Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Secondary Frequency Control Considering Optimized Power Support From Virtual Power Plant Containing Aluminum Smelter Loads Through VSC-HVDC Link  PDF

  • Peng Bao
  • Wen Zhang
  • Yuxi Zhang
the Key Laboratory of Power System Intelligent Dispatch and Control of the Ministry of Education, Shandong University, Jinan 250061, China

Updated:2023-01-25

DOI:10.35833/MPCE.2021.000072

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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.

I. Introduction

IN recent years, the growing number of renewable energy sources replacing conventional power plants has become a noticeable issue [

1]. The increasing concerns involve the loss of reserve for frequency control and the reduction in regulation capacity in power systems as conventional power plants are displaced. As a result, the ability of power system to counteract disturbances decreases, especially in load centers without sufficient regulation resources [2], [3]. Besides, there are many industrial power grids with sufficient energy supplies and abundant regulation resources [4]. A typical industrial power grid usually consists of self-owned generators, renewable energy sources, energy-intensive industrial loads, and some normal loads [5], [6]. Renewable energy sources can provide adequate or even surplus energy supplies, and self-owned generators have the same regulation capacity as the conventional units in power systems. Moreover, some industrial loads can be ideal flexible loads owing to their high controllability and energy-intensive characteristics [7]. If the industrial power grid could be included in the interconnected power system, it would have great potential to provide power support to the load centers lacking regulation capacity.

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 [

8]-[10]. The fast and accurate EDCPS can serve as the reserve for frequency control to counteract disturbances in the system requiring power support (RPS) system, which has the same function as conventional power plants.

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 [

11]. Additional local control usually adopts proportional control [12], [13], derivative control [14], [15], inertia emulation control [16], or a virtual synchronous generator (VSG) method [17]-[19]. As mentioned in [11], proportional-only droop control is easy to implement, but its performance is often inferior to control methods with an integral part. In [12], two droop control schemes (Udc-f and P-f) are compared, where the P-f scheme adopts proportional-integral (PI) control, and the Udc-f scheme adopts proportional-only droop control. Results show that the P-f scheme has better performance than the Udc-f scheme. Reference [13] investigates an industrial load supplied by a VSC-HVDC link, where PI control effectively improves the power quality supplied to industrial plants. However, the role of derivative control is not considered in these references. Reference [14] adopts synthetic inertia control, where the control signal of a derivative controller is calculated according to the derivative of the frequency, i.e., the rate of change of the frequency (ROCOF). In [15], synthetic control is divided into two types: continuous control and one-shot control. The advantage of continuous control is its high adaptiveness. However, it has higher requirements for filtering. In contrast, one-shot control is simple to implement but has low adaptiveness. In [16], a novel scheme combining inertia emulation control and synthetic inertia control is proposed, which aims to blend the energy stored in the HVDC link with the power control capabilities of wind turbines. This ensures a fast frequency response with lower requirements for capacitance volume and wind turbine performance. The VSG is emerging as an alternative approach for controlling the VSCs operating in the power system. In [19], the idea of operating an inverter to mimic a synchronous generator is proposed, where an inverter operating in this way is defined as a synchronverter. According to the literature, the main difference between the synchronverter and the VSG is that the synchronverter lacks short-term energy storage. The advantages of additional local control are lower control complexity and shorter communication delays, which ensure a fast response. However, the disadvantages are also obvious. The lack of supervision and communication might cause disproportional power support, leading to undesired frequency oscillation in the providing power support (PPS) system [20].

Coordinated control usually adopts a distributed control architecture. A coordinated strategy is presented in [

21], which allows all generators in different networks connected by an HVDC link to respond to frequency deviations in the RPS system without a centralized controller. The use of two droop operating modes is described in [22] and ensures the optimal operation of multiple terminals based on network characteristics. Note that these research works assume that there is no communication delay. However, [23] shows that the communication delay can obviously worsen the control performance.

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.

