Multi-Objective Robust Optimization of Hybrid AC/DC Distribution Networks Considering Flexible Interconnection Devices

Hybrid AC/DC distribution networks with the high-efficiency consumption and high-proportion access of new energy have become a crucial trend for future modern distribution networks. High penetration of renewable energy brings various controllable resources along with uncertainties to hybrid AC/DC distribution networks, making conventional single objective optimization fail to meet optimal operation requirements of flexibility and reliability. In this paper, considering uncertainties of distribution generations and loads, a multi-objective robust optimization model based on various controllable devices is proposed. To increase resource utilization, the proposed method comprehensively and properly models the full variety of possible control means (i.e., flexible distribution switch, voltage source converter, energy storages, et al). Utilizing the abundant control means, a multi-objective optimization which minimizes network losses, voltage deviations and operation cost simultaneously is modeled. Then, based on the multi-objective model, a two-stage robust optimization based on second order cone method and column constraint generation algorithm is proposed to achieve the solution, which can deal with the fluctuation of loads and renewable energy quickly and effectively. Outdoing traditional single objective robust optimization, our multi-objective robust optimization utilizes various controllable resources, obviously improving the safety, flexibility and economy simultaneously. Finally, compared with existing robust optimization, the proposed method is tested in numerous case studies to verify its effectiveness and advantages in a modified IEEE 33-node system.


I. INTRODUCTION
With concerns on carbon emission and cost reduction of distributed generations (DG), the utilization rate of renewable energy is increasing [1], in which photovoltaic (PV) and wind turbines have been greatly utilized [2,3]. Moreover, energy storage systems (ESS), flexible electrical loads, and almost DC devices, are continuously connected to distribution networks. These changes make hybrid AC/DC distribution networks a crucial trend for future distribution networks. Modern electronic devices, such as DG, ESS and flexible interconnection devices, have the advantages in fast response, wide control range and low operation cost, which bring flexible controllable means to distribution system operation control systems, as well as high uncertainties. Therefore, to increase energy efficiency and operation flexibility, it is necessary to study a novel operation method in hybrid AC/DC distribution systems considering multiple control means and uncertainties [4].
Due to uncertainties of solar energy, wind and consumer behavior, the predicted data cannot be accurate ideally in practice, especially in the future distribution networks with high proportion of DGs [5]. This phenomenon leads to the fact that the traditional deterministic optimization is no longer applicable [6]. To make the optimization more accurate, robust optimization has become an indispensable method in distribution network, microgrid and other fields, including voltage regulation [7], network reconfiguration [8], energy storage equipment optimization [9] and reducing operation cost [10]. In [11], a two-stage robust optimization model is proposed for microgrid system to ensure safe operation in the worst scenario by optimizing the active and reactive power of voltage source converter (VSC). In [12], "budget of uncertainty" is set to overcome the conservatism caused by only considering the worst scenarios in the system, and also improves the economy. However, most robust optimization only uses the single objective model, which cannot fulfill the requirements of flexibility, economy and safety in the AC/DC hybrid distribution network with multiple uncertainties [13]. In [14], a robust optimization model is established to co-optimize the slopes of active and reactive power droop control of VSCs with the aim to minimize the total network losses, whose objective function only includes distribution network losses. In [15], a robust optimization method is proposed to address the short-term variations of renewable energy generation and loads for the strategic investment problem. In the existing literature, a number of multi-objective optimization methods have emerged to deal with the safe, reliable and economic operation in distribution networks [16]. Therefore, to cope with the uncertainties from various renewable resources and controllable resources, the combination of multi-objective model and robust optimization can be considered as a promising methodology to solve the real operation optimization for future distribution networks under high uncertainties [17].
The continuous development of power electronic technology brings flexible controllable electronic devices into distribution networks, including flexible interconnected devices, DGs and reactive power compensation devices [18]. In hybrid AC/DC distribution networks, VSCs, as the interconnected devices between AC and DC distribution networks, can also regulate active/reactive power in AC networks and active power in DC networks. In [19], a coordinated real-time voltage regulation method is proposed to utilize converter-based controllable devices in hybrid AC/DC medium distribution networks, such as DGs, ESSs, and VSCs. Moreover, flexible distribution switch (FDS) which can regulate power between AC and AC distribution networks is a novel flexible switch with many advantages. Compared with traditional "hard" switch, FDS with fast response can accurately control the power flow of active power and reactive power between different sections in distribution networks, and can regulate power flow smoothly. FDS makes the distribution system more flexible and reliable [20]. At present, the model, control and optimization of FDS have been in-depth research in AC distribution networks [21]. In [22], the operation optimization based on FDS is modeled and analyzed, and the advantages of FDS in improving power quality of distribution networks and reducing network losses are verified. Therefore, as flexible interconnected devices, FDS and VSC can achieve flexible energy transmission between AC and DC networks, realizing realtime, fast, sensitive and smooth power control. Moreover, DGs and ESSs are also important control means in operation and control in distribution networks. However, the joint optimization of FDS, VSC and other control means is rarely considered in hybrid AC/DC distribution networks, which will certainly greatly improve the flexibility, economy and reliability of hybrid AC/DC distribution networks by the coordination of various control means.
In this paper, to target the above research gaps, a multiobjective robust optimization method in hybrid AC/DC flexible distribution networks is proposed. To resist the risk of fluctuation of renewable energy, the robust method optimizes operation cost, voltage deviation and network losses simultaneously by multiple control means. Its major contributions are as follows: (1) Various control means especially flexible interconnection devices are applied in our coordinated optimization to regulate power flow between different sections in hybrid AC/DC flexible distribution networks. (2) Considering uncertainties of DGs and loads, a multi-objective robust optimization method is proposed to operation cost, voltage deviation and network loss simultaneously. (3) A two-stage robust optimization method based on second order cone method and column constraint generation (C&CG) algorithm is proposed to deal with the fluctuation of load and renewable energy quickly and effectively.
The reminder of this paper is organized as follows: the multi-objective optimization model considering flexible interconnection devices in hybrid AC/DC distribution networks is proposed in Section 2; Section 3 presents a twostage robust optimization model and solution algorithm. Case studies are presented in Section 4, while Section 5 concludes this paper.

