Abstract
The growing number of renewable energy replacing conventional generators results in a loss of the reserve for frequency control in power systems, while many industrial power grids often have excess energy supply due to abundant wind and solar energy resources. This paper proposes a secondary frequency control (SFC) strategy that allows industrial power grids to provide emergency high-voltage direct current (HVDC) power support (EDCPS) for emergency to a system requiring power support through a voltage source converter (VSC) HVDC link. An architecture including multiple model predictive control (MPC) controllers with periodic communication is designed to simultaneously obtain optimized EDCPS capacity and minimize adverse effects on the providing power support (PPS) system. Moreover, a model of a virtual power plant (VPP) containing aluminum smelter loads (ASLs) and a high penetration of wind power is established for the PPS system. The flexibility and controllability of the VPP are improved by the demand response of the ASLs. The uncertainty associated with wind power is considered by chance constraints. The effectiveness of the proposed strategy is verified by simulation results using the data of an actual industrial power grid in Inner Mongolia, China. The DC voltage of the VSCs and the DC in the potlines of the ASLs are also investigated in the simulation.
2022.
IN recent years, the growing number of renewable energy sources replacing conventional power plants has become a noticeable issue [
A viable option to connect a load center and the industrial power grid is using voltage source converter (VSC) based high-voltage direct current (HVDC) interconnections, which can flexibly provide fast emergency HVDC power support (EDCPS) for an interconnected AC power system because of its high controllability [
The applications of a VSC-HVDC system to provide EDCPS have been investigated in many research works, in which the control strategies of EDCPS can be mainly divided into two categories: additional local control strategies and coordinated control strategies. The characteristics of the two control strategies are analyzed in [
Coordinated control usually adopts a distributed control architecture. A coordinated strategy is presented in [
From the review above, there are some gaps in the existing research. Firstly, most research works focus on EDCPS for primary frequency control (PFC), but few have studied EDCPS during the secondary frequency control (SFC) period. The short timescale of PFC leads to a contradiction: additional local control is imprecise, but the delay of coordinated control is nonnegligible. Secondly, the differences in the characteristics of the PPS system and RPS system are often not highlighted in the literature. In fact, the participation from demand side can greatly improve the flexibility of the PPS system. Thirdly, less consideration is given to the adverse effects on the PPS system caused by EDCPS such as undesired frequency oscillation and poor frequency quality.
To address the gaps, three contributions are made.
1) An model predictive control (MPC)-based SFC strategy with EDCPS is proposed. MPC controllers are included in the PPS system and RPS system to implement precise coordinated control. The communication delays are negligible for longer SFC periods.
2) An actual industrial power grid in Inner Mongolia containing self-owned generators, wind farms, and aluminum smelter loads is selected as the PPS system. In the PPS system, a model of a virtual power plant (VPP) is built to obtain high controllability.
3) The adverse effects of EDCPS on the PPS system are reduced by predictive control. The power imbalance caused by EDCPS can be counteracted in advance according to a feed-forward signal. Meanwhile, the adverse effects caused by the uncertainty associated with wind power are reduced by a chance-constraint method.
The framework of the proposed SFC strategy is shown in

Fig. 1 Framework of proposed SFC strategy.
A distributed control architecture is adopted to provide supervision and necessary communication. There are two independent MPC controllers in the interconnected power system. The objective of the two controllers is to minimize ACE1 and ACE2. In each power system, MPC controller 1 or MPC controller 2 calculates the optimal control commands according to or and ACE1 or ACE2. To distinguish the severity of disturbances, the threshold of power support is defined as . If ACE1 is detected to exceed , the RPS system will be considered suffering heavy disturbances. The AGC units are involved in regulation. An increment will be added to the rated HVDC power to provide EDCPS. Meanwhile, the VPP in the PPS system will change the power output of the self-owned generators and the power consumption of the ASLs according to the received feed-forward signal to minimize the power imbalance in the PPS system. Otherwise , EDCPS will not be triggered, and the HVDC power set point remains the rated power.
The SFC model for the two-terminal HVDC system is shown in