II. Framework of Proposed Control Strategy

The framework of the proposed SFC strategy is shown in Fig. 1. The two-terminal HVDC power system consists of an RPS system and a PPS system connected by a VSC-HVDC link. The RPS system consists of automatic generation control (AGC) for conventional power plants, non-AGC units and normal loads which are based on a regional load center with an insufficient reserve for frequency control. The PPS system consists of self-owned generators, wind farms, aluminum smelter loads (ASLs), and normal loads which are based on an actual industrial power grid in Inner Mongolia, China [

24]. The components in the industrial power grid are integrated as a VPP model. In Fig. 1, Ps,1 and Ps,2 are the power injections of VSCs; Pdc is the DC power transmitted by the HVDC network; x1 and x2 are the state variables of RPS and PPS systems, respectively; ACE1 and ACE2 are the area control errors (ACEs); and Pset,iG, Psetasl, and Psethvdc are the set points of generator power output, ASL power consumption, and HVDC transmission power, respectively.

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 x1 or x2 and ACE1 or ACE2. To distinguish the severity of disturbances, the threshold of power support is defined as ACEtrig. If ACE1 is detected to exceed ACEtrig, 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 ACE1<ACEtrig, EDCPS will not be triggered, and the HVDC power set point remains the rated power.

III. Model Formulation

A. SFC Model

The SFC model for the two-terminal HVDC system is shown in Fig. 2, where Pm,1 is the total mechanical output of AGC and non-AGC units in RPS system; Pm,2 is the mechanical output of self-owned generator in PPS system; Pnl,1 and Pnl,2 are the normal load consumptions in RPS and PPS systems, respectively; Pwg is the wind power output; K1(s), K2(s), M1(s), M2(s), and MASL(s) are the transfer functions of MPC controller, governor-turbine model, and ASL model, respectively; Pvpp is the power output of VPP; and Pasl is the power consumption of ASL. Note that some of the variables are defined when they appear in the following equations. Power injection in the RPS system consists Pm,1, Pnl,1, and Ps,1. Power injection in the PPS system consists of Pm,2, Pwg, Pasl, Pnl,2, and Ps,2.

Fig. 2  SFC model for two-terminal HVDC system.

Let the RPS system and PPS system be denoted by the subscripts i=1 and i=2, respectively. Then, the dynamics of the ith system can be expressed as:

f˙i=12HiΔPi-Di2Hifi (1)
P˙mH,i=Pv,i-PmH,iTCH,i (2)
P˙mL,i=PmH,i-PmL,iTRH,i (3)
P˙v,i=Pset,iG-ΔfiRi-Pv,iTg,i (4)
ΔP1=Pm,1+Ps,1-Pnl,1 (5)
ΔP2=Pm,2+Pwg-Pasl-Pnl,2-Ps,2=Pvpp-Ps,2 (6)
Pm,i=FHP,iPmH,i+FLP,iPmL,i (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; ΔPi 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 Δf1 and ΔP2, respectively. Therefore, ACE1 and ACE2 are selected as:

ACE1=Δf12=fs-f12 (8)
ACE2=ΔP22=Pvpp-Ps,22 (9)

B. VSC-HVDC Model

A schematic diagram of the two-terminal VSC-HVDC link is shown in Fig. 3.

Fig. 3  Schematic diagram of two-terminal VSC-HVDC link.

As widely reported in [

25] and [26], the dynamics of a VSC-HVDC system in the dq rotating frame can be expressed as follows, where the Park transformation is amplitude invariant.

diddt=-Rtid-ωLtiq+usd-udLt (10)
diqdt=-Rtiq+ωLtid-uqLt (11)
dUdc1dt=3K1id1cos δ1+iq1sin δ12Cv-idc (12)
dUdc2dt=3K2id2cos δ2+iq2sin δ22Cv+idc (13)
didcdt=Udc1-Udc2-RlidcLl (14)
Ps=1.5usdid (15)
Qs=1.5usdiq (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; δ is the angle between uc and us; ω 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 [

27]. Under this assumption, the complex control of the VSCs can be simplified to PI control, which reduces the order of the control system and changes it into a linear control system. This can greatly improve the simulation efficiency and will not affect the simulation accuracy because the period to complete an inner control process is much shorter than a simulation step in this paper.

