The traditional quantification of carbon emission levels in distribution networks mainly relies on the nodal carbon potential of the system. For a specific node, it is described as the ratio of the carbon flow into that node to its active power. Physically, it represents the carbon emission per unit of electricity consumed at the node, corresponding to the carbon emission density of the generator at that node and the weighted average of the carbon flow densities of adjacent nodes concerning the injected power from generators and adjacent nodes. The unit is kgCO₂/kW·h (Zhou et al., 2012) Eq. 1. e j = ∑ j = 1 N P j ⋅ ρ j + P G , j ⋅ ρ G , j ∑ j = 1 N P j + P G , j (1)

In the equation, e _j represents the carbon potential of Node j, P _j represents Injected power at node j, P _G,j represents the injected power from generator units connected to node j, ρ _j represents the carbon flow density at node j, ρ _G,j represents the carbon potential of generatorn units connected to node j.

The traditional calculation for dynamic carbon emission factor in distribution networks is as follow Eq. 2 (Zhou and Kang, 2019): C D i s , t = ∑ j = 1 , t ∈ T N P j , t × e j , t ∑ j = 1 , t ∈ T N P j , t (2)

In the equation, C _Dis,t represents the dynamic carbon emission factor of the urban distribution network at time t, P _j,t represents the active load magnitude of node t at time t, e _j,t represents the carbon potential magnitude of node t at time t, calculated based on the carbon emission flow theory, N represents the set of nodes within the distribution network.

The traditional dynamic carbon emission factor for distribution networks reflects the carbon emissions per unit of electricity consumption at time t. However, the improved dynamic carbon emission factor takes into account the equivalent carbon emissions generated by adding new energy sources. During these times, all electricity consumption in the distribution network comes from photovoltaic output, resulting in a dynamic carbon emission factor of zero for the distribution network. In such cases, users, when considering changes in their electricity consumption based on the dynamic carbon emission factor, cannot perceive the curtailed solar power at the moment when the carbon emission factor is zero. The closer its value is to zero, the higher the level of new energy consumption. Therefore, the equivalent reduction in carbon emissions is also different in article (Ge et al., 2022). The improved expression for the calculation of the dynamic carbon emission factor is as follows: C D i s , t ¯ = ∑ j = 1 , t ∈ T N P j , t × e j , t − P Q , t × e 0 , t ∑ j = 1 , t ∈ T N P j , t (3)

In the equation, C D i s , t ¯ represents the improved dynamic carbon emission factor for the distribution network, considering the curtailed solar power, P _Q,t represents the curtailed solar power in the distribution network at time t, e _0,t represents the carbon potential of the main grid at time t.

Based on the carbon potential matrix E formed by the carbon potential at each node in the distribution network and the active power flow matrix at load nodes, the carbon emissions at the load nodes can be determined, clearly defining the carbon emission responsibilities of different nodes.

The active power flow matrix at a specific node can be defined as a diagonal matrix, as shown below: P = P 11 0 0 0 ⋱ 0 0 0 P n n (4)

Since there are no losses in the active power transmission between nodes, the off-diagonal elements are 0, the calculation for the diagonal elements is as follows Eq. 5 (Chen et al., 2023): P j j = ∑ l ∈ N + P l , l ∉ k + ∑ l ∈ N + P l + P G , l , l ∈ k + (5)

In the equation,k⁺ represents the set of branches connected to node j that are adjacent and have generating units with power output, when a specific node is not adjacent to any generating units, that is l ∉ k , the diagonal elements of the active power flow matrix P _jj is the active power flow entering node j; when there is a generating unit on a specific branch, then the diagonal element of the node adjacent to that branch P _jj is the sum of active power flow entering node j and the outputs of generating units connected to that node.

Combining (Eqs. 3, 4), the calculation method for the node carbon potential matrix is as follows for (Chen et al., 2023) Eq. 6: E = P − P B T − 1 P G , i n T Q G (6)

In the equation, E represents the node carbon potential matrix, P _B represents the matrix of branch flow distribution, P G , i n T represents the matrix of generating unit injection flow distribution, Q _G the matrix of carbon emission intensity for generating units.