II. MULTI-OBJECTIVE OPTIMIZATION MODEL IN HYBRID AC/DC DISTRIBUTION NETWORKS
To increase energy utilization, flexible interconnection devices are attracted significant attractions in the proposed multi-objective optimization model in this paper. Therefore, FDS, VSC and ESS are optimized to pursue the safety and economy of hybrid AC/DC distribution networks. This section aims to present the overall objective and constraints of the proposed multi-objective optimization model.

A. OBJECTIVE
In order to optimize network losses, voltage deviation and operation cost simultaneously, the objective function can be written as follows: ,, 11 where f is the objective function value; λ 1 , λ 2 and λ 3 are the weight factors of network losses (P loss ), voltage deviation (ΔV) and operation cost (C op ) in all day and λ 1 +λ 2 +λ 3 =1. N ac and N dc are the number of nodes of AC and DC distribution systems respectively. Power losses Ploss: Ωi is the set of all nodes connected to node i. Rij is the resistance of the branch between node i and node j. Iij is the current from node i to node j. price(t) is the electricity price at time t. Voltage deviation ΔV: V i (t) is the nodal voltage of node i at time t. V op min and V op max are the maximum and minimum value of voltage optimization range [23]. Operation cost C op : K, N and J are the number of VSC, FDS and ESS, respectively. P VSC loss,k (t) and P FDS loss,n (t) are the power losses of kth VSC and nth FDS at time t respectively. P ESS j,ch (t) and P ESS j,dis (t) is the charge/discharge power of the jth ESS. The active power value means that ESS absorbs power from the grid and vice versa. K ess is the unit power cost of ESS which depends on investment cost and maintenance cost. In this paper, the electivity price and feed-in tariffs are the same at time t.

B. CONSTRAINTS
The multi-objective optimization model established in this paper needs to satisfy the constraints of power flow, VSC, FDS, ESS as well as constraints of secure operation. The constraints can be described as follows.

1) POWER FLOW CONSTRAINTS
The hybrid AC/DC distribution network adopts the DistFlow model. The power flow constraints are [24]: where, φ i is the branch set with node i as the head node. ϕ i is the branch set with node i as the end node. R ij and X ij are the resistance and reactance of branch ij. P i (t) and Q i (t) are the injected active and reactive power of node i at time t. P ij (t) and Q ij (t) are the active and reactive power flow from node i to node j at time t. In the DC system, the reactive power and reactance value are equal to zero in (6)- (8).

2) VSC CONSTRAINTS
The equivalent model of VSC is shown in Figure 1, which is equivalent to the form of a converter and impedance in series.

3) FDS CONSTRAINTS
FDS is a soft switching device to regulate energy transfer between AC and AC systems [25]. The equivalent model of the multi-terminal FDS is shown in Figure 2. The constraints of the nth multi-terminal FDS are: , , ,,

4) ESS CONSTRAINTS
where, for the jth ESS, P ESS j,max is the maximum charge/discharge power, U ESS j (t) is binary variable, equal to 0 when the ESS is in charge at time t, while 1 means the ESS is in discharge at time t. E ESS j,max and E ESS j,min are the maximum and minimum energy stored [26].
where, V i,max and V i,min are the allowable upper and lower limit of nodal voltage. I ij,max is the allowable upper current value in the branch.

