Over the last decade, the penetration of wind power is gradually increasing as the load diversity changes (Yang et al., 2021a). However, the inherent fluctuation of wind power and load also constrains the operating economy and safety of the unit commitment problem (UC) with the long-distance power transmission (Milligan et al., 2009). On the other hand, the flexible AC transmission system (FACTS) device can enhance the flexibility of the network side, which also affects the operating conditions (Yuan et al., 2010). The DPFC is derived from the UPFC, which will be the most powerful tools in the FACTS. It has the same external characteristics as that of the UPFC and has advantage over the transmission corridors, investment, and replaceability (Khanchi and Garg, 2013; Dai et al., 2019; Tang et al., 2020).

Generally, the flexible operating principle of the major UC problem is divided into three categories: source side, demand side, and network side. In the source and demand sides, various research methods have been studied in the UC problem to enhance the operating flexibility by tackling uncertain parameters such as stochastic optimization (SO), robust optimization (RO), and information gap decision theory (IGDT). All these methods focus on tackling the uncertainty parameters such as wind or load uncertainties. The SO optimizes the dispatch problem with various scenarios considering uncertainty samples, which can enforce the dispatch scheduling feasibility. Wang et al. (2012) and Nandi et al. (2022a) present a stochastic UC model considering the uncertainty of demand response (DR) to improve the overall social welfare, where the uncertainty of DR is functioned as the chance constrained form. Zhao et al. (2014); Nandi and Kamboj (2021); Nandi et al. (2022b); and Kamboj et al. (2022) evaluate the wind utilization in the UC problems, where the wind uncertainty is also solved by the stochastic chance constraint. Wu et al. (2019) formulated a two-stage dispatch model considering network congestion with the chance-constrained forms of wind and DR uncertainties. Dvorkin et al. (2014) presented a stochastic rolling UC model to evaluate the operating cost, considering wind spillage and load not served, where the wind is constrained by the chance-constrained form and the load is depicted as its stochastic interval form. Obviously, there are two drawbacks: the computational efficiency decreases rapidly as the scenarios increase, and the probability distribution function (PDF) is hard to obtain accurately (Shen and Raksincharoensak, 2021a; Shen and Raksincharoensak, 2021b; Shen et al., 2021; Shen et al., 2022). There is no need to obtain the exact PDF of uncertain parameters in the RO method, and only its uncertainty boundary is offered to describe the fluctuation of the uncertain parameters. An and Zeng (2014) explore the wind uncertainty by formulating a “min–max–min” robust model to research the dispatch problem, and the result verifies the effectiveness of optimal scheduling to incorporate the wind. In the work carried out by Gangammanavar et al. (2015), the worst scenarios of uncertain wind is well-distinguished with the deterministic load. In the study by Zhang et al. (2017), the authors researched the coordination of DG and elastic-price DR scheduling with uncertainty in the microgrid, which is solved by the CCG algorithm. Wang et al. (2016) propose an adjustable robust model of the building energy system to optimize the social welfare, where the PV output and load demands are uncertain. Zhao et al. (2013) assumed that the connection between elastic electricity price and load demand fluctuates within a certain range, derived the uncertainty set for demand response, and then proposed a two-stage robust model with interval sets to depict the uncertain parameters. Zhang et al. (2016) developed a robust model coordinating the energy storage system and direct-load control (DLC) considering uncertainties, which the generation/wind/PV/ESS and DLC scheduling planning satisfy for any realization of uncertainty. Based on the aforementioned research, the two-stage model for the RO method to deal with the uncertainty is mainly important in two directions. On the one hand, the reserve capacity in the first stage is optimized to adapt to the fluctuation of the uncertain parameters in the second stage, and on the other hand, the uncertain parameters are directly optimized in the second stage to ensure power balance. The conservation of the RO method challenges the operating cost of the system dispatch, where the extreme worst scenario hardly appears (Yang et al., 2021a; Yang et al., 2021b; Li et al., 2021; Yang et al., 2022a; Yang et al., 2022b). The IGDT method aims to search for the adjustable bound of uncertain parameters based on its stochastic model and robust model, which satisfy the objective function in the predefined interval. The IGDT overcomes the difficulties of acquiring distribution function in the SO problem and reduces the conservation of the RO problem. Thus, the computational time of IGDT is much lower than that of the RO and SO methods, and the conservation is also improved obviously. This approach is widely used in dealing with the uncertainty of renewable energy (Nikoobakht and Aghaei, 2017; Ahmadi et al., 2018), energy system (Ahmadi et al., 2019; Khajehvand et al., 2021), electrical vehicles (Rabiee et al., 2014), and other loads (Ahrabi et al., 2021). Nikoobakht and Aghaei (2017 present a robust model to solve the SCUC problem considering wind uncertainty; the wind absorption is optimized with flexible resources. Ahmadi et al. (2019) formulate the UC problems with the ESS uncertainty to improve the optimal capacity of ESS. A linear model by Rabiee et al. (2014); Ahrabi et al. (2021) is established to evaluate the effect of load uncertainty to the dispatch scheduling based on its stochastic model.

