Optimal control method of active distribution network considering soft open point and thermostatically controlled loads under distributed photovoltaic access

To achieve the flexible operation of the active distribution network (ADN), an optimal control method for the ADN considering the soft open point (SOP) and thermostatically controlled loads under distributed photovoltaic (PV) access is proposed. First, the mathematical model of the SOP and thermostatically controlled loads are constructed based on the flexibility of the SOP and the thermal storage characteristic of thermostatically controlled loads, respectively. Then, the mathematical model of the ADN considering the SOP and thermostatically controlled loads are further constructed. The optimal control of the ADN is realized by adjusting the transmission power of the SOP and the switching state of the electric water heater. Finally, the optimal operation analysis of the SOP and thermostatically controlled loads under various control methods is carried out, and the impact of the SOP and thermostatically controlled loads on the economic and secure operation of the ADN is further investigated. The results show that the proposed method taps the demand response potential of thermostatically controlled loads while satisfying the water consumption habits of different customers, and utilizes the network reconfiguration with the SOP to further improve the security and economy of the ADN operation.


Introduction
With the rapid development of new power generation technology, a large number of distributed generators (DGs) have been connected to the distribution network, resulting in complex and variable system currents, large voltage fluctuations at nodes, and even the overflow of node voltage, making it difficult for the traditional distribution network to meet operational requirements (Dong et al., 2022).For this reason, the concept of the ADN has been introduced.The ADN is defined as the distribution network with operation and control capabilities and with distributed or decentralized energy within it.It is a new solution for the flexible use of DG according to the characteristics of the distribution network.It can also integrate the network-side and load-side flexibility resources and make them participate in the optimal control of the whole system.Therefore, to improve the secure and economic operation of the ADN, it is crucial to study the optimal control methods of the ADN involving various types of flexible resources.
Distribution network reconfiguration is a means of optimizing control based on the switching action on the CONTACT Haidong Yu vrblack2023@proton.menetwork side.The power flow in the distribution network can be optimized using distribution network reconfiguration, thus improving the economy and security of the distribution network operation.At present, some progress has been made in the study of network reconfiguration considering DGs.The paper (Li et al., 2018) proposes a dynamic reconfiguration strategy for distribution networks with electric vehicles and DGs based on an interval number approach, which combines neighbourhood search and clone selection algorithms to achieve dynamic reconfiguration of the ADN.Based on mathematical models and node partitioning for wind power, photovoltaic (PV), and domestic energy storage, an optimal reconfiguration model for distribution networks containing DGs based on multi-objective optimization is developed in the literature (Tan et al., 2022).Although the abovementioned intelligent algorithm-based distribution network reconfiguration strategies have the advantages of high local search performance and fast solution speed, they have the disadvantage of easily falling into local optimal solutions when compared to mathematical optimization algorithms.The paper (Gao et al., 2017) extends the original distribution network structure with on-load tap changers (OLTCs) and energy storage systems (ESSs) and concludes the dynamic network reconfiguration study for the distribution network that considering demand response and active management.To reduce the cost of distribution network operation, a distribution network power flow optimal control model is built in this paper using the DistFlow power flow model and the Second Order Cone Relaxation (SOCR) method.
Network reconfiguration, as one of the important means of optimizing distribution network operation, has many advantages.However, problems such as unsmoothed transition currents in tie switches may lead to significant increases in switching losses and even equipment failures.The soft open point (SOP) is a new type of intelligent power distribution device that can replace the tie switch for feeder interconnection and can be used to flexibly control the power flow of the distribution network, improve the node voltage level and optimizing network losses (Song et al., 2018).Most of the existing research focuses on the application of SOP for optimal operation of distribution networks or power restoration.Literature (Song et al., 2018) applies SOP to the distribution network's fault restoration, using the voltage support function of SOP to extend the power restoration range of the distribution network.In the literature (Zhang et al., 2021), SOP is combined with network reconfiguration, adding demand response to the original distribution network structure, considering the electricity price elasticity coefficient, and reconfiguring the distribution network on this basis to improve the power supply recovery range.
However, the previous research only considers network-side flexibility resources, and the impact of userside flexibility resources on the optimal operation of the distribution network is not fully considered.The flexibility of thermostatically controlled loads can be used to improve the secure and economic operation of the ADN.As a typical representative of thermostatically controlled loads, electric water heaters are the most common domestic water heating devices used by residents.Users can use the heat storage capacity of the tank to take advantage of its flexibility, use demand response based on the price of electricity, and reduce electricity costs by adjusting the operating time of electric water heaters.Electric water heaters consume about 20 percent of daily residential energy demand (Lu et al., 2021), so incorporating their user-side flexibility into the optimal operation of the ADN can not only reduce electricity costs for customers, but also optimize power flow distribution to achieve peak shaving and valley filling, and reduce network losses.To achieve a two-way interaction and coordinated optimization between the network side and the user side, a bi-level optimization model for community energy is established in the literature (Lu et al., 2021), with the combined optimal cost of the user as the lower-level objective function and the benefit of the community energy operator as the upper-level objective function.A load aggregation model for electric water heaters is established in the literature (Sun et al., 2016).Based on this model, the importance of flexible operation of electric water heaters for improving the load factor and distribution system economics is analyzed.
However, the previous studies are not comprehensive enough to consider the active role of network-side flexibility resources in system optimization, which is a limitation when it comes to combined network-side and user-side optimization.
Based on the above, this paper develops a mathematical model of the ADN considering the SOP and electric water heater access, and uses SOCR to transform the non-convex nonlinear problem into a mixed integer second-order cone programming (MISOCP) problem.Optimal control of the ADN can be achieved by adjusting the transmission power of the SOP and the switching state of the electric water heater.The contributions of this paper are summarized as follows: 1) An optimization model for an Active Distribution Network (ADN) with the SOP and electric water heater has been developed.The model considers the optimal operations of the ADN, the thermal inertia of electric water heaters, and the water consumption habits of users.By using piecewise linearization and SOCR, the original model was transformed into a mixed-integer second-order cone programming (MISOCP) model and solved effectively.2) To fully utilize the demand response capacity of userside optimization, electric water heaters are used as flexible resources for the ADN, taking into consideration the comfortable water temperature range of users (Lu et al., 2021).This approach enables the flexible operation of the ADN. 3) A mathematical model for the SOP in the ADN has been established (Song et al., 2018).When the system power is insufficient, SOP can balance the load between feeders by shifting the load and regulating reactive power, which effectively improves the distribution network load distribution.Additionally, the combined use of SOP and network reconfiguration helps to reduce power losses and improve voltage imbalances.4) The proposed method exploits the demand response potential of thermostatically controlled loads while satisfying the water consumption habits of different users, and utilizes the network reconfiguration with the SOP to further improve the security and economy of ADN operation.

