Research on Robust Day-Ahead Dispatch Considering Primary Frequency Response of Wind Turbine

Featured Application

The method proposed in this paper can be applied to the dispatch of power system with the large-scale integration of wind power, under the background of proposing of virtual synchronous generator technology.

Abstract

With the large-scale integration of renewable energy sources (e.g., wind power), the system inertial response and primary frequency regulation are affected. The virtual synchronous generator technology, which makes it possible for wind power units to adjust the frequency, provides a new way of approaching this problem. In this paper, we set up the related constraints that fit the optimal dispatch framework with a primary reserve representing the primary frequency response between the conventional synchronous generator and the wind power units, and the technical parameters of the virtual synchronous generator. Meanwhile, we use the robust day-ahead optimization dispatch model considering the wind power integrated primary frequency control in the real-time operation with the scenario of the wind power output showing the uncertainty of wind power. Based on the model, we identify the key set using an iterative method and obtain the maximum power loss. Through the proposed model, we could provide the day-ahead unit output and real-time primary reserve, thus ensuring the reliable operation of the system. Finally, the effectiveness of the proposed method is verified by a computational experiment.

1. Introduction

With the rapid development of the power system and the increasing demand of countries across the globe to protect the environment, renewable energy represented by wind power and the photovoltaic turbine has received more and more attention. As a major power consumer, the Chinese government attaches great importance to the development and utilization of renewable energy. Recent years have seen remarkable progress in the wind power and photovoltaic industries and the installed capacity in China. In 2017, the installed capacity for wind power in China reached 163,670 MW, up by 10.5%, while the installed photovoltaic capacity was 130,250 MW, up by 68.7% compared with the year before [1].

It is worth noting that the large-scale integration of renewable energy to the power system poses a challenge to its frequency stability. As an example, take the wind power that cannot be dispatched; its influence on the control of the system frequency is demonstrated in the following areas [2,3]:

• The converter decouples the turbine’s rotor from the system frequency, which indicates that the mechanical power of the wind turbines and the electromagnetic power of the system are decoupled; in this way, the fan will not be able to provide an inertial response. Moreover, as the wind turbines usually run in a maximum power point tracking (MPPT) mode, the primary frequency reserve could not be retained;
• With a large-scale wind integrated power system taking the place of some conventional synchronous generator units with the frequency regulation capability, the inertia of the wind turbine will be 0 with no reserve in the case of frequency fluctuations due to power loss and other reasons, which will harm the inertial response ability and primary frequency reserve capacity.

Scholars have conducted extensive and in-depth research into this issue. In [3], the influence of the large-scale replacement of conventional synchronous generators by wind turbines is analyzed, and based on this the author proposes the primary reserve strategy for conventional synchronous generators to avoid load shedding while ensuring the system frequency stability. The work in [4] has analyzed the impact of implementing high-penetration renewable energy sources on frequency and active power. The idea of combining the energy storage with a wind farm and making full use of the features of quick response and control flexibility of the energy storage for the frequency regulation of the system through charge and discharge in the peak-to-valley period is proposed in [5,6]. However, limited by the storage technology and costs, the electric energy cannot be stored in large quantity. Some scholars and engineering experts therefore put forward the virtual synchronous generator (VSG) technology, which makes it possible for the grid connection of the new energy units using the power controller containing power electronic converters to share the features of the grid-connected synchronous generator, such as inertia, damp, frequency regulation and voltage regulation [7,8,9]. During this process, the rotor motion equations for the VSG and primary frequency regulation are also introduced. Liu, et al. [10] compares VSG with droop control and draws the conclusion that VSG can provide greater inertia and system frequency stability. Therefore, the VSG in wind turbines plays a role in the frequency regulation during the operation of the system.

The primary frequency regulation of conventional synchronous generators in the system dispatch strategy, which often appears as the primary reserve, is the second-level reserve triggered automatically by the fluctuation of the frequency. The work in [11] establishes a system frequency response model in the economic dispatch which contains a large number of nonlinear and non-convex constraints, while in [12], the primary reserve constraint is taken into account in the conventional unit commitment problem, which is later solved through the mixed integer linear programming method. However, it fails to take into consideration the transient response of the speed governor. The situation is improved in [13], which takes into consideration the system inertia, governor ramp rate and frequency modulation dead band and establishes a more accurate primary reserve constraint. Through the optimal dispatch in the real-time market time scale, the dispatch strategy which ensures the abundance of the Primary Frequency Response (PFR) is developed. Within the framework of the Optimal Power Flow (OPF), the above research draws up the dispatch strategy that meets the demands of the primary frequency response. However, with the increase in the wind power penetration rate, the number of conventional units that can be dispatched in the power grid drops, which affects the primary reserve. It is therefore of great significance to fully consider the calculation of the frequency regulation capability of the VSG in wind turbines.