Fig. 2 SFC model for two-terminal HVDC system.
Let the RPS system and PPS system be denoted by the subscripts and , respectively. Then, the dynamics of the
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
where fi, ∆fi, and fs are the frequency, frequency deviation, and standard value, respectively; Hi and Di are the synchronous machine inertia and machine damping coefficient, respectively; is the power imbalance; Ri is the speed drop; Pm,i and Pv,i are the mechanical outputs of the valve position; PmH,i and PmL,i are the mechanical outputs of the high- and low-pressure cylinders, respectively; Tg,i is the time constant of the governor; TCH,i and TRH,i are the time instances of the high- and low-pressure cylinders, respectively; and FHP,i and FLP,i are the proportionality coefficients of the high- and low-pressure cylinders, respectively.
The objectives of the RPS system and PPS system are to minimize and , respectively. Therefore, ACE1 and ACE2 are selected as:
(8) |
(9) |
A schematic diagram of the two-terminal VSC-HVDC link is shown in

Fig. 3 Schematic diagram of two-terminal VSC-HVDC link.
As widely reported in [
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
where Ps and Qs are the active power and reactive power injected into the AC system, respectively; us and uc are the voltages of the AC system and VSC, respectively; Udc is the DC voltage of the VSC; usd is the d-axis component of us; Rt and Lt are the resistance and inductance of the transformer, respectively; Rl and Ll are the resistance and inductance of the transmission line, respectively; Cv is the capacitance at the DC side; ud, id, uq, and iq are the d- and q-axis components of the voltage and current of the VSC, respectively; is the angle between uc and us; is the angular frequency of the AC system; and K is the utilization rate of the DC voltage, which is the ratio of the maximum amplitude of the fundamental wave of the AC voltage that the inverter can output to the input DC voltage (a coefficient related to the modulation method, K = 0.866 in this paper.
VSC-HVDC control consists of fast inner current control and slower outer control. The outer control supplies the reference values of the current id,ref and iq,ref. The inner control adjusts id and iq to follow the reference values. The outer control is based on PI control and has four optional control objectives for each VSC: Ps, Udc, Qs, and us.
Since the inner current control is much faster than the outer control, id and iq can be directly assumed to be equal to their reference values id,ref and iq,ref, respectively [
ASLs are typical flexible loads based on continuous power control by a self-saturable reactor. A dynamic model for an ASL with a self-saturable reactor has been detailed in our previous work [
(17) |
(18) |
(19) |
where Ibk, Ebk, Cbk, Lbk, and rbk are the current, power supply voltage, capacitance, inductance, and resistance of the buck converter in the ASL, respectively; , , and kc are the current, resistance, and control coefficient of the control winding in the ASL, respectively; , , , , and are the DC current, DC voltage, resistance, electromotive force, and time constant of the potline in the ASL, respectively; d is the duty cycle of the buck converter in ASL; and is the rated value of the DC voltage of the potline.
ASL control is based on the DC current control scheme, which aims to stabilize the DC at its reference value in smelter loads [
can be converted to by (20), which is based on the ASL model [
(20) |
In this paper, is assumed to be composed of the forecasted wind power and forecasting error . The forecasted wind output is determined by the wind farm parameters and wind speed v [
(21) |
(22) |
where Cp is the coefficient of performance of the rotor; is the mass density of air; and A is the swept area of the blades.
MPC controller 1 is designed to calculate the optimal control commands with the objective of minimizing ACE1 in the RPS system. To improve control performance, the grid constraints of the RPS system are included in the prediction model.
As mentioned in Section III-A, the aggregated single-unit model is expressed in (1)-(4). To consider the grid constraints, the aggregated model should be expanded to a model of multiple individual generators.
Let the subscript j denote the serial number of the
(23) |
where ; ; is the electrical power of the
(24) |
The state equation of all of the conventional generators in the system is:
(25) |
where ; ; ; ; ; ; and .
Considering the topological information of the power grid, the DC power flow is used for the power network equation:
(26) |
where B is the admittance matrix of the DC power grid; P is the active power in each node; and is the phase angle of the voltage in each node.
(27) |
where the subscripts e and l correspond to the parameters of the generator and load nodes, respectively.
The relationship between the phase angle of the voltage in the bus near the generator and the angle of the rotor of the generator is:
(28) |
where is the transient reactance of the unit; is the rotor of the generator; and is the bus near the generator.
Combining (27) and (28), the electrical power vector in the RPS system can be expressed as:
(29) |
(30) |
(31) |
Because EDCPS is included in the control scheme, the power injection of the VSC1 node is also a control variable. Then, (29) can be expressed as:
(32) |
where and are the coefficients of and , respectively.
Since is a state variable rather than a control variable, it can be replaced with the target value . Substituting (32) into (25), the state space model of the RPS system considering grid constraints and HVDC participation can be expressed as:
(33) |
where ; ;
; and .
Note that the grid constraints mentioned here include the DC power flow (27) and an equation for the angle of the rotor of the generator (28), where the two equations are combined to derive the power flow
Discretizing the state space model in (33) with the sample period Ts, we obtain the prediction model of the RPS system as:
(34) |
where ; and , , and are the system matrices derived from A1, B1, and in (33) after discretization with the sampling time .The notation is used throughout this paper. Since the state space model in (33) considers grid constraints, they are also included in the discretized prediction model, which means that the predicted state variables also satisfy the constraints.
(35) |
The optimization of the RPS system aims to minimize ACE1, which is determined by the frequency deviation:
(36) |
where is the diagonal matrix of standard value of rotor angular velocity.
The receding horizon optimization can be formulated as a quadratic programming (QP) problem:
(37) |
(38) |
(39) |
(40) |
The control variable is . The objective function in (37) consists of ACE1 and the penalty term for the incremental control variable, where and are the weight matrices. The constraint in (38) is an equation for the prediction model considering the grid constraints. The constraint in (39) expresses the amplitude constraints of the control variables, which consist of the lower and upper limits of the output of the generator and HVDC power modulation. The constraint in (40) is the ramping constraint of the output of the generator and HVDC power modulation.
This subsection introduces a method for determining the control step Tc1, prediction horizon Tp1, and weight matrices (, ) of MPC controller 1 in the RPS system, where the control step and prediction horizon are constant and the weight matrices are time-varying owing to feedback control deviations.
In the SFC process, the AGC period is usually set to be 4-8 s. As an AGC controller, the control step of the proposed MPC controller should also be within this range to match the remote terminal unit (RTU) in the AGC units. In this paper, the control step of the MPC controller in the RPS system is selected to be s.
Since the prediction model in (34) is obtained by the discretization of the state space model of the RPS system with the sample period Ts, the prediction is one-step, and the prediction horizon Tp1 is equal to the sample period Ts. In this study, the prediction horizon of the MPC controller in the RPS system is selected to be s.
In the objective function, the weight matrices and represent the priority of the elimination of ACE1 and the penalty on the control variables, respectively. In order to ensure transient performance, is much larger than . Under this premise, if is kept constant, the variation in can also affect the performance of the MPC controller. If becomes larger, there will be a greater penalty on the increment of the control variables, which leads to conservative control signals. On the contrary, a smaller leads to aggressive control signals.
Since and can affect control performance, they should be selected properly. In this paper, is kept constant, and varies with the feedback frequency deviations :
(41) |
where and are the standard value and droop coefficient of , respectively.