C. MODEL TRANSFORMATION
In this paper, the above model is transferred into second order cone model based on linearization and relaxation technique for rapid solution. Firstly, to realize the linearization， define (2), (4), (6)(7)(8) and (20) can be replaced by (21). To replace the absolute value item, an auxiliary variable μ i (t)=|V i,2 (t)-1| and some constraints are added: The power flow constraint (7), VSC loss constraint (13) and FDS loss constraint (16) are nonlinear quadratic constraints, which can be further relaxed to the following second-order cone constraints: The circle constraints (12) and (15) can be transferred into (26) and (27) Now, using convex relaxation and linearization, the original multi-objective optimization model is reformulated as the following SOCP model, as shown in (28)

III. MODEL AND SOLUTION PROCEDURE OF MULTI-OBJECTIVE ROBUST OPTIMIZATION
Considering the significant spatial and temporal uncertainties of DG and load output, the model and algorithm should be improved to cope with uncertainties in the practical application. In this section, a multi-objective robust optimization model based on C&CG is developed to solve the robust optimization problem of hybrid AC/DC flexible distribution networks.

A. MODEL
The deterministic multi-objective optimization model proposed in Section 2 can be expressed in a compact form, shown as (29).  (9) and (14), the 3rd line involves (9), (11), (17)- (20), (22) and (26)- (27), the 4th line indicates that the uncertainty is ignored in deterministic model, the output power of DGs and loads at each time step are equal to the predicted value, and the last line involves (23)- (25). The deterministic optimization model (29) can be solved by second order cone optimization. However, the solution may be inappropriate due to the predicted errors. Therefore, in this paper, the uncertainties of DG and load are considered in (30): where, uDG(t) and uL(t) are the actual output power of DG and

B. SOLUTION PROCEDURE
C&CG is applied to solve the two-stage robust optimization model, denoted as equation (31). Similar to Benders-dual algorithm, the method decomposes the model into a masterproblem and a sub-problem and then solves them iteratively. The formulation of the master-problem is expressed as: According to [27], the optimal solution of (34) is obtained only when u is an extreme point of the uncertainty set. In our paper, the worst scenario occurs in the upper bounds of loads and the lower bounds of DGs. Thus, the uncertainty set of DGs and loads can be rewritten as (35).

t u t B u t u t u t B t u t
Bt Bt (35) where, Г DG and Γ L , named as the "budget of uncertainty", are integers valued between 0 and T, which are used to adjust the conservatism degree of the optimal solutions. Usually, we make where, ξ DG and ξ L are the fluctuation interval coefficient of DG and load.
After the above transformations, (34) becomes a form with binary variables and continuous variables, which can be converted by linearization techniques. Then, the max form of the sub-problem will be transferred into the following mixedinteger linear optimization model: T  T  T  T  T , , , , , , , where,  is the upper bounds of dual variables π, which may take the values large enough. After the above process, the two-stage robust optimization model is decomposed into the master-problem (MP) and subproblem (SP), and solved by C&CG algorithm. The flow chart is shown in Figure 3

A. TEST SYSTEM PARAMETERS
The proposed method is verified on a modified IEEE33 node hybrid AC/DC distribution network, as shown in Figure 4. The network is divided into AC network and DC network by VSC. The reference voltage of both AC and DC parts is 12.66 kV. Node1 is the slack node, and the optimization interval of voltage per-unit value is [0.985,1.015]. In our paper, DGs are considered as PVs. There are four PVs in the system, which are connected to Node 11, 13, 20 and 31, respectively, three ESSs with the upper power limit of 1MW and the rated capacity of 5MW·h, which are connected to Node 15, 24 and 30, respectively. Moreover, a multi-terminal FDS is connected to Node 21, 25 and 33. Three 24-hour varying loads are connected to Node 18, 23 and 32. It is assumed that the power direction of the VSC is from AC part to DC part, which is positive. The sampling power of the system is 1 hour, and the system implements time-of-use electricity price. The electricity price in each period is shown in Figure 5. The power profiles of PVs and loads are shown in Figure 6. The parameters of VSC and FDS are shown in Table Ⅰ.