The work in [14] elaborates on the impact of increasing the wind turbine capacity in power systems on conventional units and the reserve level and finds that the reserve and system operating costs are increasing. Ye, et al. [15] regard wind turbines as schedulable unit resources. It proposes two reserve allocation strategies of wind turbines and analyzes their advantages and disadvantages as well as the applicable scenarios. Based on the analysis, the author puts forward the method which integrates the two mentioned reserve allocation strategies. The method used to determine the reserve capacity through the fixed reserve capacity or a certain proportion of the wind power output is not an effective one, as it might lead to an inadequate reserve capacity or a waste of resources, even though it makes use of the frequency regulation ability of the wind turbines. Moreover, the flexibility of the system primary reserve does not see an obvious increase. However, the VSG attains a faster adjustment speed, and in order to make sure that it goes with the traditional synchronous generator, the performance indexes are usually basically set to be the same as the traditional one. Therefore, it is probable to set up the primary reserve constraint that is more accurate and suitable for the optimal dispatch of the wind turbines, which will work with the other synchronous generators to enhance the frequency stability of the system by integrating the method to determine the primary reserve capacity of the traditional synchronous generator unit with the VSG technical standard.

In fact, the above analysis of wind power primary reserve capacity allocation strategies fails to consider the uncertainty of wind power. In [16], considering that the increase of the grid connection of wind power in Europe in the future will lead to a greater prediction fault, the author proposes the coordinated dispatch at different time scales. The prediction accuracy of new energy varies at different time scales. Normally, other units put aside the reserve capacity for adjustment (different from the primary reserve) within the day-ahead market time-scale dispatch strategy, to ensure the real-time power balance of the system [14]. In this regard, with the two-stage robust optimization method, we can effectively coordinate the dispatch strategies in different stages for different targets. The work in [17,18,19] applies the method to the unit commitment problem and decide on the unit start-stop strategy for the first stage and the economic dispatch strategy for the second stage. Li, et al. [20] considers wind power generation uncertainties as well as zonal reserve constraints under both normal and N-1 contingency conditions and proposes a robust model. As robust optimization is a three-layer optimization problem, it needs to be solved by decomposition algorithms such as the Bender’s algorithm and column-and-constraint generation (CCG) algorithm [17,21,22]. In [23], we locate an adaptive robust method, which uses different scenarios to represent the uncertainty of wind power, and which finds the key scenario with the maximum balance cost to represent the uncertainty information contained in all of the scenarios. Following this, the set of key scenes is considered to find out the day-ahead dispatch strategy. Since the primary reserve capacity of wind power is related to its output, it is indispensable to decide on the nearly real-time primary reserve capacity within the real-time optimization framework, considering the primary frequency response, so as to avoid the influence of a wind power forecasting error. Therefore, not only should we consider the uncertainty balance cost of the wind power during the real time operation of the system, but also the cost of the primary reserve.

In summary, this paper proposes an improved adaptive robust optimization model to address the uncertainty of wind power in the case of a large-scale grid connection of wind power. In the two-stage optimization, the unit output and power balance reserve are decided by the day-ahead dispatch, while during the real-time dispatch, after adjusting the unit output, we build up the accurate wind power primary reserve constraint within the real-time optimization framework to coordinate the primary reserve capacity between the wind power and other synchronous generators according to the frequency indexes of the system as the VSG could retain the frequency regulation reserve. The main contributions of this paper are as follows:

In order to improve the accuracy of the coordination with other synchronous generators in the system and determine the primary reserve capacity of wind power, a more reasonable and effective approach to determine the primary reserve capacity of wind power is proposed on the basis of the fixed reserve capacity mode following the norms of VSG technology as well as the forms of restraints that fit the optimal dispatch framework.

Robust optimization is of great significance for solving the uncertainty problem of wind power. In this paper, within the two-stage robust optimization framework, we take into consideration the primary reserve in the real-time operation and propose the day-ahead dispatch strategy targeting the minimum operation cost after coordinating the reserve capacity allocation between the VSG and the conventional generator. Then, we analyze the strategy with the IEEE computational experiment.

In the second part of the paper, we analyze the inertial support and primary frequency regulation of the system frequency response before and after the wind power grid connection. In Part 3, a robust optimization dispatch model considering the frequency response is established to coordinate the reserve and dispatch schemes on the day-ahead and real-time scales. The primary reserve is adopted in response to the system frequency fluctuations. Finally, the solution is given. The computational experiment is run in Part 4, during which a modified IEEE 57-node system is used to verify the effectiveness of the proposed algorithm, and the influence of the grid connection of the wind power on the system frequency regulation is analyzed. The fifth part is the conclusions drawn upon the analysis of the computational results.

2. Frequency Analysis of Wind Integrated Power System

The inertia of the power system provides a short-term power support that responds to changes in the system frequency to prevent it from dropping rapidly, which enables the system to adjust the operating strategy to reconstruct the power balance. The spinning inertia of the system is mainly provided by the synchronous generator. When the system frequency fluctuates, the synchronous generator rotor releases its rotational kinetic energy to track it. Therefore, in the analysis of the system frequency, the inertia time constant H of the synchronous generator is very important. It is the ratio of the kinetic energy of the unit rotor to the rated capacity of the generator at the synchronous angular velocity ${\mathrm{\Omega }}_0$ of the generator, as shown in the following equation [24]:

$$H=\frac{W_k}{S_N}=\frac{J{\mathrm{\Omega }}_0^2}{S_N}$$

(1)

where $W_k$ represents the rotational kinetic energy at the rated speed of the generator; $S_N$ is the rated capacity of the generator, while $J$ means the moment of inertia of the rotor.

