloy load and thermal power units are considered to participate in wind power consumption.
The wind power and load forecast curves are shown in Fig. 2. From the figure, it can be seen that the load has a large deviation from the predicted power of the wind power, and the largest deviation segments are concentrated in 0:00–5:00, 10:00–15:00, and 20:00–23:00, and the effect of the affiliated scheduling strategy will be verified later in the article.
Figure 3 shows the output of thermal units when wind power is consumed only by thermal units. In the first phase of the dispatching plan, during the 2:00–5:00 time period, wind power is in the most frequent time period, but the load demand is small, there is a great pressure of peaking, at this time section of the unit 4, 5, 7, 8, 9 has been in the shutdown state, unit 1, 2, 3, 6, 10 is in the lowest output state cannot be effective in the consumption of wind power, resulting in a 1071 MWh of abandoned wind power. In the second phase of the scheduling plan, due to the limited climbing capacity of thermal power units to cope with the wind power forecast error capability is insufficient, the second phase of the scheduling results in 1816 MWh of wind abandonment, resulting in a total of 2887 MWh of wind abandonment, the wind abandonment rate reached 7.79%.
Figure 4 shows the unit output for Scenario 2. With the participation of the alumina load in the dispatch, the peaktovalley difference of the net load curve decreases by 200 MWh, and the participation of the ferroalloy load in the dispatch can provide additional standby capacity for the thermal power units, and the unit mix is optimized. In the first phase of the scheduling plan, 546 MWh of wind abandonment is caused only in the periods of 3:00–4:00 and 23:00–24:00, and Unit 2 can be shut down all day. In phase 2, the participation of ferroalloy loads causes only 657 MWh of wind loss, resulting in a total of 1203 MWh of wind loss, with a wind loss rate of 3.40%.
Figure 5 shows the power curves of Scenario 2 after the aluminum load and ferroalloy load participate in the dispatch. During the 2:00–5:00 time period when wind power consumption is limited, the power of the aluminum load is adjusted to the maximum to maximize the consumption of wind power; during the 19:00–22:00 time period when the power of the aluminum load is adjusted to the maximum during the peak period, the power of the aluminum load is adjusted to the maximum during the peak period small to achieve the purpose of peak shaving and valley filling, thus optimizing the unit mix. When the thermal power unit rotating reserve is insufficient, the power consumption of ferroalloy load is adjusted to cooperate with the thermal power unit for wind power consumption, and a total of 971 MWh of wind power is consumed through scheduling ferroalloy load.
Comparing Figs. 3 and 4, it can be seen that the unit 2, the unit 2 can be in a state of shutdown throughout the day, and there is no need for a break in the output power of the unit 8. This also confirms that the method herein reduces the use of thermal power generation and avoids the loss of economy that is caused by the repetitive starting and stopping of the unit.
In the figure, the power below the horizontal axis is the reduction in thermal power consumption through the consumption of wind power, totaling 845 MWh, which is a reduction of about 507 tons of carbon dioxide emissions, based on the fact that in the process of electricity production, the carbon dioxide emissions per megawatthour of electricity are about 0.6 tons.
Comparative analysis of distributed robust optimal scheduling results
This section will validate the effectiveness of the datadriven distributed robust optimization model. Given the limited sample size of wind power forecasting errors, to analyze the impact of different historical sample sizes on the total system cost, it is assumed that the wind power forecasting errors follow a normal distribution with a mean of zero and a standard deviation of 10% of the forecasted wind power value. Test samples of the required size are then drawn from this distribution.
Figure 6 illustrates the trend of scheduling costs at a specific time in the system under various sample sizes, as influenced by changes in the Wasserstein ball radius.
The larger the number of samples, the smaller the reduction in costs before reaching the lowest cost point, and the smaller the radius value corresponding to this cost minimum. This is because, as the number of samples increases, the empirical probability distribution approaches the true probability distribution, consequently reducing the Wasserstein ball radius that just encompasses the true distribution. Once the selected radius exceeds the optimal radius, the scheduling costs begin to rise rapidly. This increase is due to the fact that with a larger radius, the worstcase probability distributions within the probability distribution uncertainty set deviate more from the true distribution, and more reserve resources are set aside, leading to rapidly increasing scheduling costs. Therefore, choosing a reasonable radius is essential.
As shown in Fig. 6, the curves for sample sizes of 1000 and 5000 are nearly identical, indicating that further increases in sample size have a minimal impact on the curve. For computational efficiency, a sample size of 1000 can be chosen as the maximum in practical applications.
Table 1 shows that under different sample sizes, the scheduling results using an adjusted radius are more economical than those using a calculated radius, with less wind curtailment as well. Moreover, the smaller the sample size, the greater the improvement in model scheduling results by adjusting the radius. In situations where historical data is limited, adjusting the Wasserstein ball radius can significantly optimize the scheduling outcomes.
Comparison of results of different uncertainty methods
This paper compares the sourceload coordinated DRO model with traditional SO (Stochastic Optimization) and RO (Robust Optimization) models. Figure 7 graphically contrasts the economic aspects of these three algorithms. When the sample size is small, the DRO solution tends to align with RO optimization; as the sample size increases, the DRO solution shifts towards SO optimization. This demonstrates that adjusting the sample size can balance the economic efficiency and robustness of the DRO solution.
Under varying sample sizes, as the confidence level of the Wasserstein ball radius increases, the economic cost associated with the DRO results also increases. This is because a higher confidence level in the Wasserstein ball radius leads to a larger radius, requiring more reserves to balance wind power forecasting errors, thereby raising overall operational costs. When the confidence level is 0%, the Wasserstein ball radius is zero, and the DRO becomes an SO using the empirical probability distribution as the true distribution. Conversely, when the confidence level is 100%, the Wasserstein ball radius tends towards infinity, turning the DRO into an RO that encompasses all possible probability distributions.
From this analysis, it is evident that adjusting the sample size and the confidence level of the Wasserstein ball radius can effectively balance the economic efficiency and robustness of the DRO model scheduling results.
Conclusion
In this paper, a twostage distributed robust optimization scheduling model for sourceload coordination based on joint opportunity constraints is established in the context of high wind power penetration, and the following conclusions are obtained through the validation of the arithmetic examples:

(1)
The sourceload coordinated scheduling model considering aluminum electrolysis load and ferroalloy load can optimize the combination and output of thermal power units, and achieve a better effect of local consumption of wind power.

(2)
The selection of Wasserstein ball radius has a certain impact on the DRO optimization results, and the selection of the optimal radius by using the method of limit scenario can further reduce the comprehensive operating cost of the system.

(3)
The DRO method based on joint opportunity constraints proposed in this paper can improve the conservatism of the general DRO method by adjusting the sample scale The Wasserstein spherical radius confidence achieves a better balance between economy and robustness.
In the process of rising penetration rate, highquality flexibility resources such as multiple types of energy storage have broader application prospects, and assume an important role in future flexibility scheduling, this followup will further study the multistage planning of diversified flexibility resources, to better cope with future flexibility needs brought about by a high proportion of renewable energy connected to the grid.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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Study on the Framework and Planning Path of “integrated of Source, Grid, Load and Storage” Synergistic Development of New Power System under the Carbon Perspective (GXJJJ00NYS2310022).
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J. Sun: writing—original draft, formal analysis, writing—review and editing. D. Hua, X. Song, M. Liao: formal analysis, validation. Z. Li, S. Jing: software, visualization. All author review and approval the final manuscript.
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Sun, J., Hua, D., Song, X. et al. Multisource coordinated lowcarbon optimal dispatching for interconnected power systems considering carbon capture. Energy Inform 7, 61 (2024). https://doi.org/10.1186/s42162024003677
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DOI: https://doi.org/10.1186/s42162024003677