ADN framework with integrated SOP and thermostatically controlled loads
As shown in Figure 1, this section introduces the operational framework of the ADN with integrated SOP and thermostatically controlled loads.The framework consists of an ADN and a Building Energy Management System (BEMS).Through the BEMS, the network and building sides form an information communication network that allows the exchange of 'network and load' information as well as the transfer of optimal control methods.Distributed PV power sources are connected to building clusters and distribution networks via power transmission lines, and thermostatically controlled loads (electric water heaters) are connected to the distribution network management system via BEMS, forming the basic framework of the ADN.Flexible resources such as PV, SOP, and thermostatically controlled loads allow for the investigation of optimal control methods for the ADN.Based on this framework, the BEMS can send incentive signals based on user-side demand, analyze and respond to the incentive signals from the distribution network management system, and then transmit the dispatch strategy to the user-side in order to achieve demand response.The combination of the two can achieve a comprehensive optimization of the 'gird and load' total cost.

Mathematical model of the SOP
The main component of the SOP is fully controlled power electronics.Through a variety of control modes, the SOP can achieve functions such as improving voltage, optimizing power flow distribution, and reducing network losses due to the flexible and varied control methods of the power electronics.In the normal mode, it operates with a voltage source converter (VSC) on one side to control the transmitted power and VSC on the other side to control the DC voltage, i.e.PQ − V dc Q control.The following is the mathematical model of the SOP in the normal mode of operation, taking into account the converter losses. (2) Where 2 are the active and reactive power injected into the converters on both sides of the SOP at the SOP access node i, j.P SOP i,t , P SOP j,t are the converter losses at the SOP access node i and access node j, respectively.γ SOP is the loss factor of the SOP converter.And S SOP i,t 2 , S SOP j,t 2 are the converter capacities connected to node i, j respectively.Each variable is a parameter at time t.

