Abstract
As synchronous generators (SGs) are gradually displaced by renewable energy sources (RESs), the frequency stability of power systems deteriorates because RESs, represented by utility-scale solar and wind power sources, do not provide the inertial response, primary frequency response, secondary frequency response, and tertiary frequency regulation. As a result, the remaining SGs may not be sufficient to maintain the power balance and frequency stability. The concept and control strategies of virtual synchronous generators (VSGs) enable the inverter-based wind and solar power sources to emulate the outer characteristics of traditional SGs and participate in the active power and frequency control of power systems. This paper focuses on the automatic generation control (AGC) with virtual synchronous renewables (VSRs). First, the VSR strategy that enables the RESs to participate in AGC is introduced. Second, based on the interval representation of uncertainty, the output of RES is transformed into two portions, i.e., the dispatchable portion and the stochastic portion. In the dispatchable portion, the RESs can participate in AGC jointly with SGs. Accordingly, a security-constrained economic dispatch (SCED) model is built considering the RESs operating in VSR mode. Third, the solution strategy that employs the slack variables to acquire deterministic constraints is introduced. Finally, the proposed SCED model is solved based on the 6-bus and 39-bus systems. The results show that, compared with the maximum power point tracking (MPPT) mode, VSRs can participate in the active power and frequency control jointly with SGs, increase the maximum penetration level of RESs, and decrease the operating cost.
base Superscript for base generation
D Subscript for demand
DS Subscript for dispatable source
fore Superscript for forecasting value
G Subscript for synchronous generators (SGs)
head Superscript for regulation headroom
i, j, k, l, n Indices for dispatchable sources, renewable energy sources (RESs), demands, lines, and buses
MPPT Superscript for the maximum power point tracking (MPPT)
pene Subscript for penetration
percent Subscript for utilization percentage
R Subscript for RESs
ref Superscript for reference values
t Index for time slots
upward Superscript for upward reserves
VSR Subscript for dispatchable portion of RESs in VSR mode
The row of matrix
, and matrices with all elements equal to 1
Power deviation of stochastic portion of RES j from forecasting power
, Power deviation of RES j and demand k from forecasting power
, Power deviation bounds of RES j and demand k
, Actual and forecasting power of stochastic portion of RES j
Standard deviation of forecasting error of RES j
,, Fuel cost coefficients of generator i
NDS×NDS identity matrix
, , Relevance parameters of generator i, RES j, and demand k to bus n
Power transfer distribution factor of bus n to line l
,, Numbers of generators, RESs, buses, and dis- , patchable sources
, Actual and forecasting power of demand k at time t
Actual generation of dispatchable source i at time t
, The maximum and minimum outputs of dispatchable source i at time t
Actual generation of SG i at time t
, Power flow and transmission capacity of line l
, The maximum power point and power reference of RES j
, Actual and forecasting outputs of RES j at time t
, Downward reserves and the minimum output of VSR j
, Actual and base generation of VSR j at time t
Virtual upper limit for dispatchable portion of RES j at time t
, Upward reserves and regulation headroom of VSR j
Percentage of standard deviation relative to forecasting value of RES
, Upward and downward ramping limits of dispatchable source i
Length of reserve response time
T Number of time slots
Length of one schedule interval
, Cost coefficients of upward and downward reserves
Participation factor of dispatchable source i at time t
Base output of dispatchable source i at time t
Base output of SG i at time t
, Upward and downward reserves for RES and load provided by dispatchable source i at time t
, Upward and downward contingency reserves provided by dispatchable source i at time t
, Positive and negative slack row vectors for demand
, Positive and negative slack variables for demand k
, Positive and negative slack row vectors for RESs
, Positive and negative slack variables for RES j
DRIVEN by the energy crisis, sustainable development, and climate change, the renewable energy sources (RESs) represented by utility-scale wind power and solar energy have been widely utilized [
The high-proportion inverter-based generation will deteriorates the frequency stability of power system. In traditional power systems, the power supply is made up of tens of or hundreds of synchronous generators (SGs), which possess frequency response capabilities and can be dispatchable on a long-term time scale. SGs provide the inertial response (IR), primary frequency response (PFR), and secondary frequency response (SFR), restoring the power balance and nominal frequency [
It is noted that the fundamental discrepancy between RESs and SGs is the capability of providing frequency responses and the dispatchability in the domain of frequency stability. Therefore, the concept and control strategies of virtual synchronous generator (VSG) are prompted to incorporate asynchronous RESs into the frame of active power and frequency control [
On the formulation of the economic dispatch model, endogenous approaches are gaining popularity [