C. ASL Model

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 [

7]. Therefore, the dynamics of an ASL can be characterized by the state space model in [7]:

I˙bkI˙caslI˙dasl=-rbkLbk-RcLbk01RcaslCbk-1RcaslCbk00-kcτdRdasl-1τdIbkIcaslIdasl+EbkLbk00d+EbkLbk0Ud0asl-EdaslτdRdasl (17)
Udasl=Ud0asl-kcIcasl (18)
Pasl=UdaslIdasl (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; Icasl, Rcasl, and kc are the current, resistance, and control coefficient of the control winding in the ASL, respectively; Idasl, Udasl, Rdasl, Edasl, and τd 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 Ud0asl 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 Idasl at its reference value Irefasl in smelter loads [

28], [29].

Psetasl can be converted to Irefasl by (20), which is based on the ASL model [

30]. Then, Idasl can be controlled to Irefasl through the DC control scheme. Consequently, Pasl will also be controlled to Psetasl.

Irefasl=Psetasl2+4RdaslPsetasl-Edasl2Rdasl (20)

D. Wind Farm Model

In this paper, Pwg is assumed to be composed of the forecasted wind power Pwgfor and forecasting error ewg. The forecasted wind output is determined by the wind farm parameters and wind speed v [

31]. The forecasting error obeys the standard normal distribution N(0, σ2) [32].

Pwg=Pwgfor+ewg (21)
Pwgfor=12CpρAv3 (22)

where Cp is the coefficient of performance of the rotor; ρ is the mass density of air; and A is the swept area of the blades.

IV. MPC-based Control Schemes

A. MPC Controller in RPS System

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.

1) State Space Model of Generators

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 jth individual generator. The state equation of the jth generator can be expressed as:

x˙Gj=-1/Tg,j0001/Tg,jRj1/TCH,j-1/TCH,j00001/TRH,j-1/TRH,j000000ωs0FHP,j/2HjFLP,j/2Hj0-Dj/2HjxGj+1/Tg,j0000uGj+0000-1/2HjPej+-ωs/Tg,jRj00-ωs0 (23)

where xGj=[Pvj,PmHj,PmLj,δj,ωj]T; uGj=Pset,jG; Pej is the electrical power of the jth generator; and ωs is the base value of the angular speed. The other parameters have the same meaning as those in the aggregated model.

Equation (23) can be written in condensed form as:

x˙Gj=AGjxGj+BGjuGj+CGjPej+DGj (24)

The state equation of all of the conventional generators in the system is:

x˙G=AGxG+BGuG+CGPe+DG (25)

where xG=[xG1T,xG2T,,xGnT]; uG=[uG1T,uG2T,,uGnT]; Pe=[Pe1,Pe2,,Pen]; AG=diag(AGj); BG=diag(BGj); CG=diag(CGj); and DG=diag(DGj).

2) Prediction Model Considering Grid Constraints

Considering the topological information of the power grid, the DC power flow is used for the power network equation:

P=Bθ (26)

where B is the admittance matrix of the DC power grid; P is the active power in each node; and θ is the phase angle of the voltage in each node.

Equation (26) can be expanded as:

PePl=BeeBelBleBllθeθl (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:

Pe=δ-θeXd' (28)

where Xd' is the transient reactance of the unit; δ is the rotor of the generator; and θe is the bus near the generator.

Combining (27) and (28), the electrical power vector Pe1 in the RPS system can be expressed as:

Pe1=F1δ1+G1Pl1 (29)
F1=I1+Bee,1-Bel,1Bll,1-1Ble,1Xd1'-1Bee,1-Bel,1Bll,1-1Ble,1 (30)
G1=I1+Bee,1-Bel,1Bll,1-1Ble,1Xd1'-1Bel,1Bll,1-1 (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:

Pe1=F1δ1+G1Pl,1=F1δ1+[Ghvdc,1Gnl,1][Ps,1Pnl,1]T (32)

where Ghvdc,1 and Gnl,1 are the coefficients of Ps,1 and Pnl,1, respectively.