For large-scale wind power grid-connected systems, the system equivalent inertia time constant can be obtained by referring to Equation (1):

$$H_s=\frac{W_{k,s}}{S_{N,s}}=\frac{{\displaystyle \sum _{i=1}^m\frac{1}{2}{J}_i{\mathrm{\Omega }}_0^2}+{\displaystyle \sum _{j=1}^n{W}_{w,j}}}{S_{N,s}}$$

(2)

where $W_{k,s}$ is the total rotational kinetic energy of the system, including that of the conventional synchronous generator and wind turbines; $S_{N,s}$ indicates the total rated capacity of the system; $J_i$ is the moment of inertia of the synchronous generator i; and $W_{w,j}$ is the rotational kinetic energy of the fan j.

In fact, during the grid connection of the large-scale wind turbines represented by the Doubly-fed Induction Generator (DFIG), the number of the conventional synchronous generator units is reduced to accommodate the wind power and cut the wind curtailment while maintaining the system power flow balance [25]. Because of the converter, though the wind turbine rotor has a certain rotational kinetic energy, it cannot respond to the frequency change. Consequently, in Equation (2), $W_{w,j}\approx 0$. When the total rated capacity of the system remains unchanged, the responsible kinetic energy decreases and the system equivalent inertia time constant drops too. At the same time, because the wind turbines could retain the frequency regulation reserve for the primary frequency regulation and because they usually run in the MPPT mode, if we do not consider the energy storage there will not be power reserve for the primary frequency regulation. The MPPT mode indicates that, through the self-optimizing process, the operating point voltage or current is adjusted to make sure the wind turbines operate near the peak work point.

The grid connection of large-scale wind turbines can result in a decrease in the system’s inertial response and primary frequency regulation ability. The frequency regulation of different systems is affected by the grid connection to different extents due to factors like the wind power grid-connected capacity, synchronous generator installed capacity and network topology. Generally, the inertial frequency response may be affected when the wind power exceeds 10% of the system [26]. With the increasing grid-connected capacity of renewable energy, many countries require an inertia response and power support capabilities of grid-connected wind power in order to ensure a stability of frequency for the system [27,28,29]. To meet the above requirements, wind turbines can be involved in the system frequency response by energy storage or VSG technology.

Through the parallel connection of the energy storage devices and the AC bus at the wind turbines, or at the wind farm outlet and the joint control, it is possible to obtain similar external characteristics of the frequency regulation of the synchronous generator. In this case, the combined system of the wind turbine and the energy storage device could provide an inertial response, and the specific inertial response time constant could be worked out from the dynamic characteristics of the system.

The VSG-based wind power units share a similar operating mechanism and external characteristics to the conventional synchronous generators, which is of great significance to the frequency support in the synchronous grid. Taking the primary frequency response as an example, the VSG of the wind turbine can provide a primary reserve through the control of the pitch angle [7]. In a normal operation, the pitch angle is controlled at 0 to ensure the maximum power, and the pitch angle for the primary reserve will be above 0. When the system frequency changes, the VSG adjusts the pitch angle to control the storage or release of the reserve, providing power support. Therefore, in a short time, Equation (2) can be formulated as:

$$H_s=\frac{{\displaystyle \sum _{i=1}^{m+n}{H}_i{S}_i}}{S_{N,s}}$$

(3)

where H_i and S_i represent the inertia time constant and capacity of unit i, including the synchronous generators and wind power units participating in the PFR.

3. Optimization Model Considering Primary Frequency Response

We aim to optimize the reserve capacity for the PFR. With the increase in the prediction time scale and decrease in the accuracy of the wind power output prediction, it is necessary to take into consideration the uncertainty of the wind power prediction. Therefore, a two-stage robust optimization model is proposed. On the one hand, in order to cope with the power imbalance caused by the wind power prediction error in the real-time operation, a certain spinning reserve for the conventional generator units should be included in the day-ahead dispatch strategy; on the other hand, during the real-time operation, the primary reserve for the conventional synchronous generator units and VSG of the wind turbines is determined based on the actual output of the wind power. In addition, this paper describes the uncertainty of the wind power using representative scenarios [30,31].

3.1. Primary Frequency Response

We consider that in the real-time market, the wind turbines using VSG technology could provide an inertial response and PFR. The method employed to determine the reserve capacity through the fixed reserve capacity or a certain proportion of the wind power output is not a widely applicable one. Therefore, we will establish a more detailed primary reserve model with the optimization dispatch framework, in accordance with the VSG technical standards and system frequency requirements.

According to the grid operation code, there is a certain lower limit of the system frequency (this paper only considers the frequency drop). Taking the state grid as an example, the rated frequency is 50 Hz, with a frequency offset of 0.2–0.5 Hz. For the 5–30 s PFR time frame, the power adjustment after the frequency drop involves a power loss, system inertia, ramp response rate of the governor and frequency dead band. Based on this, we depict the governor model as in Figure 1:

Figure 1. Governor Model.

Figure 1. Governor Model.

where P_m(t) and P_e(t) represent the mechanic and electromagnetic power of the system; P_loss is the loss of the active output caused by a fault in the units; $R_i^F$ is the primary reserve of the unit i, while C_N is the ramp rate of the governor; f₀, f_d, f_N and f_M represent the initial frequency, the governor’s dead band, the nadir-based frequency of the unit after the fault, and the lower limit of the system frequency, respectively; and t_d and t_N are the time when the frequency reaches the dead band and the lowest point.