In distribution networks with a high proportion of photovoltaics, the carbon potential at a node is affected by both photovoltaic power sources and energy storage elements in discharge mode. If the distribution network lacks generating units and energy storage components, then in such a scenario, the carbon potential at all nodes equals the carbon potential of the main network. However, if the distribution network includes generating units and energy storage components, given the main network’s carbon potential, distribution network load, and generating unit outputs, it becomes possible to calculate the distribution of branch power flows and branch carbon flow density. Subsequently, node carbon potentials can be determined based on the carbon flow density of the branches.

If the dynamic carbon emission factor matrix C T = C D i s , 1 , C D i s , 2 , , . . . , C D i s , T for the given time period T is provided, the improved dynamic carbon emission factor matrix for the system C T ¯ = C D i s , 1 ¯ , C D i s , 2 ¯ , . . . , C D i s , T ¯ , combining the matrix of active power load for the system P L , t = P L , 1 , P L , 2 , . . . , P L , T , the total carbon emissions R _T for the entire distribution network during that time period can be obtained as follow: R T = C T ¯ ⋅ P L , t T ⋅ Δ T (7)

Traditional incentive-based demand response (Xu and Guo, 2023) mainly involves users adjusting or reducing electricity consumption during peak demand periods to receive discounts or compensation. This paper proposes an incentive-based demand response load considering the dynamic carbon emission factor. In this approach, agreements are pre-signed between the power supplier and the consumer. The dispatch center issues instructions the day before the demand response, and based on the dynamic carbon emission factor, the load is transferred from high carbon periods to low carbon periods. The load remains unchanged during neutral carbon periods. According to the agreement, users receive compensation after load transfer. The expression for compensation is as follows for Eq. 8 (Ge et al., 2022): W I D R i n d = ∑ t = 1 N β i n d P i n , t i n d = ∑ t = 1 N β i n d P o u t , t i n d W D R E c o m p = ∑ t = 1 N γ c o m p P o u t , t c o m p (8)

In the equation, W I D R i n d represents compensation for incentive-based load, β i n d represents compensation coefficient for transferable load, P i n , t i n d represents the amount of load transferred in at time t, P o u t , t i n d represents the amount of load transferred out at time t, W D R E c o m p represents compensation for reducible load, γ c o m p represents compensation coefficient for reducible load, P o u t , t c o m p represents the amount of load reduction at time t.

Price-based demand response in (Li et al., 2023) involves changing the original electricity consumption behavior based on changes in electricity prices. It mainly includes two methods: time-of-use pricing demand response and tiered pricing demand response. This paper primarily adopts time-of-use pricing demand response, aiming to improve the economic and low-carbon performance of the distribution network by responding to variations in electricity prices.

For electricity consumption periods under different carbon emission intensities, users will adjust their electricity consumption structure (Kaile et al., 2020). The electricity price level will not only affect the load of that period, but also affect the load of other periods, which is a multi period demand response. Based on the relationship between electricity quantity and price demand balance, and the multi-period electricity quantity-price elasticity matrix, users’ response behavior can be comprehensively characterized. The ratio of electricity quantity to price change rate can be described as the electricity quantity-price elasticity matrix, representing the sensitivity of electricity consumption in each period to prices. The expression is as follows for Eq. 9 (Zhou et al., 2016): χ = Δ E Δ P ⋅ P E (9)

In the equation, χ represents electricity quantity-price elasticity indicator, E and Δ E the quantity of electricity and its variation respectively, P and Δ P represent the price and its variation respectively.

The expression for the electricity quantity-price elasticity matrix M under time-of-use pricing is Eq. 10 (Zhou et al., 2016): M = δ h h δ h m δ h l δ m h δ m m δ m l δ l h δ l m δ l l (10)

In the equation, δ h h 、 δ m m 、 δ l l represent the self-elasticity coefficients of electricity quantity to price for high-carbon, normal-carbon, and low-carbon periods, the rest are cross-elasticity coefficients.