Although the FACTS has an advantage over the flexibility of the network side, few studies have been proposed to investigate its impacts on operating performance. Ziaee et al. (2017) optimized the TCSC device to improve the absorption of wind based on the stochastic method, evaluating the positive effect of wind spillage and considering optimal location and allocation of TCSC simultaneously. Nasri et al. (2014) formulate a two-stage model to minimize wind spillage and load shedding considering optimal TCSC with a fixed scenario. All these research studies focus on a single-time phase sample, which only indicates the aspects of improving operating safety considering the optimization of the FACTS. There are several studies which focus on the areas of the UC problem, where the FACTS location is predefined. Li et al. (2018) investigate the effect of the UPFC to the operating cost with a fixed wind scenario. Sang et al. (2017) reduced the wind spillage by optimizing the location–allocation of TCSC considering various wind scenarios. Considering the past research studies, there is no evidence of evaluating the inter-connection between the generator status and optimal FACTS. At the same time, there is no research study on the robust UC problem considering the optimal FACTS, which may be the best way to locate the FACTS successfully.

This study develops a two-stage robust model with optimal DPFC based on its PIM model considering wind and load uncertainties, which can not only hold the internal characteristics of multiple DPFCs but also enforce the feasible horizon with the uncertain parameters. We solve the status of generators and location of the DPFC in the master problem and obtain the dispatch solution and compensation level of the DPFC in the subproblem. The main contribution in this study can be summarized as follows:

1) The DPFC scheduling planning maintains the consistence, which is easy to adopt for the uncertain environments based on the proposed model.

We demonstrate the effectiveness of the proposed two-stage robust dispatch problem in the IEEE 24 bus system and provide insight into the influence on the performance of the DPFC. This article is organized as follows: Section 2 introduces the power injection model of the DPFC and a relaxed AC-SOCP power flow model. Section 3 presents the two-stage robust model of the optimal location–allocation problem. Section 4 describes the procedure of the CC&G method. Section 5 shows the results and discussion, while the conclusion is represented in Section 6.