Compared with constant weight matrices, time-varying weight matrices make the control more flexible and reasonable. When the deviation is large, the control is more aggressive to obtain better transient performance. When the deviation becomes small, the control is more conservative to obtain high stability, which smoothens the frequency recovery curve and reduces the adverse effects on the VSC-HVDC system caused by dramatic changes.
The industrial power grid in the PPS system is integrated as a VPP model. MPC controller 2 is designed to calculate the optimal control commands with the objective of minimizing ACE2. The adverse effects caused by the urgency of EDCPS are reduced by including a feed-forward signal in the prediction model. The uncertainty associated with wind power is considered by a chance-constraint method [
According to the derivation in Section IV-A, the electrical power vector in the PPS system can be expressed as:
(42) |
where , , , and are the coefficients of , , , and , respectively.
To minimize the power imbalance in the PPS system caused by EDCPS, the EDCPS command calculated in the RPS system will be sent to the PPS system in advance as a feed-forward signal. Then, the power injection Ps,2 can be replaced by the EDCPS command signal . In this way, the EDCPS command is included in the prediction model as a feed-forward signal.
According to (33), the state space model of the PPS system can be expressed as:
(43) |
where
; and .
The control variable is the power set point of the generators and ASLs. , , and are included in the matrix D2.
Discretizing the state space model in (43) with the sample period Ts, we can obtain the prediction model of the PPS system as:
(44) |
where ; and .
(45) |
In the PPS system, self-owned generators, wind farms, ASLs, and normal loads can be regarded as a VPP with Pvpp. ACE2 is decided by the difference between Pvpp and Ps,2. Here, Ps,2 is replaced with to include the feed-forward signal in ACE2, which can be reformulated as:
(46) |
As mentioned in Section III, Pwg consists of and rwg, where . The prediction model should be modified to include rwg.
Substituting (21) into (41) and (42), the prediction model involving the random variable rwg can be reformulated as:
(47) |
(48) |
where and .
Including , ACE2 can be formulated as:
(49) |
The receding horizon optimization in the PPS system can be formulated as a QP problem with a random variable:
(50) |
(51) |
(52) |
(53) |
(54) |
(55) |
The control variable is . The objective function in (50) consists of ACE2 and the penalty term for the incremental control variable. The constraint in (51) is an equation for the prediction model considering the grid constraints. The constraint in (52) is the power imbalance equation of the PPS system. The constraint in (53) expresses the amplitude constraints of the control variables, which consist of the lower and upper limits of the output of the generator and the power consumed by the ASL. The constraint in (54) is the ramping constraint of the output of the generator and the ASL power regulation. The constraint in (55) represents the lower and upper bounds of the frequency considering , where and are the upper and lower bounds, respectively.
ewg is included in the objective function in (50) and the constraint in (55). Therefore, (50)-(55) are a stochastic optimization problem and is not tractable in its current form. To convert this stochastic optimization problem into a deterministic optimization problem, a formulation for the chance constraints is adopted. Since ewg obeys , ACE2 in (50) can be reformulated using the mathematical expectation:
(56) |
where is the mathematical expectation of ; and is the variance of .
(57) |
According to probability theory, the frequency prediction in (57) also obeys a normal distribution
The probabilistic constraint in (47) needs to be reformulated into a deterministic constraint. This can be done by interpreting it as a tightened version of the original constraint, where tightening represents a security margin against uncertainty:
(58) |
where is the uncertainty margin.
Then, (50)-(55) can be reformulated as a deterministic QP problem:
(59) |
This subsection introduces a method for determining the control step Tc2, prediction horizon Tp2, and and of MPC controller 2 in the PPS system.
Similar to the MPC controller in the RPS system, the control step and prediction horizon of the MPC controller in the PPS system are also selected to be s and s, respectively.
Correspondingly, is kept constant, and is time-varying, which is determined by the feedback power deviation :
(60) |
where and are the standard value and droop coefficient of , respectively.
Flowcharts of the proposed control schemes of the RPS system and PPS system are shown in