The active output decrease can result in a decreased system frequency. At the same time, the governor responds within the range of the reserve. The mechanical power of the system increases, which leads to the balance of the mechanical power and the electromagnetic power. If we ignore the load damping, the relationship between the frequency variation and power could be shown as follows [32,33]:

$$M_H\frac{df\left(t\right)}{dt}=-\Delta P=P_m\left(t\right)-P_e\left(t\right)$$

(4)

where $M_H$ indicates the system inertia after the fault with the MWs/Hz unit; and through the integration of both sides of the Equation (4), we get:

$${\displaystyle \underset{0}{\overset{t_N}{\int }}\frac{df\left(t\right)}{dt}dt=}{f}_N-f_0=\frac{1}{M_H}{\displaystyle \underset{0}{\overset{t_N}{\int }}\left[P_m\left(t\right)-P_e\left(t\right)\right]}dt=-\frac{1}{M_H}\left(P_{loss}{t}_d+\frac{P_{loss}^2}{2C_N}\right)$$

(5)

Using the governor model in Figure 1, Equation (5) equals:

$$f_N-f_0=-\frac{1}{M_H}\left(P_{loss}{t}_d+\frac{P_{loss}^2}{2C_N}\right)$$

(6)

If C_M, the maximum ramp rate of the governor, exists, f_M corresponds to it. f_N and C_N could be replaced by C_M and f_M; then, Equation (6) would become:

$$C_M=\frac{0.5P_{loss}^2}{M_H\left(f_0-f_M\right)-P_{loss}{t}_d}\stackrel{M_H=\frac{P_{loss}{t}_d}{f_d}}{=}\frac{0.5P_{loss}^2}{M_H\left(f_0-f_M-f_d\right)}$$

(7)

The power fluctuation of the system is supported by inertia for a short time within the dead band of the governor, so the system inertia time constant can be expressed by Equation (7): $M_H=\frac{P_{loss}{t}_d}{f_d}$.

Therefore, if we only consider the synchronous generator in the system, for P_loss, the primary reserve of the system satisfies the following restraints:

$${\displaystyle \sum R_i^F}\ge P_{loss}$$

(8)

$$0\le R_i^F\le 2C_i\frac{M_H\left(f_0-f_M-f_d\right)}{P_{loss}}$$

(9)

where $C_i$ is the ramp rate of the governor of unit i.

From Equations (8) and (9), we can see that when the power imbalance is small, a sufficient reserve capacity can be retained in the optimization dispatch through the above constraints. Equation (8) indicates that the primary reserve of the system is sufficient to cover the power loss and maintain the power balance, while through Equation (9) we determine the reserve capacity based on the ramp rates of different governors.

From Equation (9), we can see that between $t_N$ to $t_d$, assuming that the power variation of the unit i is $\Delta P$, $C_i=\frac{\Delta P}{t_N-t_d}$, whereas if we consider the power imbalance of P_loss after the fault in the unit, $C_{MIN}=\frac{P_{loss}}{t_N-t_d}$ is the minimum required ramp rate of the governor; so, $\Delta P=C_i\frac{P_l}{C_{MIN}}$. Therefore, in order to meet the demand of the reserve (8) without causing any waste, the primary reserve of unit i, $R_i^F\le \Delta P=C_i\left(t_N-t_d\right)=P_{MIN}$ where $P_{MIN}$ is the adjustment power of i. In fact, the primary reserve of unit i shall be within the range of the upper limit of the unit’s output and comply with the constraint $P_{MIN}$, while releasing before $t_N$. Readers can read it in [13] for ease of understanding.

The above discussion demonstrates the PFR constraint for the economic dispatch (ED) based on the synchronous generator parameters. In this paper, we also consider the grid connection of large-scale wind power. In order to ensure a sufficient frequency response of the grid, it is necessary to involve wind power in the primary frequency control. Research into the frequency considers mainly the real-time time scale. The scenario of the wind power output during the real-time operation is fixed; based on this, the work in [15] proposes the constant power reserve mode of the wind power, as shown in Figure 2a. And we propose primary reserve constraint of the wind power, as shown in Figure 2b.

Figure 2a shows that when the wind power output is below a certain level, the wind power could not provide the primary reserve. When it is above that, the reserve capacity is fixed. This is a technically reliable mode and has been applied [34]. However, it is undeniable that this mode is not flexible enough in the ED, especially when it joins other synchronous generators in the frequency response. The primary reserve in wind power indicates some wind curtailment, so the fixed reserve capacity does not prove to be an effective approach considering the dispatch strategy and costs. Therefore, a better form of constraint is needed within the framework of the optimal dispatch.

As we have mentioned, the VSG control technology enables the wind turbine to share similar external frequency regulation characteristics to the synchronous generator. When the system frequency deviation is above f_d and the active output of the VSG is greater than 20% of the rated capacity, the VSG active output can be adjusted to participate in the PFR of the system. When the system frequency drops, the VSG active output rises, and the maximum active output increase will be at least 10% of the rated capacity. However, in order to ensure a sufficient power generation of the wind farm, the number should not be too large. In this paper, assuming that it is 20%, when the real-time wind power output scene is determined, the question of whether to retain the primary reserve depends on whether the output power is more than 20% of the rated capacity. The primary reserve constraint is shown in Figure 2b in the following form:

$$0\le R_W^F\le \min \left(\epsilon P_N,2C_W\frac{M_H\left(f_0-f_M-f_d\right)}{P_{loss}}\right)$$

(10)

$$P_W^r+R_W^F\le P_W^m$$

(11)

where $R_W^F$ indicates the primary reserve of the wind power unit i, and P_N is its rated installed capacity; $C_W$ represents the ramp rate of the governor after the simulation of the virtual synchronous generator units; ε is the upper limit for the active output increase greater than 10%; and $P_W^r$ and $P_W^m$ indicate the actual wind power output and the wind power output in the MPPT mode.