The power system planners aim to determine the location–allocation of the DPFC considering wind and load uncertainties, which can enhance the management efficiency of power flow and decrease the investment of the DPFC. However, the operators desire to minimize the operation cost of injected DPFCs and improve the operating level of the system. Therefore, optimal location–allocation of the DPFC in the power system must consider the operational cost, investment of installing the DPFC, curtailment of wind spillage, and load shedding. The optimal model is represented by Eqs 9–23 m i n   ∑ t ∑ i ∈ G i [ S U i + S D i + c g , i P i , t G ] + ∑ t ∑ i j ∈ G i j π D P F C P i j D P F C + ∑ t ∑ i ∈ G w M C u r t P i , t W , c u r t + ∑ t ∑ i ∈ G n b M s h e d P i , t D , s h e d (9) { S U i ≥ C i s u u i , t S D i ≥ C i s d v i , t (10) { u i , t − v i , t = I i , t − I i , t − 1 u i , t + v i , t ≤ 1 (11) { ∑ h = 0 T i o n + h − 1 I i , t ≥ T i o n ( I i , t − I i , t − 1 ) ∑ h = 0 T i o f f + h − 1 I i , t ≥ T i o f f ( I i , t − 1 − I i , t ) (12) I i , t P i , t G , m i n ≤ P i , t G ≤ I i , t P i , t G , m a x I i , t Q i , t G , m i n ≤ Q i , t G ≤ I i , t Q i , t G , m a x (13) P i , t G − P i , t − 1 G ≤ ( 2 − I i , t − I i , t − 1 ) I i , t P i , t G , m i n + ( 1 + I i , t − 1 − I i , t ) R U i P i , t − 1 G − P i , t G ≤ ( 2 − I i , t − I i , t − 1 ) I i , t P i , t G , m i n + ( 1 − I i , t − 1 + I i , t ) R D i (14) P i , t G − ∑ j ∈ ψ ( i ) P i j − ∑ j ∈ ϕ ( i ) P i j , r e v + ∑ l ∈ ψ ( i ) P i j D P F C − ∑ l ∈ ϕ ( i ) P i j D P F C − P i , t W , s h e d + P i , t D , c u r t = P i , t D + P i , t D , u − P i , t W Q i , t G − ∑ j ∈ ψ ( i ) Q i j − ∑ j ∈ ϕ ( i ) Q i j , r e v − Q i , t W , c u r t + Q i , t D , c u r t = Q i , t D + Q i , t D , u − Q i , t W (15) 0 ≤ P i , t D , s h e d ≤ P i , t D + P i , t D , u 0 ≤ Q i , t D , s h e d ≤ Q i , t D + Q i , t D , u (16) 0 ≤ P i , t W , c u r t ≤ P i , t W 0 ≤ Q i , t W , c u r t ≤ 0.95 ∗ P i , t W (17) 0 ≤ P i j , t D P F C ≤ δ i j , t P D P F C m a x ∑ i j δ i j , t ≤   α L (18) ‖ P i j − P i j D P F C Q i j ‖ 2 ≤ S i j (19) ‖ P i j , r e v + P i j D P F C Q i j , r e v ‖ 2 ≤ S i j (20) T i j ≈ θ i − θ j { P i j = g i j U i − g i j R i j − b i j T i j Q i j = − b i j U i − g i j T i j + b i j R i j P i j , r e v = g i j U j − g l R i j + b i j T i j Q i j , r e v = − b i j U j + g i j T i j + b i j R i j ‖ 2 R i j 2 T i j U i − U j ‖ 2 ≤ U i + U j (21) θ i m i n ≤ θ i ≤ θ i m a x (22) ( V i m i n ) 2 ≤ U i ≤ ( V i m a x ) 2 . (23)

The objective function is to minimize the generation cost, investment cost of the DPFC, and curtailment of wind spillage and load shedding as shown in Eq. 9. Eq. 10 constrains the start-up and shut down cost of the thermal unit; Eq. 11 distinguishes the operating state from the start-up and shut down state of generators. The minimum ON/OFF time limits are shown in Eq. 12, the active and reactive output of generators is limited in Eqs 13, 14 and shows the ramp-up and ramp-down limitation of thermal units. The active and reactive power balance is depicted in Eq. 15. Eqs 16–18 constrain wind spillage, load shedding, and location–allocation of the DPFC. The transmission network security constraint is formulated in Eqs 19–20 with line forward and reverse power flow. The relaxed AC-SOCP power flow model is introduced in Eqs 21–23.