B. SIMULATION ANALYSIS
A 24h simulation has been carried out here. In order to verify the performance of the proposed two-stage robust optimization method in our paper, simulations are carried out in five different cases. The five cases are set as follows:  Table Ⅱ. Comparing the results of Case 1 and Case 2, it can be found that the addition of FDS deceases the network losses and improves voltage profiles significantly. Taking Node 33 as an example, the voltage profiles with and without FDS (Case 1 and Case 2) are shown in Figure 7. Compared with Case 1, voltage profiles in Case 2 get improved during 1-8h and 19-24h because FDS transmits power from terminal T1 and T2 to T3 during this period. The addition of FDS makes the system power flow mode more flexible, and can better distribute the power between AC and DC networks to increase economy and stability of the hybrid AC/DC distribution networks. Compared with Case 2 and Case 3, although ESSs increase the operation cost, the regulation effect of ESSs on voltage deviation is obvious. This is because ESSs can absorb power when the power is relatively sufficient to make up for the serious voltage drop caused by power defect. It can be seen that VSCs, ESSs and FDS can greatly improve the system optimization effect efficiently. In order to verify the effectiveness of the two-stage robust optimization method, the worst scenario of the above 5 cases is selected for verification. The minimum nodal voltage value in 5 cases is shown in Figure 8. It should be noted that Case 3 is a deterministic case which is the same as the situation that the predicted deviation is equal to 0 in Case 4. As shown in Figure 8 and Table Ⅱ, comparison between Case 3, Case 4 and Case 5 shows that ignoring the predicted deviation, the optimization performance of Case 3 is better than that of Case 4 and Case 5. In other words, the strategies of robust optimization are more conservative than that of deterministic optimization. Comparison between Case 4 and Case 5 shows that the greater budget of uncertainty causes the more conservative optimization result and a certain increase in objective. The performance of the proposed robust optimization under different predicted deviations is shown in Figure 9. Comparing deterministic optimization and robust optimization, the larger the predicted deviation is, the more significant the effect of robust optimization is. Moreover, under the same predicted deviation, the case with higher uncertainty budget has higher objective value, meaning that its optimization strategies are more conservative. In practical applications, it can be considered to comprehensively select the budget of uncertainty according to the actual situation, while ensuring economy and safety. The performance of our proposed robust optimization under different predicted fluctuations and actual deviations is shown in Figure 10. The predicted fluctuation is 0, 5%, 10%, 15% and 20%, respectively and the actual deviation is 0, 10% and 20%. It can be seen that the robust optimization has better performance than the deterministic optimization (ξ=0) when there are uncertainties of PVs and loads. Moreover, the closer the predicted fluctuation range of robust optimization is to the actual deviation, the better the robust optimization performs. Therefore, the predicted fluctuation range ξ in the robust optimization should be selected reasonably according to the practical situation. In order to highlight the advantages of the proposed multiobjective robust optimization, different scenarios are compared between our proposed optimization and traditional single objective robust optimization [14,15]. Firstly, whether voltage deviations are considered or not in the optimization objective influences voltage profiles obviously. As shown in Figure 11, taking Case 3 and Case 5 as examples, it can be seen that the minimum voltage is significantly lower at 1-6 and 23-24 hours when ignoring voltage deviations. Although sacrificing the network economy to some extent when considering voltage deviations, voltage profiles have been significantly improved. From the perspective of safety, especially in the hybrid AC/DC distribution networks with high penetration of renewable energy, strong volatility makes it necessary to keep voltage value in a safe and reasonable range with a certain margin. Moreover, comparing with [28] whose objective function doesn't include losses and arbitrage of ESS, our proposed optimization decreases the ESS cost and increases ESS arbitrage income significantly in Case 3-5 (shown in Table III). In future distribution network electricity market, it is an important way for distributed resources, especially ESS, to get more profits by optimizing power outputs according to the electricity price.

V. CONCLUSION
In this paper, a multi-objective optimization model of hybrid AC/DC distribution networks considering DGs and loads uncertainties is established based on a two-stage robust optimization algorithm. To optimize operation cost, voltage deviation and network losses simultaneously, a multiobjective coordinated optimization model based on various controllable means is proposed. The considered means in our paper are PVs, loads, FDS, ESS, and VSCs. Compared with the previous single objective optimization, the proposed multi-objective robust optimization takes into account both economy, flexibility and security of hybrid AC/DC distribution systems. FDS and VSCs are utilized to transfer energy between AC and DC distribution systems flexibly and to increase energy efficiency in hybrid AC/DC flexible distribution networks. Then, the proposed robust model considers the uncertainties of DGs and loads. Through solving the two-stage robust optimization model, the distribution networks are able to obtain an optimal solution under the "worst" scenario. Compared with the traditional deterministic method, the larger the actual deviations are, the more obvious the control effect of the proposed robust method is. Moreover, the proposed scheme has stronger robustness and can effectively resist the risk of fluctuation of renewable energy.