In the economic dispatch model considering AGC, the participation factors are employed to determine the allocation of generation adjustment once the variation of renewable generation is revealed [
As the inverter interfaces provide RESs with fast and accurate control capabilities, the optimal power flow is investigated based on practical control strategies of wind power [
In this paper, we focus on incorporating VSRs into active power and frequency control. Based on the concept of VSR and RO, a security-constrained economic dispatch (SCED) model is proposed to schedule the generation and reserves jointly. The contributions of this paper are as follows.
1) Based on the interval representation of uncertainties, the output of RES is transformed into a combination of a dispatchable portion with the virtual upper limit (VUL) and a stochastic portion with the original uncertainty. After the transformation, the RESs can be scheduled within the dispatchable portion. Also, there is no curtailment before the scheduling.
2) The control loops of determining the output reference of RESs in the VSR mode are designed. As the dispatchable portion provides guaranteed regulation capabilities between the VUL and the lower output limit, the base generation and upward reserves of VSRs are deterministic regarless of how the uncertainty of RES reveals. The permanent curtailment of RESs is analyzed, and the utilization and penetration of the generation capacity are derived.
3) Focusing on AGC, a new formulation of a robust SCED model is proposed, where the sources of power systems are divided according to their dispatchability. The dispatchable sources include the VSRs and SGs, while the undispatchable sources include the stochastic portion of the RESs. The case studies on 6-bus and 39-bus systems verify the correctness and effectiveness of the VSR mode compared with the MPPT mode.
The rest of this paper is organized as follows. Section II introduces the necessity of RESs participating in AGC and control strategy of VSR. Section III describes the formulation of the SCED model. Section IV introduces the solution strategy of handling nonlinear constraints. Section V discusses the case studies. Finally, Section VI concludes this paper.
In the normal operation, the power systems are continually subjected to the disturbances caused by the uncertainties from the generation and load sides. In traditional power systems, the active power control and frequency control are multiple temporal stages of power balance and frequency regulation. When a persistent power deficiency or surplus causes the frequency to deviate for more than a predetermined duration, the SFR activates and restores the frequency to the nominal value by regulating the output of SGs selected by system operators, also known as AGC [
The SGs and RESs vary greatly in the generation profiles and output characteristics. First, the outputs of wind and solar power are fluctuant and stochastic, primarily determined by instant meteorological conditions. The large-scale integration of RESs brings new uncertainty to the generation side and aggravates the worst imbalance scenario. Second, the wind and solar generation are designed to exploit the maximum available power. The RESs do not respond to frequency deviations, and the power electronic interfaces decouple the physical connection between the RESs and the power grid. Therefore, with the increasing penetration of RESs and the consequent displacement of traditional SGs, the frequency stability of synchronous power systems deteriorates. Focusing on the SFR stage, multiple SGs are selected to satisfy the required regulation capacity and speed of AGC from the technical and economic perspective, and the power flow limits of transmission lines are also considered. When the regulation capability of the selected SGs is depleted, other SGs activate. However, the dynamic SFR capability is limited by the generation limits and the ramp rate. For example, the maximum continuous rating of SGs is typically 2% per minute [
The wind and solar power sources are also classified into inverter-based generation due to their asynchronous characteristics requiring inverter interfaces [
In the RO, the output of RESs and load demand can be modeled by a nominal power and a power deviation limited by a given uncertainty set, which is expressed as:
(1) |
To eliminate the impact of different representations, we model the variable representing the output of RESs using the Gaussian probability density function [
(2) |
For RES j, we have and the upper and lower bounds of the uncertainty set can be defined as:
(3) |
The uncertainty set covers 99.74% quantile of the output of RES. In the same way, the uncertain parameter of the demand k can be modeled by a forecasting power with an uncertainty set and we have .
Based on the forecasting generation and uncertainty set, the output of RES can be transformed into dispatchable and stochastic portions.
The dispatchable portion is restricted by a VUL in (4), which indicates the maximum guaranteed output of RES regardless of how its uncertainty realizes.
(4) |
The stochastic portion is a Gaussian uncertainty variable, which inherits the original stochastic characteristics. The expected generation of is .
The synthesis of the two portions is given by:
(5) |
It is noted that the synthesis of the RES output is still stochastic. Based on the transformation, there is no curtailment before solving the SCED model, and the original uncertainty remains the same. The dispatchable and stochastic portions of the RES output is shown in