With enough of an active output, the upper limit of the primary reserve of the wind turbine VSG will be the smaller value between the maximum increase of the output in the technology guidelines and the maximum value calculated by the VSG parameters, as shown in Equation (10). In addition, the wind curtailment other than the primary reserve is not considered here, as shown in the shaded part of Figure 2b. As the maximum power of the theoretical unit includes the real-time output and the retained primary reserve, Equation (11) changes into an equation. Considering the participation of VSG in the system inertia support and PFR, in order to coordinate with other synchronous generators, the total amount of the primary reserve of the system in Equation (8) should include the reserve of the wind turbine.

The ramp rate of the governor in the above equation can be obtained through the response rate of the unit under the large frequency deviation. In this paper, the corresponding values are drawn according to the technology guidelines, as shown in Appendix A.

3.2. Two-Stage Robust Optimization Model

Since it is necessary to take into consideration the day-ahead reserve for the power imbalance in the optimization dispatch framework [35], we propose a three-layer robust optimization model involving the day-ahead and real-time stages. Assuming that the load is forecast at a certain value, under the expected value, this paper determines the power imbalance reserve by the day-ahead scheduling to cope with the error between the wind power output and the wind power forecast in the real-time operation. Under the real-time scale, in different wind power output scenarios, the unit output needs to be adjusted again within the output determined in the day-ahead dispatch and within the power imbalance reserve constraint. On this basis, the primary reserve of the unit should also be retained.

Therefore, in this model, the first stage is the day-ahead economic optimization dispatch, which decides the output and power imbalance under the expected value of the wind power output. The second stage targets the minimization of the balancing cost of the wind power output scene of the set $\mathrm{\Omega }$, which considers the readjustment of the output of the conventional units and primary reserve in the real-time scene s. In the model, the basic day-ahead scene is indicated by the superscript 0, and the real-time scene is indicated by the superscript s. The mathematical expression is as follows:

$$\begin{array}{l}\min {\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}\left(C_{G,i}{P}_{G,i,t}^0+C_{R,i}^U{R}_{i,t}^{U,0}+C_{R,i}^D{R}_{i,t}^{D,0}\right)}}+\\ \underset{s\in \mathrm{\Omega }}{\max }\min {\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{C}_{G,i}\left(r_{i,t}^{U,s}-r_{i,t}^{D,s}\right)}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G+N_W}{C}_{R,i}^F{R}_{i,t}^{F,s}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_B}{C}_{L,i}{P}_{i,t}^{non}}}\end{array}$$

(12)

where T is the dispatch period, 24 h; $N_G$, $N_W$ and $N_B$ indicate the number of traditional synchronous generators, the wind power and the system power nodes respectively; $P_{G,i,t}^0$, $R_{i,t}^{U,0}$ and $R_{i,t}^{D,0}$ represent the output of unit i, and the upper/lower reserve for the power imbalance during the t period in the basic scenario, while $C_{G,i}$, $C_{R,i}^U$ and $C_{R,i}^D$ indicate the coefficients of the output cost, upper reserve cost and lower reserve cost of unit i; $r_{i,t}^{U,s}$ and $r_{i,t}^{D,s}$ represent the adjustment of the active output of unit i during the t period in the real-time scenario s; $R_{i,t}^{F,s}$ is the primary reserve, and $C_{R,i}^F$ its cost coefficient; $P_{i,t}^{non}$ refers to the load shedding and $C_{L,i}$ to its cost coefficient.

In the day-ahead dispatch strategy, the expected value of the wind power output is set as the basic scenario, and we target the minimum unit output and reserve cost. For the optimization goal specified in Equation (12), the constraints are as follows:

$${\displaystyle \sum _{i\in m}{P}_{G,i,t}^0}+{\displaystyle \sum _{w\in m}{P}_{W,w,t}^{e,0}}-P_{L,m,t}-{\displaystyle \sum _{l\in m}{P}_{l,t}^{mn,0}}=0$$

(13)

$$P_l^{mn,0}=\frac{{\theta }_{m,t}^0-{\theta }_{n,t}^0}{x_{mn}}$$

(14)

$${\underset{\_}{P}}_{G,i}\le P_{G,i,t}^0\le {\overline{P}}_{G,i}$$

(15)

$$R_{D,i}\le P_{G,i,t}^0-P_{G,i,t-1}^0\le R_{U,i}$$

(16)

$${\underset{\_}{P}}_l^{mn}\le P_{l,t}^{mn,0}\le {\overline{P}}_l^{mn}$$

(17)

$${\displaystyle \sum R_{i,t}^{U,0}\ge R_t^{U,\min }},{\displaystyle \sum R_{i,t}^{D,0}\ge R_t^{D,\min }}$$