Fig. 4 Flowchart of proposed control scheme.
The control of the two systems operates independently. The end of control in one system does not affect the continuation of the control in the other system. If is below , the control of the RPS system will be stopped, but the control of the PPS system can continue if . Accordingly, if is below , the control of the PPS system will be stopped, and the control of the RPS system can continue if . When both and are controlled to be lower than , the whole algorithm stops.
In this section, simulation results are presented to demonstrate the effectiveness of the proposed MPC-based SFC strategy. The two-terminal HVDC power system shown in

Fig. 5 Schematic diagram of test power system.
The RPS system is modified from the New England 39-bus power system, where the generator in bus R39 is replaced by HVDC power injection from the PPS system. The total load power of the PPS system is 6196 MW, and detailed data of the New England 39-bus power system can be found in [
Description | Value | Description | Value |
---|---|---|---|
AC voltage of VSC1 | 220 kV | Line impedance |
0.1+j4×1 |
AC voltage of VSC2 | 220 kV | Capacitor | 1000 μF |
DC voltage level | 400 kV | Rated power | 800 MW |
Number | Ud (kV) | Id (kA) | Ed (V) | Rd (mΩ) |
---|---|---|---|---|
ASL1 | 2.26 | 160.000 | 864 | 8.10 |
ASL2 | 1.54 | 624.000 | 576 | 2.88 |
ASL3 | 1.60 | 1.248 | 600 | 2.25 |

Fig. 6 Wind speed and total forecasted wind power of WF1 and WF2.
Three cases are studied. In Case 1, the RPS system experiences a small disturbance with EDCPS, which is not triggered to demonstrate the effectiveness of the proposed MPC controller. In Case 2, two scenarios in which the RPS system experiences a large disturbance (power shortage and power surplus) are considered to verify the effectiveness of EDCPS and ASL participation. In Case 3, the internal characteristics of the VSC-HVDC system and ASLs are presented.
This case is used to test the performance of the proposed MPC controllers with time-varying weight matrices. For comparison, traditional PI control, fuzzy PI control, and the fixed-weight MPC method are also evaluated in the same simulation environment. For the proposed varying-weight MPC method, and are diagonal matrixes consisting of 0.8. The standard values and are diagonal matrixes consisting of 0.2. For the fixed-weight MPC method, the weight matrixes are equal to the standard values of the corresponding parameters in the proposed MPC method. The parameters of traditional PI control are tuned by trial and error.
Firstly, the load power in bus R8 increases by 100 MW (about 1.6% of the total load power) at 100 s to create a power shortage in the RPS system. Then, the frequency of the RPS system drops because of the power shortage. The primary frequency control and secondary frequency control of the RPS system are triggered in succession. Since ACE1 is small, EDCPS is not triggered, and only AGC units participate in frequency control.
The frequency curves of RPS system under different control methods are shown in

Fig. 7 Frequency curves of RPS system under different control methods.
In the PPS system, since EDCPS is not triggered, the power transmitted between the RPS system and the PPS system remains constant. The main disturbance in the PPS system comes from the fluctuation in the wind power. In the control process, ASLs maintain normal production, and only generators participate in VPP power control. As shown in

Fig. 8 Performance analysis. (a) VPP control performance with different control methods. (b) Frequency fluctuations in PPS system under different control methods.
Frequency fluctuations in PPS system under different methods are shown in
Method | VPP deviation (MW) | Frequency deviation (Hz) |
---|---|---|
Traditional PI control | 218.2485 | 0.1196 |
Fuzzy PI control | 128.4429 | 0.0751 |
Fixed-weight MPC | 101.5253 | 0.0592 |
Proposed MPC | 94.0372 | 0.0564 |
This case is used to verify the effectiveness of EDCPS and ASL participation in SFC when there is a large power shortage or power surplus in the RPS system. For comparison, the no-EDCPS scenario in the RPS system and the no-ASL scenario in the PPS system are evaluated in the same simulation environment. Fuzzy PI control is shown to have better performance than traditional PI control, so fuzzy PI control is used as a comparison for the proposed proposed MPC method.
In the power shortage scenario, a fault is set on bus R35 at 100 s, which causes a 650 MW power shortage (about 10.5% of the total load power) in the RPS system. In the power surplus scenario, the load power in bus R8 decreases by 500 MW (about 8% of the total load power) at 100 s. As a result, the frequency of the RPS system drops or rises sharply, and EDCPS is triggered. The HVDC transmission power rapidly increases or decreases to provide frequency control support.
The frequency control performance in RPS system with different control methods are shown in