Since Ps,1 is a state variable rather than a control variable, it can be replaced with the target value Psethvdc. Substituting (32) into (25), the state space model of the RPS system considering grid constraints and HVDC participation can be expressed as:

x˙1=A1x1+B1u1+D1 (33)

where x1=[Pv,1PmH,1PmL,1δ1ω1]T; u1=[Pset,1GPsethvdc]T;

A1=-1/Tg,10001/Tg,1R11/TCH,1-1/TCH,100001/TRH,1-1/TRH,1000000ωs0FHP,1/2H1FLP,1/2H1-F1/2H1-D1/2H1;

B1=1/Tg,100000000-Ghvdc,1/2H1; and D1=-ωs/Tg,1R100-ωs-Gnl,1Pnl,1/2H1.

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 equation (29). The power flow equation includes the grid constraints owing to the presence of the admittance matrix B1 in the coefficient matrices F1 and G1. Then, because F1 and G1 are implicit in the matrices A1 and B1 in (33) via the power flow equation, the grid constraints are also included in the state space model.

Discretizing the state space model in (33) with the sample period Ts, we obtain the prediction model of the RPS system as:

x1[k+1]=Ad1x1[k]+Bd1u1[k]+Dd1 (34)

where Ad1=eA1Ts;  Bd1=0TseA1TsdtB1;  Dd1=0TseA1TsdtD1; and Ad1, Bd1, and Dd1 are the system matrices derived from A1, B1, and D1 in (33) after discretization with the sampling time Ts.The notation f[k]=f(kTs) 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.

Equation (34) is the prediction model of all state variables for the generators. Accordingly, a prediction of the frequency ω1 (f=ω in p.u.) can be obtained with the output matrix Cd1=[0,0,0,0,1]:

y1[k+1]=Cd1x1[k+1]=Cd1Ad1x1[k]+Cd1Bd1u1[k]+Cd1Dd1 (35)

3) Receding Horizon Optimization

The optimization of the RPS system aims to minimize ACE1, which is determined by the frequency deviation:

ACE1=ωs-y1[k+1]Tωs-y1[k+1] (36)

where ωs is the diagonal matrix of standard value of rotor angular velocity.

The receding horizon optimization can be formulated as a quadratic programming (QP) problem:

min J1=ωs-y1[k+1]TQ11ωs-y1[k+1]+u1[k]-u1[k-1]TQ12u1[k]-u1[k-1] (37)
s.t.               y1[k+1]=C1Ad1x1[k]+C1Bd1u1[k]+C1Dd1 (38)
u1,minu1[k]u1,max (39)
u1[k]-u1[k-1]Δu1,max (40)

The control variable is u1=[Pset,1G,Psethvdc]T. The objective function in (37) consists of ACE1 and the penalty term for the incremental control variable, where Q11 and Q12 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.

4) Determination of MPC Parameters

This subsection introduces a method for determining the control step Tc1, prediction horizon Tp1, and weight matrices (Q11, Q12) 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 Tc1=5 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 Tp1=10 s.

In the objective function, the weight matrices Q11 and Q12 represent the priority of the elimination of ACE1 and the penalty on the control variables, respectively. In order to ensure transient performance, Q11 is much larger than Q12. Under this premise, if Q11 is kept constant, the variation in Q12 can also affect the performance of the MPC controller. If Q12 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 Q12 leads to aggressive control signals.

Since Q11 and Q12 can affect control performance, they should be selected properly. In this paper, Q11 is kept constant, and Q12 varies with the feedback frequency deviations Δf1:

Q12=Q12s-Kq1Δf1 (41)

where Q12s and Kq1 are the standard value and droop coefficient of Q12, 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.

B. MPC Controller in PPS System

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 [

33], [34].

1) Prediction Model Including a Feed-forward Signal

According to the derivation in Section IV-A, the electrical power vector Pe2 in the PPS system can be expressed as:

Pe2=F2δ2+G2Pl,2=F2δ2+[GaslGhvdc,2GwgGnl,2][PaslPs,2PwgPnl,2]T (42)

where Gasl, Ghvdc,2, Gwg, and Gnl,2 are the coefficients of Pasl, Ps,2, Pwg, and Pnl,2, respectively.

