An energy market interaction framework that considers energy management issues between DNO and multiple prosumer is shown in Figure 1. The figure includes the high-voltage (HV) grid, which provides the required electrical energy and receives excess energy, the local energy interaction market that distributes electrical energy and transmits energy demand and price signals, and two key participants: the DNO, which serves as an intermediary for transactions between the HV grid and prosumers, and the photovoltaic prosumers who generate and consume their own electricity.

As the energy interaction intermediary, the DNO acts as the operator of the distribution network (DN) and the internal price setter. Its responsibilities include: 1) overseeing the operation of the distribution network to ensure system safety and stability while meeting high-quality power requirements from users; 2) participating in the energy market and engaging in energy transactions with the HV grid to balance supply and demand when internal imbalances occur; 3) coordinating internal energy transactions by dynamically setting internal transaction prices based on economic and safety considerations, referencing users’ electricity demand and feed-in tariffs, and allocating internal energy to improve consumption capacity.

As the main participants in the energy market, prosumers consist of photovoltaic (PV) systems, loads (both fixed and flexible), and energy storage systems (ESS). Among these, flexible loads, also referred to as transferable loads, are controllable loads that do not affect basic living needs, enabling demand response. This allows prosumers to adjust their loads based on the transaction prices set by the DNO and develop their own electricity consumption strategies. Additionally, to enhance the network optimization level of the DNO, Multi-Terminal Soft Open Points (MOP) are installed within the distribution network (DN).

As shown in Figure 2, this paper explores a bi-level energy trading framework based on the Stackelberg game, which shows the leader-follower relationship between DNO and prosumers, and both play interest games with exchanged power through clearing prices.

The upper-level leader, the DNO, can determine the operational strategy of the MOP and the internal clearing prices based on time-varying grid prices, feed-in tariffs, the exchange power between prosumers and the DNO, and the operational status of the distribution network (DN), aiming to maximize social welfare. Meanwhile, the lower-level follower, the prosumer, can adjust its electricity consumption and ESS charging/discharging strategy according to the clearing prices set by the DNO, and determine the exchange power with the DNO, in order to minimize its own costs.

It can be observed that this is a bi-directional decision-making problem, where the clearing price and exchange power serve as the communication bridge between the upper and lower levels. The DNO sets the clearing price based on the exchange power with the prosumers, while prosumers respond to the DNO’s clearing price by determining their own energy scheduling plans. Therefore, although the DNO holds priority in the decision-making process, it must take into account the demand response (DR) of each prosumer, thus forming a leader-follower game of interests.

In summary, the proposed energy market interaction and game framework effectively solves the complexity of energy trading in flexible distribution networks involving multiple production-consumers. By integrating dynamic clearing prices, DR, and MOP-based network optimization, bidirectional coordination between DNO and prosumers is achieved, ensuring joint achievement of economic and operational objectives.

Even after linearization in Equation 32, quadratic nonlinear terms in the form of l t , i j v t , i still exist. Therefore, further processing is required, and the convex relaxation of the function is expressed as follows: 2 P t , i j 2 Q t , i j l t , i j − v t , i 2 ≤ l t , i j + v t , i j , ∀ t (39)

Similarly, Equation 14 also contains quadratic nonlinear terms, and the transformed second-order cone constraint is as follows: P t , i V S C 2 + Q t , i V S C 2 ≤ 2 S t , i V S C 2 S t , i V S C 2 (40)

Equation 41 is defined to verify the constraint effect. If the gap value is sufficiently small, it is considered that the accuracy after relaxation is reasonable, which also means that the initial model can be transformed into a model that can be solved by commercial solvers using the two processing methods described above. gap = l t , i j − P t , i j 2 + Q t , i j 2 v t , i ∞ (41)

Based on the Stackelberg leader-follower game, and combining the above framework and model, the following bi-level optimization problem is formulated:

Upper level: min ⁡ F D N O ;

For the two-layer optimization model, the traditional iterative method is adopted, and the solving process is relatively complicated.

Furthermore, considering user privacy and security, and to avoid unnecessary information exchange between prosumers and the DNO, this paper applies the KKT conditions to transform the above bi-level model, thereby improving computational efficiency and protecting user privacy.