Mathematical model of thermostatically controlled loads
Take a storage-type electric water heater with an automatic heating function as the representative.When the user uses domestic hot water, the tank is automatically replenished with cold water, resulting in a lower water temperature in the tank.The electric water heater's heating function automatically activates when the water temperature in the tank drops below the lower limit of the temperature setting due to cooling or user activity.If the water temperature remains within the set range, the electric water heater will operate in the holding mode.The dynamic process described above is illustrated in Figure 2.
The following mathematical model of the electric water heater can be constructed: Where T d t represents the water temperature in the electric water heater tank at time t, T d t+1 represents the water temperature in the electric water heater tank at time t + 1, T s t represents the ambient temperature at time t, and P d represents the active power of the water heater during heating.R d and C d represent the thermal resistance and heat capacity of the water heater.L represents the total capacity of the electric water heater tank.t represents the time scale of water heater operation, which is one hour.And l t represents the user's unit water consumption at time t.ρ d t denotes the state of the electric water heater at time t.ρ d t = 0 indicates that the electric water heater is in holding mode, while ρ d t = 1 indicates that the electric water heater is in heating mode.

Distribution network power flow model
The optimal operation control of the ADN is basically the optimal power flow control of the distribution network, so the construction of the optimal power flow model is essential in the process of optimal control of the distribution network (Liu et al., 2015).A general branch dynamic optimal power flow model is constructed as follows (Farivar & Low, 2013).
, ∀ij ∈ B (7) Where p ij,t + q ij,t represent the active and reactive power injected by node j at moment t, respectively.δ(j) represents the set of parent nodes of node j in the distribution network, and π(j) represents the set of children nodes of node j in the distribution network.P ik,t + Q jk,t represent the active power and reactive power flowing through the branch jk at moment t, respectively.P ij,t + Q ij,t represent the active power and reactive power flowing through the branch ij at moment t, respectively.V i,t and V j,t represent the voltages of node i and node j at moment t, respectively.I ij,t represents the current flowing in the branch ij at moment t. ¯Iij , I ij represent the upper and lower current limits of branch ij, respectively.Vj , V j represent the upper and lower voltage limits of node j, respectively.g j,t and b j,t represent the equivalent conductance and equivalent dana of the load connected to node j at moment t, respectively.r ij and x ij represent the resistance and reactance of the branch ij, respectively.The latter Eqs (6) to ( 9) are the distribution system power flow constraint, the power balance constraint, and the upper and lower security constraints, in that order.
The above distribution network optimal power flow model is a non-linear programming model, now let ˜Iij = I 2 ij and Ṽj = V 2 j , the optimal power flow model after relaxation can be displayed as follows.

The objective function
In this paper, the operating cost of the SOP is considered in the modelling process, and the objective functions are the network-side integrated cost (the sum of network loss, SOP loss, and switching operation cost) and the controllable cost of thermostatically controlled loads on the user side. min Where F denotes the total cost.C loss , C sop , C switch , and C sell denote the network loss cost factor, SOP loss cost factor, and user-side electricity price, respectively.N SOP is the set of nodes in the system that are connected to the SOP.B is the set of all branches in the distribution network.
The thermostatically controlled loads are located at node N h in the system.I ij,t is the current flowing in branch ij at time t.And r ij is the equivalent resistance connected to node.P SOP,L i,t represents the converter loss generated by the converter on one side of SOP access node i at time t.P h i,t represents the total power of the thermostatically controlled loads used by users in node i at time t.The switching states of the branch ij after and before network reconfiguration are α ij and α ij,0 , respectively.When α ij = 0, the branch is open, and when α ij = 1, the branch is closed.T represents the time scale of system operation, which is one hour.

The constraints
The constraints of the ADN optimization model considering thermostatically controlled loads and the SOP are as follows.
(1) SOP capacity constraints Where S SOP i,t and S SOP j,t are the capacities of the converters connected to nodes i and j respectively.
(2) Upper and lower temperature limits for electric water heaters To maintain the water temperature within a comfortable range during domestic hot water consumption, electric water heaters can generally set their own upper and lower water temperature limits.Such a function can be expressed as a mathematical model as follows.
Where T d t represents the current water temperature in the electric water heater tank.T d.max and T d.min represent the upper and lower limits of water temperature.
(3) The ADN power flow constraints are shown in Eqs ( 10)-( 12).(4) The ADN security constraints are shown in Eqs ( 13) and ( 14).(5) Constraints on PV power output considering reactive power Based on the current research on the form of DG active management modelling, the DG active output can be set as a variable, and after the main active management method of active output regulation has been studied and matured, part of the DG can have some reactive power impact on the grid (Tan et al., 2022).The PV model with a constant power factor is as follows (Liu et al., 2014).
Where P pv,pre i,t and P pv i,t represent the predicted and actual active power outputs of the PV source connected to node i at time t, respectively.Q pv i,t represents the actual reactive power output of the PV source connected to node i at time t.And ∂ DG j represents the power factor of the reactive power output relative to the active power output at node j.
(6) Constraints on the number of switch actions Each switching action in the network reconfiguration process has an associated operating cost.In addition, the switch in the distribution system has a limited lifetime.As a result, the total number of switching actions in the distribution system is limited as follows in order to achieve optimal switching action and optimal network operation economy.
Where α ij and α ij,0 represent the switching states of branch ij after and before network reconfiguration, respectively.When α ij = 0, the branch is open, and when α ij = 1, the branch is closed.η total b is the maximum number of times all tie and segment switches in the distribution system are allowed to open and close during the full operation p.
(7) Radial operational constraints of the ADN The reconfigurations in the distribution network change the network topology, where the following constraints are required to maintain the radial operation of the distribution network and to avoid the existence of islands and ring networks.
Where β ij is a 0-1 variable, 1 if node i is the parent of node j, 0 otherwise.η B is the total number of branches in the distribution network.