To minimize the power imbalance in the PPS system caused by EDCPS, the EDCPS command Psethvdc 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 Psethvdc. 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:

x˙2=A2x2+B2u2+D2 (43)

where x2=[Pv,2PmH,2PmL,2δ2ω2]T;  u2=[Pset,2GPsetasl]T;

A2=      -1/Tg,20001/Tg,2R1/TCH,2-1/TCH,200001/TRH,2-1/TRH,2000000ωs0FHP,2/2H2FIP,2/2H2-F2/2H2-D2/2H2;

D2=-ωs/Tg,2R200-ωs-Ghvdc,2Psethvdc-GwgPwgfor-Gnl,2Pnl,2/2H2; and B2=1/Tg,200000000-Gasl/2H2.

The control variable u2=[Pset,2G,Psetasl]T is the power set point of the generators and ASLs. Psethvdc, Pwgfor, and Pnl,2 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:

x2[k+1]=Ad2x2[k]+Bd2u2[k]+Dd2 (44)

where Ad2=eA2Ts;  Bd2=0TseA2TsdtB2; and Dd2=0TseA2TsdtD2.

Equation (44) is the prediction model of all state variables of the generators. The prediction of the frequency f2 can be obtained by the output matrix Cd2=[0,0,0,0,1]:

y2[k+1]=Cd2x2[k+1]=Cd2Ad2x2[k]+Cd2Bd2u2[k]+Cd2Dd2 (45)

2) Receding Horizon Optimization Including a Chance Constraint

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 Psethvdc to include the feed-forward signal in ACE2, which can be reformulated as:

ACE2=Pvpp-Psethvdc2=Pwg+Pm,2-Pasl-Pl,2-Psethvdc2 (46)

As mentioned in Section III, Pwg consists of Pwgfor and rwg, where rwgN(0,  σ2). 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:

x˜2[k+1]=x2[k+1]+RNdewg (47)
y˜2[k+1]=y2[k+1]+Cd2RNdewg (48)

where RNd=eA2TsdtRN and RN=0,0,0,0,-Gwg/2H2T.

Including ewg, ACE2 can be formulated as:

ACE2=Pwgfor+Pm,2-Pasl-Pl,2-Psethvdc-ewg2=ΔP2for-ewg2 (49)

The receding horizon optimization in the PPS system can be formulated as a QP problem with a random variable:

min  J2=Q21ΔP2for-ewg2+u2[k]-u2[k-1]TQ22u2[k]-u2[k-1] (50)
s.t.               y2[k+1]=Cd2Ad2x2[k]+Cd2Bd2u2[k]+Cd2Dd2 (51)
ΔP2for=Psethvdc-Pwgfor+Pm,2-Pasl-Pnl,2 (52)
u2,minu2[k]u2,max (53)
u2[k]-u2[k-1]Δu2,max (54)
fminy2[k+1]+Cd2RNdewgfmax (55)

The control variable is u2=[Pset,2G,Psetasl]T. 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 ewg, where fmax and fmin 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 N(0, σ2), ACE2 in (50) can be reformulated using the mathematical expectation:

ACE2=EΔP2for-ewg2=EΔP2for2-2EΔP2forEewg+Eewg2=ΔP2for2-2ΔP2forEewg+E2ewg+Dewg=ΔP2for2-2ΔP2for0+02+Dewg=ΔP2for2+σ2 (56)

where E(·) is the mathematical expectation of (·); and D(·) is the variance of (·).

Equation (55) can be expressed as a probabilistic constraint with a predescribed probability β:

Prfminy2[k+1]+Cd2RNdewgfmaxβ (57)

According to probability theory, the frequency prediction y2[k+1]+Cd2RNdewg in (57) also obeys a normal distribution N(f0, (Cd2RNd)T(Cd2RNd)σ2).