(18)

$$P_{G,i,t}^0+R_{i,t}^{U,0}\le {\overline{P}}_{G,i},P_{G,i,t}^0-R_{i,t}^{D,0}\ge {\underset{\_}{P}}_{G,i}$$

(19)

where $P_{W,w,t}^{e,0}$ is the expected output of the wind power unit w at the moment t; $P_{L,m,t}$ and $P_{m,t}^{non}$ indicate the load and load shedding at node m at the moment t, while $P_{l,t}^{mn,0}$ is the power of line mn at t; ${\theta }_{m,t}^0$ and ${\theta }_{m,t}^0$ refer to the phase angle between node m and n at t; $x_{mn}$ is for the reactance of line mn, ${\overline{P}}_{G,u}$ and ${\underset{\_}{P}}_{G,u}$ represent the range of output of unit u; $R_{U,u}$ and $R_{D,u}$ show the upper and lower limits of the ramp rate; ${\overline{P}}_l^{mn}$ and ${\underset{\_}{P}}_l^{mn}$ represent the limit of line mn, while $R_t^{U,\min }$ and $R_t^{D,\min }$ are the minimum constraints for the upper and lower reserve.

Equations (13)–(17) represent the grid power balance, unit output, ramping constraints, and line capacity constraints. Equations (18) and (19) show the spinning reserve used to cope with the power imbalance caused by the wind power prediction error. The reserve meets the upper and lower limits of the unit output.

In the real-time operation, we adjust the output for each scenario in the wind power output scene of the set $\mathrm{\Omega }$, and we look for the key scene, whose constraints are as follows:

$${\displaystyle \sum _{i\in m}\left(P_{G,i,t}^0+r_{i,t}^{U,s}-r_{i,t}^{D,s}\right)}+{\displaystyle \sum _{w\in m}{P}_{W,w,t}^{r,s}}-\left(P_{L,m,t}-P_{m,t}^{non}\right)-{\displaystyle \sum _{l\in m}{P}_{l,t}^{mn,s}}=0$$

(20)

$$P_l^{mn,s}=\frac{{\theta }_{m,t}^s-{\theta }_{n,t}^s}{x_{mn}}$$

(21)

$${\underset{\_}{P}}_{G,i}\le P_{G,i,t}^s=P_{G,i,t}^0+r_{i,t}^{U,s}-r_{i,t}^{D,s}\le {\overline{P}}_{G,i}$$

(22)

$$R_{D,i}\le P_{G,i,t}^s-P_{G,i,t-1}^s\le R_{U,i}$$

(23)

$${\underset{\_}{P}}_l^{mn}\le P_{l,t}^{mn,s}\le {\overline{P}}_l^{mn}$$

(24)

$$0\le r_{i,t}^{U,s}\le R_{i,t}^{U,0},0\le r_{i,t}^{D,s}\le R_{i,t}^{D,0}$$

(25)

$$0\le P_{m,t}^{non}\le {\overline{P}}_m^{non}$$

(26)

$${\displaystyle \sum _{i=1}^{N_G+N_W-1}{R}_{i,t}^{F,s}}\ge P_{loss}$$

(27)

$$0\le R_{i,t}^{F,s}\le 2C_i\frac{M_H\left(f_0-f_M-f_d\right)}{P_{loss}}$$

(28)

$$0\le R_{W,t}^{F,s}\le \min \left(\epsilon P_N,2C_W\frac{M_H\left(f_0-f_M-f_d\right)}{P_{loss}}\right),P_{W,w,t}^{r,s}+R_{W,t}^{F,s}=P_{W,w,t}^{m,s}$$

(29)

where $P_{W,w,t}^{r,s}$ is the actual output of unit w at t; the meaning of the other variants is the same as in the basic scenario, and for the real-time scenario we use the superscript s.

From Equations (20)–(24), we find that in the real-time operation, the constraints in different scenarios are similar to those in the day-ahead dispatch, which represent the grid power balance, the unit output and ramping constraints, and the line capacity constraints, respectively; but in the power conservation Equation (20), the power of the wind turbines is no longer the expected value of the wind power output. Moreover, the unit output is re-adjusted on the basis of the day-ahead dispatch before it meets the restraint of Equation (25). The adjusted unit output also meets Equation (22). Since we only consider the power imbalance reserve in the basic scenario in the model, and skip the primary frequency response constraint, the load shedding is provided in the real-time scenario for the sufficient primary reserve and complete wind power accommodation, as in Equations (20) and (26). Equations (27)–(29) depict the relevant constraints for the real-time scenario considering the primary frequency response, whose meanings are shown in Equations (8)–(11). Equation (27) contains all of the units except the fault unit, Equation (28) is for the traditional synchronous generators, while Equation (29) applies to virtual synchronous wind turbines.

In the above calculation, P_loss and some other variables about the primary reserve are constant. In most cases, the operator wants to obtain the maximum possible power loss due to the failure of the unit when the system frequency requirement is met. Therefore, when P_loss is a variable, the target can be as follows:

$$\max P_{loss}$$

(30)

When P_loss is not fixed, the system inertia will no longer be a constant after the failure, and Equation (3) becomes a new constraint, indicating the inertia of the system after P_loss. In addition, other constraints of the model include (13)–(29), indicating the maximum dispensable power of the system in view of the wind power uncertainty in the set $\mathrm{\Omega }$ and in the primary reserve. Though the model is for a single-layer optimization, it is a difficult one as the real-time operating constraints corresponding to all of the scenarios should be considered. It is also worth noting that by solving this model, we could obtain the maximum dispensable power. However, as in real-life situations, the power loss is demonstrated through a unit failure; the result does not refer to the capacity of a certain unit, but shows that the inertial support and PFR of the system could still meet the frequency regulation requirements when the unit with the capacity or below the capacity is at failure.