Fig. 9 Frequency control performance in RPS system with different control methods. (a) Power shortage scenario. (b) Power surplus scenario.
In the PPS system, the target value of the VPP varies with the feed-forward signal, as shown in

Fig. 10 VPP control performance with different control methods. (a) Power shortage scenario. (b) Power surplus scenario.
Scenario | VPP deviation (MW) | Frequency deviation (Hz) | ||
---|---|---|---|---|
ASL | No-ASL | ASL | No-ASL | |
Power shortage (proposed MPC) | 94.3811 | 196.2015 | 0.0643 | 0.1280 |
Power shortage (fuzzy PI) | 292.5823 | 358.2382 | 0.1677 | 0.2074 |
Power surplus (proposed MPC) | 99.7494 | 118.0885 | 0.0608 | 0.0716 |
Power surplus (fuzzy PI) | 133.2693 | 172.7302 | 0.0853 | 0.1010 |

Fig. 11 Frequency fluctuations in PPS system with different control methods. (a) Power shortage scenario. (b) Power surplus scenario.
In this case, the effects of the proposed SFC strategy on the VSC-HVDC system and ASLs are discussed. The internal characteristics of the VSC-HVDC system and ASLs with the proposed MPC method in the power shortage and surplus scenarios mentioned in Case 2 are presented.
The DC voltage of the VSC is an important index to evaluate the safe operation of the HVDC system. The variation of HVDC transmission power is shown in

Fig. 12 Variation of HVDC transmission power. (a) Power shortage scenario. (b) Power surplus scenario.

Fig. 13 DC voltages of VSC1 and VSC2. (a) Power shortage scenario. (b) Power surplus scenario.
As mentioned in [
The DC variations of ASL1 to ASL3 are shown in

Fig. 14 DC variations of ASL1 to ASL3. (a) Power shortage scenario. (b) Power surplus scenario.
We propose an SFC strategy involving EDCPS from an industrial power grid through a VSC-HVDC link. In the proposed strategy, two MPC controllers are set in the RPS system and PPS system with a distributed architecture. Time-varying weight matrices are adopted in the two MPC controllers to improve the transient performance. MPC controller 1 in the RPS system can shorten the frequency recovery time by obtaining the optimized EDCPS value. MPC controller 2 in the PPS system can reduce the adverse effects of EDCPS using a feed-forward signal in its prediction model. The industrial power grid serving as the PPS system is integrated as a VPP to obtain intuitive external characteristics. Simulations are performed using the data of an actual industrial power grid in Inner Mongolia, China. The following conclusions are drawn.
1) The EDCPS can alleviate the problem of a reserve shortage in the RPS system by sharing the reserves between the RPS system and the PPS system. Simulation results show that the participation of EDCPS can shorten the frequency recovery time of the RPS system in the large disturbance scenario.
2) The demand response of the ASLs improves the flexibility and controllability of the VPP model. Simulation results show that the control deviation of the power output of the VPP in the ASL scenario is much lower than that in the no-ASL scenario. Correspondingly, the undesired frequency oscillation caused by EDCPS is also reduced for the participation of the ASLs.
3) The proposed MPC controllers can improve the control performance of the RPS system and PPS system. A comparison with traditional PI control, fuzzy PI control, and fixed-weight MPC shows that the proposed MPC method with time-varying weight matrices has the best performance (the shortest frequency recovery time in the RPS system and the lowest frequency deviations in the PPS system).
4) The regulation of the VSC-HVDC and ASLs using the proposed control strategy will not cause a security problem. The DC deviations of the VSCs are small. The DCs in the potlines of the ASLs are also kept above the security margins.
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