The expressions for the self-elasticity coefficient and cross-elasticity coefficients for the high-carbon period are as follows for Eqs 11, 12 (Zhou et al., 2016): δ h h = − a h e h − a h e h + b h (11) δ h m = a m e h − a h e h + b h (12)

In the equation, a h , b h , a m , b m , a l , b l represent the parameters of the electricity quantity-price curve for the high-carbon, normal-carbon, and low-carbon periods, respectively.

The expression for electricity consumption in the high-carbon, normal-carbon, and low-carbon periods after multi-period demand response is as follows Eq. 13 (Ge et al., 2022): E 1 = E 0 + E 0 , h 0 0 0 E 0 , m 0 0 0 E 0 , l ⋅ M ⋅ Δ P h / P 0 , h Δ P m / P 0 , m Δ P l / P 0 , l (13)

In the equation, E ₁ represents electricity consumption in multiple periods after demand response, E ₀ represents electricity consumption in the periods before demand response, E _0,h、E _0,m、E _0,l represent electricity consumption in the high-carbon, normal-carbon, and low-carbon periods before price-based demand response, P _0,h、P _0,m、P _0,l represent the fixed electricity prices in the high-carbon, normal-carbon, and low-carbon periods before price-based demand response, respectively.

With a daily cycle and a 1-h sampling interval, assuming the set of dynamic carbon emission factors for all time periods on a given day is C 24 = C D i s , 1 , C D i s , 2 , , . . . , C D i s , 24 , Utilizing fuzzy membership functions to determine the degree to which each point belongs to peak or off-peak time periods, Among them, employing a large-type trapezoidal membership function to calculate the membership degree μ1 for peak periods, Using a small-type trapezoidal membership function to calculate the membership degree μ2 for off-peak periods, as shown below for Eq. 14 (Shao et al., 2021): μ 1 C D i s , i = C D i s , i − ⁡ min C 24 max C 24 − ⁡ min C 24 μ 2 C D i s , i = max C D i s , i − C 24 max C 24 − ⁡ min C 24 (14)

The peak and off-peak membership degrees for the maximum point on the dynamic carbon emission factor curve are 1 and 0, respectively, while for the minimum point, they are 0 and 1, Assuming a membership degree threshold of σ 1 , σ 2 ∈ 0 , 1 , dividing the periods μ 1 C D i s , i > σ 1 into high carbon periods, dividing the periods μ 2 C D i s . i > σ 2 into low carbon periods and categorizing the remaining periods as moderate carbon periods.

As shown in Figure 1, for different user scenarios, first determine the load power of industrial and commercial loads. For industrial loads, there are two main scheduling methods. One is to arrange production based on the high and low electricity prices, and the other is to respond to transferable load scheduling instructions and report to the dispatch center the load arrangement after demand response. Unlike industrial loads, commercial loads cannot transfer loads and are not affected by electricity prices. The dispatch center mainly issues instructions to reduce loads, and the load reduction is completed in the specified time period. The dispatch center provides compensation according to the agreement, ultimately reducing the dispatch cost and carbon emissions of the distribution network (Song et al., 2023b; Shi et al., 2024).

For the modified IEEE 33-node system in the commercial load scenario, the locations of various photovoltaic (PV) units and energy storage are shown in Figure 2. In this study, three distributed PV units with a capacity of 400 kW each are installed at nodes 14, 19, and 28. Additionally, an energy storage system with a capacity of 400 kW/400 kWh is added at node 32.

Schematic diagram of the modified IEEE 33-node system for the industrial load scenario as shown in Figure 3. Distributed photovoltaic units with capacities of 1,000 kW are installed at nodes 6, 15, 20, and 28. Additionally, an energy storage device with a capacity of 1000 kW/1,000 kWh is added at node 32.

The carbon potential of the main network has been predetermined, as referenced in [4], and its temporal distribution is illustrated in Figure 4.