3.3. Solution

It would be very difficult to solve the model directly. For the min-max-min problem, we introduce the auxiliary variable β for the max-min problem, so that:

$$\min {\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}\left(C_{G,i}{P}_{G,i,t}^0+C_{R,i}^U{R}_{i,t}^{U,0}+C_{R,i}^D{R}_{i,t}^{D,0}\right)}}+\beta$$

(31)

We also add the constraint:

$$\underset{s\in \mathrm{\Omega }}{\beta }\ge {\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{C}_{G,i}\left(r_{i,t}^{U,s}-r_{i,t}^{D,s}\right)}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G+N_W}{C}_{R,i}^F{R}_{i,t}^{F,s}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_B}{C}_{L,i}{P}_{i,t}^{non}}}$$

(32)

The model after the introduction of the auxiliary variable β is equivalent to that from before the change. The three-layer optimization problem is reduced to a single-layer one. However, if all of the scenes in the set Ω are taken into consideration with the corresponding real-time operation constraints, the problem would become intimidating and could hardly be solved. Therefore, Chen, et al. [23] proposes an iterative method where the master problem focuses on the key subset Ω_m, which is found through iteration in the sub-problem. As Ω_m contains all of the information of Ω, the dispatch strategy and reserve considering Ω_m will meet the output re-adjustment demands of all of the scenes. The process is as follows:

• Define the starting key subset Ω_m = {$P_{W,w,t}^{m,s1}$};
• Master problem: solve the above model to obtain the dispatch strategy X_m and the adjustment cost β_M for Stage 2;
• Sub-problem: based on X_m, calculate the goals for each and every scenario in Ω\Ω_m;

  $$\min {\beta }_s={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}{C}_{G,i}\left(r_{i,t}^{U,s}-r_{i,t}^{D,s}\right)}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G+N_W}{C}_{R,i}^F{R}_{i,t}^{F,s}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_B}{C}_{L,i}{P}_{i,t}^{non}}}$$

  (33)

  The optimization goal during the real-time operation is restricted by Equations (20) to (29). For the results obtained in 3, the maximum ${\beta }_S=\underset{s\in \mathrm{\Omega }\backslash {\mathrm{\Omega }}_m}{\max }{\beta }_s$, and the corresponding scenario is $P_{W,w,t}^{m,s^{\prime }}$;
• If ${\beta }_S>{\beta }_M$, Ω_m = {Ω_m, $P_{W,w,t}^{m,s^{\prime }}$}, then we go back to 2; otherwise, we get the dispatch strategy for the master problem, and the iteration ends.

Through the above process, we can obtain the dispatch strategy and reserve considering the key subset Ω_m. As Ω_m contains all of the information of the scenes of Ω, the maximum dispensable power caused by the unit failure could be obtained by solving the key subset Ω_m, if the operator needs that. The flow chart is shown in Figure 3.

4. Results

In this paper, the IEEE 57 modified test system is used to test the proposed model and algorithm. The unit at Node 3 is a wind turbine with a rated capacity of 140 MW, and the total installed capacity of the system is about 2000 MW. The daily power load curve and the curve of the wind power generated energy are shown in Figure 4. 50 wind power output scenarios are selected in the test to show the uncertainty information. According to China’s grid operation guideline, the rated frequency of the grid is 50 Hz, and the minimum frequency is 49.8 Hz. Here, we adopt the YALMIP optimization tool [36] and use CPLEX and IPOPT to solve the problem.

4.1. Cost Analysis

The grid connection of large-scale wind power led to the shutdown of some conventional synchronous generator sets to meet the wind power accommodation. In this example, we use the wind power to replace the synchronous generator for a simulation. The capacity of the power loss is 20 MW, the initial frequency of the system is 50 Hz, and the economic dispatch results are shown in

Table 1. Dispatch costs in different situations; PFR: primary frequency response.

Key Subset Ωm	Cost/$
	Without Primary Reserve	PFR not Involving the Wind Power	PFR Involving the Wind Power
{6;49}	5,375,256	5,508,326	5,508,085

.

From Table 1, we can see that the key scene set screened through the iteration is {6;49}. At this time, the dispatch cost increases dramatically after considering the PFR in the large-scale wind integrated power system, especially for the cost of the primary reserve or the shedding load. Since the uncertainty in the wind power scene set {6;49} is considered in all three cases, the day-ahead dispatch costs are basically the same under the same condition, including the cost of the unit output and the reserve cost for the power imbalance retained for the wind power prediction error. For the remaining real-time operating costs, the classification is as follows:

The three states in Figure 5 correspond to the three situations in Table 1, respectively. In Figure 5, we find that the adjustment costs of the unit output are the same in the different states in the real-time operation. The real-time unit output adjustment is not only related to the error between the wind power prediction and actual operation, but also to the primary response. Therefore, it is necessary to consider both the cost of unit output adjustment and that of the primary reserve. The actual operating cost changes when we consider the failure of the unit. Figure 5 shows that the impact of the frequency response of the wind power on the primary reserve is demonstrated in the following way: without the frequency response, the total cost stands at 132,257$; when wind power is involved in the frequency response process, the cost will be 132,036$, which is related to the cost coefficient of the wind power primary reserve.