Fig. 1 Dispatchable and stochastic portions of RES output.
The stochastic portion can be treated as a stochastic source, of which the output is represented as:
(6) |
From
The renewable energy prediction provides available forecasting power in each time interval, and the MPPT strategy tracks the available power in real time, which is realized within the dispatchable portion.
In the MPPT or VSR mode with no reserved headroom, the output of RESs can be represented as the combination of the whole dispatchable portion and the stochastic portion as:
(7) |
In the dispatchable portion, if the upward reserves are scheduled, the headroom between the base generation and the VUL provides this regulation capability, which can be expressed as:
(8) |
It is noted that the upward reserves may not fully utilize the capability of the regulation headroom. The scheduled base generation of dispatchable sources satisfies the power balance between generation and demand, while the continuous and discrete uncertainties largely influence the scheduled reserves, so we have:
(9) |
(10) |
Based on (10), the power reference of RES, i.e., the reference of sub-optimal power point, is given by:
(11) |
The control loops of determining the power reference of RESs are accordingly designed and shown in

Fig. 2 Control loops of determining power reference of RESs.
From
(12) |
Similarly, the expected penetration level of RES is defined as:
(13) |
When taking RESs as undispatchable sources, the conventional formulation of the joint scheduling of generation and reserves has been well addressed such as the affinely adjustable robust optimal power flow in [
(14) |
where .
In (14), the stochastic portion of RESs is considered as separated stochastic sources. It is noted that VSRs have two merits: ① the generation cost is assumed to be zero as they do not consume fossil fuels [
The operating cost is comprised of generation and reserve costs. The generation cost can be represented as a quadratic cost function, while the reserve cost can be represented as a linear function. The optimization goal of the proposed SCED model is to minimize the overall operating cost:
(15) |
1) Constraint for forecasting power balance
(16) |
Constraint (16) states that the power balance should be satisfied before the realization of uncertainties.
2) Constraint for generation capacity limit
(17) |
Constraint (17) states that the output of the dispatchable source i should not exceed the upper and lower limits. It is noted that for SGs, the minimum and maximum limits are constant, while for VSRs, the maximum limits are the VULs and fluctuant in different time intervals.
3) Constraint for line capacity limit
(18) |
Constraint (18) states that the power flow of transmission lines should not exceed the bidirectional capacity limits.
4) Constraint for ramp limit of SGs
(19) |
Constraint (19) states that the variation of generator i is limited by the ramp rate over any two consecutive time slots.
5) Constraints for ramp limits of reserves
(20) |
(21) |
Constraints (20) and (21) state that the provision of upward and downward reserves is limited by the ramp rate within the reserve response time.
6) Constraints for generation capacity limit considering reserves
(22) |
(23) |
Constraints (22) and (23) state that the output of dispatchable source i is limited by the upper and lower limits considering both the base generation and the provision of upward and downward reserves, respectively.
7) Constraints for reserve requirement considering uncertainties of RES and demand
(24) |
(25) |
Constraints (24) and (25) state that the total scheduled upward and downward reserves dealing with the uncertainties of RES and demand should satisfy the requirements in the worst scenario.
8) Constraint for reserve requirement considering the criterion
(26) |
As the outage of two or more generators rarely happens, we only consider the criterion [
9) Constraints for participation factor
(27) |
(28) |
Constraints (27) and (28) state that the participation factors should be non-negative and sum to one.
In order to solve the SCED model, the constraints should retain the linearity [
(29) |
(30) |
The matrices , , , and are defined as:
(31) |
Therefore, (29) and (30) can be expressed as:
(32) |
The RO seeks to find the optimal solutions for all possible realization of uncertainties within prescribed intervals. For the row of (32), (33) is satisfied
(33) |
By making use of slack row vectors , can be expressed as:
(34) |
where the elements in the slack row vector satisfy , , , and ; and is the column vector comprising the maximum variations of the stochastic portion of RESs.
In the same way, can be expressed as:
(35) |
where the elements in the slack row vector satisfy , , , and ; and is the column vector comprising the maximum variations of the demand.
From (34) and (35), the linearity of the constraints is ensured. Then, the SCED model can be solved by mixed-integer programming.
To verify the validity of the proposed model, we carry out case analysis on 6-bus and 39-bus systems and perform the SCED model for 16 periods representing the solar fluctuations in one day. The generation and load are based on historical data of State Grid Corporation of China. Four typical days are selected to represent four seasons in one year. The conventional SCED model, in which the solar generation operates in the MPPT mode, is also solved as a comparison. The formulation of SCED model is programmed in MATLAB and solved using Gurobi, and the test environment is Intel i7 3.6 GHz with 16 GB of memory.
The parameters of generators and lines in the 6-bus system are shown in [