SOCR transformation for SOP and distribution network power flow constraints
Except for Eqs. ( 2), (3), ( 12), ( 16), and ( 17), all constraints in the distribution network reconfiguration model and the SOP mathematical model are linear.Eqs. ( 2), (3), ( 12), ( 16), and ( 17) are non-linear constraints with strong non-convexity.This model must take into account the switching state of the branch, the operating state of the electric water heater, and so on.It is a mixed integer nonlinear programming (MINLP) model.In addition, the constraints shown in Eq. ( 11) are only for closed branches and do not consider the effect of branch opening and closing on the distribution network power flow distribution, which cannot be used in distribution network reconfiguration optimization.In this paper, the Big-M method is used to perform the relaxation transformation of Eq. ( 11), and on this basis, use the second-order cone optimization algorithm based on the mathematical optimization algorithm (Liu et al., 2014) to perform the SOCR for Eqs.

Case studies
In this paper, the optimization period is set to one day and the time step is set to one hour.In terms of simulation environment, IEEE 33 bus system are now widely used in modelling and simulation studies of ADN (Kashem et al., 2000;Tolabi et al., 2015;Yuan et al., 2016).Therefore, we applied a modified IEEE33 node example for case studies, adding elements of the SOP, PV, and thermostatically controlled loads to the base network, and the system structure is shown in Figure 3.The root node of the distribution system is node 33, with a reference voltage level of 12.66 kV and a power reference value of 100 MW.The  node voltage has an upper limit of 1.05 (SD) and a lower limit of 0.95.(SD).GAMS is used to build the model, and GUROBI is used to optimally solve the proposed MISOCP model.The computer hardware environment used for the solution includes an Intel(R) Core(TM) i5-9300H CPU 2.40 GHz, 16GB RAM, the Windows 11 operating system.
The detailed parameters of the additional flexibility resources in the distribution system are shown in Tables 1-3 and Figure 4.
The thermostatically controlled loads have the characteristic of random periods, and the user groups can be classified into three types based on their different water consumption habits: Type A users spend the entire day at home and start using water earlier in the morning and evening peak hours, corresponding to elderly users.Type B users have no hot water demand between 9:00 and 16:30, corresponding to office users.Class C users combine both user types A and B, corresponding to a mix of users in between the two cases (Lu et al., 2021).
In this paper, four different optimization methods are tested and compared to validate the effectiveness of the proposed active distribution network optimal control method considering thermostatically controlled loads and SOP under distributed PV access.
• M1: Two-stage optimization.The first stage aims at the optimal cost on the user-side and the second stage aims at the optimal cost on the network-side, and without considering the role of the SOP.

6.
Operating hours of electric water heaters under different methods.
• M2: Two-stage optimization.The objective of user-side cost optimization in the first stage and network-side cost optimization in the second stage, considering the role of the SOP.• M3: Combined optimization.The objective is to optimize both user-side and network-side costs, and the role of the SOP is not considered.• M4: Combined optimization.The objective is to optimize both user-side and network-side costs, and the role of the SOP is considered.

Thermostatically controlled loads optimization results
Figure 5 depicts the temperature profiles inside the electric water heater for users A, B, and C. Since M1 and M2 use the same method on the user side, the temperature variation profiles are the same for both.
As the optimization results show, all four methods can meet the dynamic process changes of the electric water heater and keep the water temperature within the human comfort range when the user is using water.M4 can further optimize the switching action strategy of the electric water heater by meeting the above requirements and keeping the water temperature in a lower range, ensuring the user's comfort with the water temperature, and reducing the total operation time of the electric water heater, as shown in Figure 6.
The user-side cost under M1 and M2 is $521.06,which is the limit if other parts of the system are not considered.
M3 has a user-side cost of $530.91, while M4 has a userside cost of $522.57.
As shown in Figure 7, the load curves for M1 and M2 are more volatile, while M3 and M4 achieve a some peak-shaving and valley-filling in the electric water heater loads, thereby improving the level of power flow distribution in the system after accounting for the flexibility of the thermostatically controlled loads.