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:

fmin+Ωβy2[k+1]fmax-Ωβ (58)

where Ωβ is the uncertainty margin.

Then, (50)-(55) can be reformulated as a deterministic QP problem:

min J2=Q21ΔP2for2+σ2+u2[k]-u2[k-1]TQ22u2[k]-u2[k-1]s.t.y2[k+1]=Cd2Ad2x2[k]+Cd2Bd2u2[k]+Cd2Dd2ΔP=Psethvdc-Pwgfor+Pm,2-Pasl-Pnl,2 u2,minu2[k]u2,max u2[k]-u2[k-1]Δu2,max fmin+Ωβy2[k+1]fmax-Ωβ (59)

3) Determination of MPC Parameters

This subsection introduces a method for determining the control step Tc2, prediction horizon Tp2, and Q21 and Q22 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 Tc2=5 s and Tp2=10 s, respectively.

Correspondingly, Q21 is kept constant, and Q22 is time-varying, which is determined by the feedback power deviation ΔP2for:

Q22=Q22s-Kq2ΔP2for (60)

where Q22s and Kq2 are the standard value and droop coefficient of Q12, respectively.

C. Flowchart of Proposed Control Schemes

Flowcharts of the proposed control schemes of the RPS system and PPS system are shown in Fig. 4. In the two systems, the MPC controllers collect ACE signals in real time for SFC. In small disturbance scenarios, the control variable of the RPS system is the set point of the AGC units, and it is the set point of the self-owned generators in the PPS system. However, if ACE1 exceeds the preset threshold ACEtrig (heavy disturbance scenarios), EDCPS will be triggered. In the RPS system, the set points of the AGC units and the power injection of VSC1 will be taken as the control variables. Meanwhile, the calculated Psethvdc will be sent to the PPS system as a feed-forward signal. In the PPS system, if EDCPS is triggered, the ASLs will also participate in SFC. The integrated power of the VPP will be controlled to track the feed-forward signal, which aims to counteract the power imbalance caused by EDCPS. The SFC process of the RPS system and PPS system continues until the ACE signals are reduced to the required value ACEreq. If an ACE signal (ACE1 or ACE2) is below ACEreq, the control of the corresponding system will stop.

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 ACE1 is below ACEreq, the control of the RPS system will be stopped, but the control of the PPS system can continue if ACE2>ACEreq. Accordingly, if ACE2 is below ACEreq, the control of the PPS system will be stopped, and the control of the RPS system can continue if ACE1>ACEreq. When both ACE1 and ACE2 are controlled to be lower than ACEreq, the whole algorithm stops.

V. Case Study

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 is used as the test system. All simulations are performed using MATLAB. The optimizations invoke the Gurobi solver [

35]. Given the long time scale of SFC, the detailed behaviors of the electronic power converters are ignored in the simulation.

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 [

36]. The PPS system is based on an actual industrial power grid in Inner Mongolia, China [24]. The industrial power grid consists of 15 buses. The generated power of G1-G4 is 350, 150, 540, and 600 MW, respectively. The normal load in bus P7 is 40 MW. The HVDC power exploration in bus P5 is 800 MW. The power consumed by ASL1-ASL3 is 350, 440, and 610 MW, respectively. The parameters of the VSC-HVDC system are listed in Table I. The data for ASL1-ASL3 are listed in Table II. We assume that wind farms WF1 and WF2 have the same wind speed, so they are simply considered to have the same power output. According to statistical data of the wind power output in [37], typical wind speed data and the probability distribution of the forecast error can be obtained, where the forecast error ewg~N(0, 0.036). Wind speed and total forecasted wind power of WF1 and WF2 are shown in Fig. 6. The initial value of the total wind power of WF1 and WF2 is rated at 600 MW.

TABLE I  Parameters of VSC-HVDC System
DescriptionValueDescriptionValue
AC voltage of VSC1 220 kV Line impedance 0.1+j4×10-4 Ω
AC voltage of VSC2 220 kV Capacitor 1000 μF
DC voltage level 400 kV Rated power 800 MW
TABLE II  Data for ASL1-ASL3
NumberUd (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.