Fig. 3 Forecasting solar generation in scenario 1 and load demand in 6-bus system.
To verify the robustness and superiority of the proposed approach, the comparisons are carried out between the VSR mode and MPPT mode.
In the MPPT mode, the SCED model is infeasible when the forecasting solar generation increases to the level of scenario 4, whereas the VSR mode is feasible in all scenarios.
The scheduled generation and reserves in the VSR mode in the 6-bus system are shown in

Fig. 4 Scheduled generation and reserves in VSR mode in 6-bus system. (a) Scenario 1. (b) Scenario 2. (c) Scenario 3. (d) Scenario 4. (e) Scenario 5. (f) Scenario 6.
In scenarios 1-3, the expectation of solar generation equals the forecasting value. This is because the solar plant does not need to curtail or keep headroom for the upward reserve provision. Therefore, the MPPT and VSR modes are both feasible. In scenarios 4-6, the expectation of solar generation is less than the forecasting value because of the headroom, which indicates the solar generation needs to be deloaded and provide upward reserves for the continuous uncertainties. In scenarios 5 and 6, the upward reserves do not fully utilize the headroom because of the lower limits of other SGs. Also, the VSR provides downward reserves for RES and load in all the scenarios. From the results, it can be concluded that the solar plant in the VSR mode can actively participate in the power balance regulation.
VSRs can actively curtail in case of excessive generation in scenarios 4-6, and the participation factors in the 6-bus system are shown in

Fig. 5 Participation factors in VSR mode in 6-bus system. (a) Scenario 4. (b) Scenario 5. (c) Scenario 6.
The total operating costs of the MPPT and VSR modes in the 6-bus system are compared, as shown in

Fig. 6 Comparison of operating costs between MPPT and VSR modes in 6-bus system.
Considering four seasons in a whole year, the MPPT and VSR modes are both feasible in scenarios 1-3, and there is no curtailment or scheduled upward reserves in the VSR mode. Hence, the expectation of the utilization percentages of MPPT and VSR modes are equal. As the solar generation capability increases, only the VSR mode is feasible. The scheduled regulation headroom causes a permanent curtailment in spring and summer.
The utilization percentages of solar capacity in spring and summer in the 6-bus system are shown in