The formulated MISOCP problem aims to improve the operating level with optimal location and ratings of the DPFC. However, there are uncertainties of wind and load, as shown in Eqs 24–25. P i , t D R = { P i , t D R | P i , t D R ∈ [ P i , t D R , F − μ i , t D R △ P i , t D R , P i , t D R , F + μ i , t D R △ P i , t D R ] ,   μ i , t D R ∈ { 0,1 } } . (24) P i , t W = { P i , t W | P i , t W ∈ [ P i , t W , F − μ i , t W △ P i , t W , P i , t W , F + μ i , t W △ P i , t W ] ,   μ i , t W ∈ { 0,1 } } . (25)

Once the DPFC is injected into the grid, the device should offer its functions considering the uncertainty circumstance of wind-load with fixed locations. We developed a two-stage robust approach to obtain the robust dispatch solutions with the PIM model of the DPFC, which can easily adapt to the uncertain environment. The robust model is shown as m i n   ∑ t ∑ i ∈ G i [ S U i , t + S D i , t ] + m a x   m i n [ ∑ t ∑ i ∈ G i c g , i P i , t G + ∑ t ∑ i ∈ G i π D P F C P i j D P F C . + ∑ t ∑ i ∈ G i M C u r t P i , t W , c u r t + ∑ t ∑ i ∈ G i M s h e d P i , t D , s h e d ] . s . t . { ( 10 ) − ( 23 ) ( 24 ) − ( 25 ) (26)

The column-and-constraint generation method is introduced to solve the proposed two-stage robust problem (Zeng and Zhao, 2013). For simplicity, the robust problem can be reformulated in the following compact matrix form: m i n x   c T x + m a x λ m i n y   d T y + e T λ . s . t .    A x ≤ b , x ∈ { 0,1 } . Y = { y | C y ≤ f G x + D y ≤ g E y = λ ‖ Q i y + q i ‖ 2 ≤ h i y + d i , i = 1 , ... , n } . (27)

The objective described in (27) corresponds to constraint (26), which is modeled in a “min–max–min” optimization form. The outer “min” is to minimize the start-up and shut down costs of generators considering the optimal locations of the DPFC; the decision variable { x } is a binary variable, which represents the state variables including the thermal generator operating state and optimal locations of the DPFC. The “max” is to find the worst uncertainty scenario under the uncertainty circumstance; the decision variable { λ } refers the wind and load operating level, which is shown in Eqs 24–25. The inner “min” is to obtain the solutions under the worst uncertainty case; the decision variable { y } represents the continuous variables in the second stage, which is described in Eqs 13–23. It can be observed that the decision variable { λ } is optimized in the second stage by maximizing the minimal second stage costs, which can easily improve the robustness.

The details of the CC&G method are shown as follows:

This work presents a two-stage robust dispatch method with optimal location–allocations of the DPFC considering wind-load uncertainties. In the model, we mainly optimize the scheduling state of thermal units and location–allocation of the DPFC to tackle the uncertainties. Case studies are performed to demonstrate the effectiveness of the proposed method. The conclusions are summarized as follows:

1) The relaxed AC-SOCP model can easily simulate the nonlinear AC power flow and has an advantage of solving speed and difficulties.

However, the robust dispatch with the optimal DPFC may face a conservative challenge because of the overall intervals of uncertainties. Some studies have developed a distributed robust optimization to overcome the conservation, which combine the priorities between stochastic and robust optimization. In addition, the DPFC has shown great advantages over the management on the network side. We will conduct more research studies on the control capabilities of the DPFC in the future. Such advantages may be effective in dispatching and operating the principle of integrated energy system (IES) due to the electric characteristics; we can relieve the couple conjunction in the gas turbine (GT) and CCHP with the energy storage system (ESS) by optimal coordination of the control capabilities of the DPFC. Another coordinate research on the TEP and DPFC has been in process to tackle the high wind-load conditions to improve robustness and flexibilities in the network side, which render more capability of available transfer power. Obviously, there is an advantage of the DPFC to be adopted in N-k contingency analysis, which may be the best performance in the application areas of the DPFC.