SOP optimization results
M2 and M4 incorporate the SOP into the optimized operation of the ADN.Figures 8 and 9 show the transmitted power curves of the SOP for M2 and M4, respectively.As can be seen, SOP provides less reactive power in the valley of low power consumption and more reactive power to improve voltage levels during peak power consumption periods (10:00-14:00 and 19:00-21:00).This means that SOP can control the flow of active power to reduce network losses while also providing some reactive power to raise the voltage levels.
The network reconfiguration results of the two methods are different, but both achieve the desired results.For both, the number of switching actions is four.Table 5 compares the network reconfiguration results.
M4 has lower SOP operating costs in its optimization results than M2, indicating that the combined optimization approach fully exploits the flexibility resources on both the user and network sides.The reconfigured network topology is shown in Figure 10.
After optimization, the system disconnects the tie switches between nodes 4-5 and 12-13, closes the tie switches between nodes 24-28, and activates the SOP.The remaining switches remain in their pre-optimization state.
Table 6 compares the operating results of the four different methods described in the previous section.It is easy to conclude that the combined control of the SOP and the thermostatically controlled loads is more beneficial in terms of both resource flexibility and system operating economy.
The user-side cost of M4 is significantly lower than that of M3 but slightly higher than that of M1 and M2.This is due to the fact that M1 and M2 ignore the operating costs of other parts of the system and instead focus on the optimal user-side cost as the objective function, with $521.06 as the limit value.The method proposed in this paper, on the other hand, significantly reduces the total   cost of operating the system, and the user-side cost is so close to the optimal result that the increase is negligible.
Figure 11 depicts the voltage profile at each distribution system node at 20:00 for each operating method.The voltage level of M4 at each node improves and the voltage quality improves when compared to the other three methods.Therefore, the optimal control method used in this paper not only improves the economy of system operation but also improves the security and reliability of system operation.The effectiveness of the method has been fully validated.
In addition, the differences between M2 and M1 and between M4 and M3 show the significant role of the SOP in improving the safety of ADN operation.After considering the role of the SOP, the system voltage quality is significantly improved and the voltage fluctuation is significantly reduced, thus preventing the occurrence of safety problems such as voltage violation.We set 1.06 as the lower voltage limit, and the Table 7 shows the comparison of the voltage quality of each method at 20:00.

Conclusion
This paper proposes an optimal control method of the ADN that takes into account the flexibility of SOP and thermostatically controlled loads under distributed PV access, and achieves a comprehensive optimization of the 'net-load' cost while satisfying the system operation constraints and customer comfort, thus significantly reducing the total cost of the ADN system operation.The key findings are summarized below.
1) The flexibility of thermostatically controlled loads can be utilized to optimize power flow distribution, achieve peak-shaving and valley-filling in the load curve, and reduce the total operating time of equipment while still maintaining customer comfort.
2) Applying the SOP to the reconfiguration of ADNs with thermostatically controlled loads can further improve system power flow distribution, smooth out system power oscillations, and reduce the amplitude of node voltage fluctuations.
3) The combined use of 'net and load' flexibility resources reduces overall system costs by 22.9 percent, improves voltage quality, and increases the economy, safety, and reliability of the ADN system operation while maintaining customer comfort.
In engineering applications, the addition of SOP can reduce the amount of power compensation equipment and energy storage equipment, thereby reducing the investment cost of the power distribution system.The combined use of SOP and electric water heater provides a better solution while maintaining the existing energy architecture.

Figure 1 .
Figure 1.Schematic diagram of the ADN.

Figure 2 .
Figure 2. Electric water heater dynamic process diagram.

Figure 4 .
Figure 4.The curve of the change in the price of electricity sold in the community during one day.

Figure
Figure SOP transmission power diagram under M2.

Table 5 .
Comparison of network reconfiguration results under two methods.

ADN optimized control results considering thermostatically controlled loads and the SOP
According to the results optimized by the optimization method proposed in this paper (i.e.M4), the total net-side cost of this modified IEEE 33 node system in one day is $181.86, of which the network loss cost is $155.63, the system net loss rate is 2.76 percent, the SOP operation cost is $18.22, and the switch action cost is $8 for four actions.

Table 6 .
Comparison of optimization results under different methods.

Table 7 .
Comparison of the voltage quality.