Fig. 7 Utilization percentages of solar capacity in different seasons in 6-bus system. (a) In spring. (b) In summer.
By summarizing the seasonal results, the annual curtailment and penetration levels of solar generation in the 6-bus system are shown in

Fig. 8 Annual curtailment and penetration levels of solar generation in VSR mode in 6-bus system.
The forecasting error is an important factor influencing the renewable uncertainty. The SCED model is solved considering various standard deviations, including 0%, 1%, 3%, and 5% of the forecasting value. The operating costs and utilization percentages considering various forecasting errors in different scenarios in the 6-bus system are shown in

Fig. 9 Operating cost and utilization percentage considering various forecasting errors in 6-bus system. (a) Operating cost. (b) Utilization percentage.
The parameters of generators and lines of the 39-bus system are shown in [

Fig. 10 Forecasting solar generation in scenario 1 and load demand in 39-bus system.
The scheduled generation and reserves in the VSR mode in the 39-bus system is shown in

Fig. 11 Scheduled generation and reserves in VSR mode in 39-bus system. (a) Scenario 1. (b) Scenario 2. (c) Scenario 3. (d) Scenario 4. (e) Scenario 5. (f) Scenario 6.
The participation factors of SGs (G1-G10) and VSRs(VSR1-VSR4) in scenarios 4-6 in the 39-bus system are shown in

Fig. 12 Participation factors in VSR mode in 39-bus system.
The total costs of the MPPT and VSR modes in the 39-bus system are compared, as shown in

Fig. 13 Comparison of operating cost between MPPT and VSR modes in 39-bus system.
Considering the operation in a whole year, the VSR mode only causes the curtailment in spring and summer. The utilization percentages of solar capactity in spring and summer in the 39-bus system are shown in

Fig. 14 Utilization percentages of capacity in different seasons in 39-bus system. (a) In spring. (b) In summer.
By summarizing the seasonal results, the annual curtailment and penetration levels of solar generation in the 39-bus system are shown in

Fig. 15 Annual curtailment and penetration levels of solar generation in VSR mode in 39-bus system.
Considering various forecasting errors identified by standard deviations, including 0%, 1%, 3%, and 5%, the operating cost and utilization percentages in different scenarios are shown in

Fig. 16 Operating cost and utilization percentage considering various forecasting errors in 39-bus system. (a) Operating cost. (b) Utilization percentage.
In conclusion, the VSR mode can increase the maximum penetration level of RESs, sustain security, and operation requirements, and decrease the operation cost.
To increase the AGC regulation resources in power systems with high penetration of inverter-interfaced RESs, an SCED model to realize the joint scheduling of generation and reserves is presented considering that RESs operate in VSR mode. The conclusions are summarizes as follows:
1) Based on the interval representation of renewable uncertainty, the dispatchable portion of the RES output is decomposed. The permanent curtailment is ascribed to the headroom, which can provide upward reserves.
2) According to the realization of uncertainties, VSRs can participate in AGC jointly with SGs by providing the upward and downward reserves.
3) Compared with the MPPT mode, the VSR mode can increase the maximum penetration level of RESs and decrease the operating cost.
4) From the annual data, the penetration level increases with the growing renewable capacity, but the curtailment also increases under the assumption of constant load profiles.
5) The operating cost and renewable utilization are influenced by the forecasting accuracy of the renewable generation, so the reduction of forecasting errors is thus necessary. In conclusion, the VSR mode provides a new operation mode for sustaining the power balance and frequency stability.
As energy storage systems provide the flexibility, the unit commitment is another important topic, and the day-ahead scheduling considering the start-up and shut-down of SGs and flexible control of storage systems is in the scope of our future work.
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