Let μ be the dual variable for the inequality constraints of the lower-level optimization problem, and λ be the dual variable for the equality constraints of the lower-level optimization problem. As shown in Equations 42, 43, the general form of the KKT condition obtained by the transformation is: ∇ L = P t , i L , m o v e , P t , i e s s , c , P t , i e s s , d , S t , e , μ i , λ i = 0 (42) 0 ≤ μ ⊥ g x ≥ 0 (43) where, ∇ L is the Lagrange function written using the KKT conditions (Zhu et al., 2022), g x ≥ 0 representing the inequality constraints in the optimization problem. The specific expression is as follows:

1) Introduce the optimization objectives and constraints of the lower prosumers to write the Lagrange function:

Since the complementary slack variables in the KKT condition have nonlinear terms of the form μ i g i x = 0 , a Boolean variable ε and a maximum positive number M are introduced to construct the following linear inequalities: 0 ≤ P t , i e s s , c ≤ ε 1 e s s , c M   0 ≤ μ 1 , t , n e s s , c ≤ 1 − ε 1 e s s , c M   t ∈ 1 , T (52) 0 ≤ u t , e e s s P e c , r a t − P t , i e s s , c ≤ ε 2 e s s , c M 0 ≤ μ 2 , t , n e s s , c ≤ 1 − ε 2 e s s , c M     t ∈ 1 , T (53) 0 ≤ P t , i e s s , d ≤ ε 1 e s s , d M   0 ≤ μ 1 , t , n e s s , d ≤ 1 − ε 1 e s s , d M   t ∈ 1 , T (54) 0 ≤ 1 − u t , e e s s P e d , r a t − P t , i e s s , d ≤ ε 2 e s s , d M 0 ≤ μ 2 , t , n e s s , d ≤ 1 − ε 2 e s s , d M     t ∈ 1 , T (55) 0 ≤ S t , e − S e min ≤ ε 1 S M 0 ≤ μ 1 , t , n S ≤ 1 − ε 1 S M   t ∈ 1 , T (56) 0 ≤ S e max − S t , e ≤ ε 2 S M 0 ≤ μ 2 , t , n S ≤ 1 − ε 2 S M   t ∈ 1 , T (57) 0 ≤ P t , i L , m o v e − L min ≤ ε 1 L , m o v e M 0 ≤ μ 1 , t , n L , m o v e ≤ 1 − ε 1 L , m o v e M     t ∈ 1 , T (58) 0 ≤ L max − P t , i L , m o v e ≤ ε 2 L , m o v e M 0 ≤ μ 2 , t , n L , m o v e ≤ 1 − ε 2 L , m o v e M     t ∈ 1 , T (59)

4) Single layer optimization model

Substituting Equations 45–51 into Equation 44 can be obtained as follows: ∇ L = ∑ t = 1 T ∑ n = 1 N λ t P t , i L − P t , i P V Δ t − ∑ t = 1 T ∑ n = 1 N μ 2 , t , n e s s , c u t , e e s s P e c , r a t − ∑ t = 1 T ∑ n = 1 N μ 2 , t , n e s s , d 1 − u t , e e s s P e d , r a t + ∑ t = 1 T ∑ n = 1 N μ 1 , t , n S S e min − ∑ t = 1 T ∑ n = 1 N μ 2 , t , n S S e max + ∑ t = 1 T ∑ n = 1 N μ 1 , t , n L , m o v e L min − ∑ t = 1 T ∑ n = 1 N μ 2 , t , n L , m o v e L max (60)

Equation 60 is combined with the upper DNO optimization objective Equation 1 to obtain the optimization objective of the single-layer optimization model as shown in Equation 61: min ⁡ F = a o f G r i d + f l o s s + f s w i t c h − ∑ t = 1 T ∑ n = 1 N λ t P t , i L − P t , i P V Δ t + b v f v d + ∑ t = 1 T ∑ n = 1 N μ 1 , t , n L , m o v e L min − μ 2 , t , n L , m o v e L max + ∑ t = 1 T ∑ n = 1 N − μ 2 , t , n e s s , c u t , e e s s P e c , r a t − μ 2 , t , n e s s , d 1 − u t , e e s s P e d , r a t + ∑ t = 1 T ∑ n = 1 N μ 1 , t , n S S e min − μ 2 , t , n S S e max (61)

Subject to: Equations 7, 8, 11–13, 15–19, 29–31, 33–34, 36–40, 45–59;

Variables: λ t , P t , i V S C , Q t , i V S C , P t , n P , P t , i L , m o v e , P t , i e s s , c , P t , i e s s , d , S t , e ;

By the above methods, the two-layer game optimization problem has been transformed into a single-layer optimization problem.

In the simulation tests, the modified IEEE-33 bus distribution system is used for analysis, which includes five prosumers with PV systems. Among them, the prosumers at buses 10 and 29 are equipped with ESS, as shown in Figure 4. The whole DN contains five PVS, two ESS and one four-feeder MOP. The relevant parameters are shown in Table 1, and the other bus branch parameters are the standard IEEE-33 bus system.

The DNO is set from the HV power purchase price reference (Qiao et al., 2025), and the DNO selling price is set to 400 ¥/MWh, without considering the reactive power influence of renewable energy. Other parameter Settings are shown in Table 2 (Yang et al., 2023).

The PV output and load demand forecast of each prosumer are shown in Figures 5A, B.

In the energy trading market, DNO, as the leader, has the pricing power, and the settlement price determined is shown in Figure 6.