A. Case 1

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, Q11 and Q21 are diagonal matrixes consisting of 0.8. The standard values Q12s and Q22s 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. In the PFC period, the four methods have the same control performance because PFC is spontaneously carried out by the governor rather than the controller. In the SFC period, it is obvious that the proposed MPC method has better performance than that of the fixed-weight MPC method, and fuzzy PI control also has better performance than that of the traditional PI control. This is because the self-tuned characteristic of the varying-weight MPC and fuzzy PI controllers ensure that the parameters can be flexibly adjusted according to the signal error. However, even the fixed-weight MPC controller has superior performance to that of the fuzzy PI controller, which means that the MPC method can greatly reduce the period over which the frequency is restored in the RPS system.

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(a), the target value of the VPP power output is kept constant (800 MW). For the two MPC methods, the VPP power output can be controlled near the target value even if the wind power greatly fluctuates. Further, the proposed MPC controller has better performance. By contrast, the performance of traditional PI control and fuzzy PI control is not good enough.

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 Fig. 8(b). The frequency for the proposed MPC method fluctuates less than that for other methods. This result corresponds to the control performance of the VPP power output. If the VPP power output is controlled near the target value, the power imbalance in the PPS system can be reduced. Accordingly, the frequency devotion will be reduced. This means that the proposed MPC method can improve the wind power accommodation capacity of the PPS system. The peak values of deviations with different control methods are listed in Table III.

TABLE III  Peak Values of Deviations with Different Control Methods
MethodVPP 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

B. Case 2

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. From the results, the proposed MPC method has better performance than fuzzy PI control in both the EDCPS and no-EDCPS scenarios. Moreover, for the same control method, the participation of EDCPS can effectively shorten the frequency recovery time.

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. Then, the power output of the VPP is controlled to track the target value. The peak values of the power deviations of the VPP and the frequency deviations in the PPS system in different scenarios are listed in Table IV. From the results, the power output of the VPP with ASL participation using the proposed MPC method has the best performance than those of the other scenarios. Further, the performance of the two scenarios with ASL participation is better than that of the two no-ASL scenarios. This is because the flexibility of the ASL allows its participation to greatly improve the regulation ability of the VPP. Accordingly, the frequency fluctuations in Fig. 11 are related to the control performance of the VPP, where the proposed MPC method in the ASL scenario has the lowest frequency deviation, and fuzzy PI control in the no-ASL scenario has the highest frequency deviation.

Fig. 10  VPP control performance with different control methods. (a) Power shortage scenario. (b) Power surplus scenario.

TABLE IV  Peak Value of Deviations in Different Scenarios
ScenarioVPP deviation (MW)Frequency deviation (Hz)
ASLNo-ASLASLNo-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.

C. Case 3

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. The power injection of VSC2 is adjusted to follow the EDCPS command signal. Accordingly, the power injection in VSC1 is also changed due to DC voltage control. The variations in the DC voltages of VSC1 and VSC2 are shown in Fig. 13. During the transient process of EDCPS, the DC voltage increases or decreases because the active power injected into the inner DC system is not balanced. Then, as the transmission power stabilizes, the active power flowing into and out of the inner DC system quickly reaches a balance under the control of the VSC. Therefore, the DC voltages Udc1 and Udc2 are stabilized at new steady-state values. In the two scenarios, the peak value of the DC voltage deviation in the transient and steady-state processes are 17 kV (4.25%) and 0.61 kV (0.15%), respectively, which will not threaten the safe operation of the VSC-HVDC system.

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 [

7], the electrolytic state of the ASLs is decided by Idasl in the potline.

The DC variations of ASL1 to ASL3 are shown in Fig. 14. During the control process, the ASLs adjust Idasl to change the power consumption. From the results, the DCs of the ASLs are always kept above 70% of the rated value, which is the security margin of aluminum electrolysis. Therefore, the safety of the ASLs can be guaranteed in the regulation process.

Fig. 14  DC variations of ASL1 to ASL3. (a) Power shortage scenario. (b) Power surplus scenario.

VI. Conclusion

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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