Distributed photovoltaic power generation is mainly influenced by the intensity of sunlight and is also affected by the temperature and humidity of the photovoltaic power generation equipment (Wu et al., 2019; Li et al., 2022b). In this paper, we mainly consider standalone photovoltaic power generation systems and do not consider the issue of power flow feedback. And energy storage and photovoltaic only provide output and do not participate in actual scheduling operations. The photovoltaic output is based on the power generated by a typical day during the summer solstice in a certain area of Jiangsu Province, sampled at 1-h intervals. The load data and photovoltaic output are shown in Figure 5.

On the summer solstice day, the load exhibits a classic double-peak pattern. The commercial load output accounts for approximately 10% of the industrial load output. The photovoltaic (PV) output starts at 6 a.m., peaks around noon at 12 p.m., and concludes by 10 p.m. in the evening.

The time period for reducing commercial loads is mainly concentrated from 12:00 to 19:00, while the load output remains unchanged in other time periods, as shown in Figure 6.

Combined with Cplex solution, the results of traditional commercial load optimization scheduling are shown in Table 1. After traditional commercial load demand response on the summer solstice, it decreased by 4.71%, the scheduling cost decreased by 5.2%, carbon emissions decreased by 10.37%, and the compensation cost was 131.17 yuan.

On the other hand, we also compared and analyzed the scheduling results using traditional carbon emission factors and improved carbon emission factors, as shown in Table 2. Due to the fact that scheduling costs do not involve carbon emission factors, whether traditional or improved carbon emission factors are used, scheduling costs remain unchanged; The improvement of the carbon emission factor takes into account the equivalent reduction in carbon emissions from photovoltaic waste, resulting in a decrease of 475.81 kg in carbon emissions compared to before the improvement.

Combining the distribution network and photovoltaic output data, the dynamic carbon emission factors of the distribution network before and after the improvement can be obtained, as shown in Figure 7. The positive half of the dynamic carbon emission factor on the Y-axis represents the overall carbon emission level of the distribution network, with values closer to zero indicating lower carbon emissions at that time. From 11 to 16 o’clock, when the photovoltaic output in the distribution network is high, the traditional carbon emission factor is zero, indicating the occurrence of curtailed photovoltaic power. However, it cannot reflect the difference in curtailed photovoltaic power at different times. The improved dynamic carbon emission factor shows negative values during curtailed periods, with higher absolute values indicating higher curtailed amounts, and values closer to zero indicating higher levels of renewable energy integration.

The paper divides the 24 h of a day into three electricity consumption periods: high carbon, medium carbon, and low carbon, based on the magnitude of carbon emissions. Each period consists of 8 h. The decision variable in this study is the time-of-use electricity price. The electricity price influences the amount of electricity consumed during different periods. Additionally, considering the variation in curtailed solar power during periods with negative carbon emission factors, the time-of-use electricity prices also differ accordingly. The time-of-use electricity prices after demand response are illustrated in Figure 8.

After demand response of industrial load, the distribution grid load is shifted from periods with higher carbon emission factors to those with lower carbon emission factors, as shown in Figure 9. The optimization results of demand response for industrial load are presented in Table 3. Compared to the situation before demand response, the carbon emissions have decreased by 1976.70 kg, and the curtailed solar energy rate has reduced by 4.73%. Additionally, the reduction in carbon emissions before and after demand response based on traditional carbon emission factors is 1250.38 kg. Considering the improved carbon emission factors after demand response, there is an increase of 36.7%. On the other hand, similar to the commercial load scenario, we also compared and analyzed the demand response scheduling results using traditional and improved carbon emission factors, as shown in Table 4. The scheduling cost remained unchanged, and the carbon emissions decreased by 1448.05 kg.

In addition, to verify the flexibility and transferability of the model, in a 96 node system, as long as the output of distributed power sources and energy storage is given, Still, like 33 nodes, demand response is achieved by adjusting electricity prices for different periods of carbon emission intensity, thereby optimizing the objective function of the entire distribution network (Yang et al., 2023). On the other hand, in the scenario examples involved in this article, due to the relatively simple 10 kV voltage level network, which does not involve many substations and distributed power sources, there is no need for a lot of nodes. Only 33 nodes are needed to simulate the distribution network in the real world and perform basic operation scheduling operations.