In fact, under certain circumstances, the output of the unit might drop, and load shedding might occur in real-time operation. Figure 6 depicts the classification of the real-time operating costs, including the load shedding cost.

Figure 6 demonstrates the adjustment cost of applying a real-time operation strategy on a day-ahead basis, in view of the actual output of the wind power. From this figure, we can see that the load shedding cost is relatively high in the real-time operation. Although the load shedding ensures that we can obtain the new optimal solution of the real-time operation strategy through optimization on the basis of the day-ahead dispatch strategy, in view of the great impact it has on the user experience and economic cost, the system dispatcher should avoid sacrificing the user experience to ensure an ample primary reserve. In fact, to avoid using the load shedding as a method of adjustment, a higher load shedding cost is set in this study; so we find a lower load shedding in Figure 6.

4.2. Wind Power Primary Reserve

The primary reserve is provided by the conventional synchronous generator and the wind power units. When the wind power is not involved in the frequency response, the synchronous generator will respond to the frequency drop caused by the power imbalance of the system. When the wind power unit displays the external frequency regulation features, it could also shoulder the pressure of the PFR of the conventional synchronous generator. The wind power primary reserve is shown as Figure 7 for Scenario 6:

Retaining the primary reserve indicates that the wind power units could not run at MPPT and that the wind curtailment is inevitable. The cost of the wind curtailment will in turn have a negative impact on the dispatch strategy aiming to cut the cost.

Table 2. Primary reserve cost and capacity of the wind power units.

Cost Coefficient/($/MW)	Cost/$	Primary Reserve Capacity of Wind Power Units/MW	Cost Coefficient/($/MW)	Cost/$	Primary Reserve Capacity of Wind Power Units/MW
0	5,513,209	10.9061	200	5,513,664	9.6847
100	5,513,436	10.5484	300	5,513,891	8.2178

shows the economic dispatch cost under different cost coefficient of the wind power primary reserve in the case of a serious unit failure (P_{los s}= 20.6 MW).

The change of the cost coefficient of the primary reserve has an impact on the dispatch strategy of the system. Overall, the total cost rises with the increase in the cost coefficient of the primary reserve. At the same time, the higher the cost coefficient, the lower the primary reserve capacity of the wind turbine. Table 2 shows that when the wind power reserve cost coefficient is lower than that of other synchronous generators, that is, when the cost coefficient is 0 and 100, the wind power reserve sees a smaller change; and when the cost goes up, the primary reserve capacity drops significantly. In fact, the virtual synchronous generator is available for rapid adjustment, and in situations like a failure and DC blocking, the VSG could respond quickly and retain enough primary reserve to stop the frequency from dropping sharply after the failure, thus gaining more time for other adjustment measures.

In addition, the pressure of the primary frequency response of the system goes up with the increase of the lost power. At this time, the importance of the frequency regulation of the wind power units is obvious. In this study, we consider the key subset {6;49}, and solve it with Equation (30) as the target. When the wind power has no frequency regulation capability, the maximum power lost at the time of the system frequency constraint is 20.4181 MW. However, when the wind power is involved in the primary reserve, the primary frequency response capability strengthens and the power loss stands at 21.3198 MW. It is worth noting that, as the lost power increases and the frequency drops, the selected key subsets may change, and load shedding might occur during the dispatch process.

In addition, we compare the calculation results of the reserve mode (10% P_N) with the proposed wind power fixed capacity and constraint model. For the adopted fixed-capacity reserve mode, the wind power frequency regulation capability can be fully exploited, and the total reserve of the 24 time periods reaches 336 MW, far exceeding the results from the model proposed in this paper; additionally, the economic cost also reaches $5,659,286. Generally speaking, the high cost for the fixed reserve capacity poses a great demand on the VSG technical parameters and entails a low flexibility. If we are to fully exploit the adjustment capability of renewable energy to accommodate the new energy in the grid, this mode is not an applicable one.

5. Discussion

In this study, we establish a wind power reserve constraint model for real-time operations with the VSG technology based on previous research on the wind power primary reserve. Then, we coordinate the day-ahead and real time operation and apply a robust optimization to the problem of the power imbalance caused by the grid connection of large-scale wind power. For the real-time operation stage, we determine the primary reserve capacity of the VSG and synchronous generator, which ensures that the drop of frequency after the unit failure does not exceed the limit. In order to solve this problem, the critical scene set is identified by an iterative method, which improves the computational efficiency, and the maximum dispensable power is obtained through solving the key set.

In fact, although the validity of the proposed algorithm is verified by calculation, the IEEE example used is different from the actual power system. The influence of wind power participating in the primary frequency response has not been fully investigated. In future research, we plan to further explore the inertial support of the system and PFR through the simulation of the grid data and structure. Nowadays, HVDC transmission enjoys a rapid development. With the expanding wind power grid connection, the significant frequency drop caused by faults such as a bipolar lockout poses a severe threat to the safe and stable operation of the system. At this time, the frequency response capability of wind power is particularly important, preventing the frequency from dropping dramatically and rapidly and providing time for the recovery of the system. This will also be the focus of the author’s following work.