As followers, prosumer adjust their electricity consumption strategies according to the internal settlement price. The transferable load of each prosumer is shown in Figure 7A, and the total exchange power with DNO is shown in Figure 7B. The charging and discharging power of ESS1 connected to prosumer 2 is shown in Figure 8A, and the energy storage of each ESS is shown in Figure 8B.

As observed from Figures 5–8, the formulation of the electricity consumption strategy of the prosumer is affected by the clearing price set by the DNO, which is specifically shown as follows:

Between 0:00 and 5:00, with zero PV output and low prosumer loads, prosumers purchase electricity to maintain normal operations. Clearing prices remain high due to economic principles but are capped by time-of-use pricing. During this period, the ESS charges and the prosumer increases the transferable load. From 3:00 to 5:00, as the electricity price decreases, the ESS charging increases, resulting in a sharp rise in the exchanged power. Between 6:00 and 8:00, the PV generation and consumer load gradually rise, but the supply is still insufficient, leading to an increase in the clearing price. At the same time as the ESS discharges, the prosumer reduces the transferable load, which reduces the exchanged power. Between 9:00 and 10:00, higher PV generation and less load can achieve energy balance through energy interaction and resource adjustment without purchasing DNO, thus achieving price reduction and zero exchanged power. From 11:00 to 13:00, peak PV output exceeds demand, enabling prosumers to charge ESS, consume transferable loads, and sell surplus energy to the DNO. Exchange power becomes negative, and clearing prices drop to the minimum limit. Between 14:00 and 17:00, as PV generation and load are reduced, the supply temporarily meets the demand and maintains the minimum price. After that, the energy internal supply exceeds the demand leading to the clearing price increase, prompting the prosumer to reduce the load and the ESS discharge, so that the exchange power is positive. From 18:00 to 20:00, insufficient PV generation and gradually rising electricity demand lead to the maximum clearing price. The prosumer reduces the transferable load and the ESS discharges, at which point, the exchanged power reaches its peak. Between 21:00 and 24:00, demand remains high, but prices stabilize due to time-of-use pricing. Prosumers remaining transferable loads and discharge surplus ESS energy, reducing costs and exchange power as load demand declines.

The voltage situation of each bus within 24 h of the test system is shown in Figure 9. It can be seen that the per unit voltage value of each bus is within the expected range (0.96 p.u.–1.04p.u.), which meets the safety of the system operation.

The gap value of 24 h is shown in Figure 10. Combined with the definition of Equation 41, it can be found that the gap value of each time period is at the level of 10-6, so it can be proved that convex relaxation is accurate.

In order to verify the effectiveness and superiority of the proposed strategy, the following three scenarios are set in this section:

Scenario 1: Only the economy of the system is considered, and safety issues such as MOP power flow optimization, power loss, and voltage deviation are not considered.

As can be seen in Figures 11A, B and Figures 12 A, B, the voltage quality of scenario 3 is better than that of the comparison scenario, indicating that the proposed strategy can improve the stability of the system. Moreover, the action of MOP is consistent with that of energy storage, which indicates that MOP can optimize the power flow distribution and improve power quality by adjusting the active and reactive power of the connected feeders.

The test results are compared as shown in Table 3, including the system power loss cost ( f l o s s , including line power loss and MOP power loss), the converted voltage deviation cost ( f v d ), the total operating cost of DNO ( F D N O ), the total operating cost of the prosumer ( J n ) and voltage over-limit rate. The voltage compliance range specified in this paper is 0.95p.u.–1.05p.u.

As shown in Table 3, the daily total operating cost of the DNO in Scenario 1, which considers only economic benefits, is $13.95, representing a reduction of approximately 83.9% compared to $86.72 in Scenario 3. However, the voltage deviation cost increases from $0.13 to $104.21, an increase of nearly 800 times, and the voltage over-limit rate increases from 0% to 26.39%. This indicates that in scenarios where safety performance is not considered, all controllable resources are allocated to maximize profits. While this approach effectively reduces system operating costs, it leads to significant grid fluctuations, poor power quality, and substantial safety risks, which are detrimental to the reliable operation of the system.

Compared to Scenario 2, Scenario 3 introduces the MOP, which increases the power loss cost of the MOP by $30.94. However, both network loss and voltage deviation costs are reduced, resulting in a decrease in the total operating cost of the DNO from $193.04 to $86.72, a reduction of approximately 55.1%. Moreover, the voltage over-limit rate is reduced from 65.03% to 0. This demonstrates a significant improvement in both economic efficiency and operational safety.

In summary, the proposed strategy not only addresses the energy transaction challenges in multi-prosumer distribution systems and ensures the economic benefits of all participants but also enhances the safety performance of the system. Additionally, it provides a novel solution for promoting local renewable energy utilization and optimizing energy interactions.