Two-Layer Robust Distributed Predictive Control for Load Frequency Control of a Power System under Wind Power Fluctuation

Abstract

The frequency stability of interconnected power systems becomes quite challenging when incorporating renewable energy sources (mostly wind power). Distributed model predictive control (DMPC) is an effective method to maintain stable grid frequency and realize power system load frequency control (LFC). This paper proposes a two-layer robust DMPC for the LFC of an interconnected power system. In the scheme, the wind power penetrating the power grid is largely affected by the environment condition, and it is taken as an uncertain disturbance to the power system. The two-layer robust DMPC consists of a nominal DMPC controller and an ancillary DMPC controller. The nominal DMPCs coordinate with each other in achieving the systemwide LFC objective, where the systemwide objective is a strict convex combination of the local LFC objectives. The nominal optimization problems are solved supposing the wind power fluctuation is zero. The ancillary DMPC generates the actual control signal for each generation unit based on signals which are transmitted from the nominal DMPC controller. The simulation on a four-area interconnected power system demonstrates the effectiveness of the proposed algorithm in alleviating the frequency deviation caused by varying the load and uncertain wind power fluctuation.

1. Introduction

In recent years, wind power has become one of the fastest-growing renewable energy sources worldwide, with a rapidly increasing installed capacity. By the end of 2021, the global wind power capacity reached 837 GW, and it is expected to continue to grow rapidly in the coming years. This growth has led to significant integration of wind power into power systems, which poses new challenges for frequency control. Due to the random and intermittent nature of wind, wind power output fluctuates and introduces uncertainty into the system, making it difficult to maintain a stable and reliable power system. Furthermore, the fast and dynamic changes in wind power output require advanced control strategies for power system load frequency control (LFC).

LFC is a critical function in power systems that aims to maintain the system frequency and power balance under changing load conditions. The control of LFC is typically achieved by adjusting the power output of thermal and hydro generators to match the load demand, with the control signals derived from measurements of frequency and power imbalance. LFC has been an essential research topic of power system control for over forty years, along with significantly improving modern control theory. Various control strategies have been developed for LFC, including fractional-order PID [1], variable structure control [2,3], state feedback control [4], intelligent control [5,6], and model predictive control (MPC) [7]. Comprehensive review articles on LFC in power systems have been prominently featured in numerous prestigious international journals to show the research focus and development directions [8,9].

By applying a state-space model as the system prediction model, MPC conducts an optimization procedure at each sampling time. Due to its constraint-handling ability, MPC has been widely applied in power system control. Paper [10] presents a multiple-MPC strategy for working with multiple operation conditions of an electric vehicle (EV) incorporated power system. Through the optimization of the EV power and the battery state of charge (SoC), the proposed method can effectively improve frequency stabilization. Paper [11] presents an explicit MPC strategy for the LFC problem of an isolated power system, enabling operators to validate the controller’s performance in advance. Paper [12] applies the bat-inspired algorithm to search for optimal MPC parameters, coping with system nonlinearities and time delays. Paper [13] combines a virtual inertia emulator (VIE) with the MPC strategy so that the output power of an energy-storage-system-based VIE can be flexibly regulated to compensate for the fluctuation in wind power.

Distributed model predictive control (DMPC) has been increasingly applied in load frequency control due to its advantages over traditional centralized control methods. DMPC can handle large-scale power systems by dividing the system into smaller subsystems, and each subsystem can perform control tasks independently. Paper [14] originally presents a DMPC framework for controlling a large-scale interconnected power system, where the MPC-based subsystems work iteratively and cooperatively to achieve the control goal of the entire system. In [15], a DMPC strategy is incorporated with the economic objective to be applied in the economic load dispatch of an interconnected power system. Paper [16] proposes coordinated DMPC for the frequency control of a power system with wind turbines (WTs). Considering different wind speed conditions, two optimization modes for the DMPC were used and the fluctuation in wind power was compensated for by conventional power plants. A hierarchical DMPC strategy was implemented in [17] for standalone renewable energy power system control. In this strategy, the coordination of power dispatch was realized in the upper layer through a distributed algorithm, while the power balance between generation and load demand was maintained in the lower layer.

According to the literature, researchers have focused their efforts on LFC study considering wind power incorporated into the grid. The operation of WTs is seriously affected by the natural environment. Furthermore, the inertia of WTs cannot contribute to the inertia of the grid, as the kinetic energy in the rotating mass is decoupled from the grid frequency using a power electronic converter. Thus, the incorporation of wind power into the power grid is naturally taken as an uncertain disturbance. Robust control can provide an effective means to maintain system stability by designing controllers that can handle uncertainty and disturbances. In [18], a tube-based DMPC strategy is proposed, achieving the plug-and-play design of a large-scale system. The robustness of the proposed approach for regulating the frequency of an interconnected power network is illustrated. In [19], PI frequency controller parameters and vehicle-to-grid control are optimized simultaneously using particle swarm optimization based on the mixed $H_2/H_{\infty }$ control. In [20], the robust distributed model predictive control (RDMPC) strategy based on linear matrix inequalities (LMIs) for power system LFC is proposed, where a min–max optimization problem was effectively solved using the LMI method. Paper [21] proposes a robust two-layer MPC control scheme to incorporate the frequency regulation capabilities of battery energy storage systems in LFC.

This paper presents a two-layer robust DMPC strategy for power system LFC by integrating the tube-based robust control method with a DMPC scheme. The two-layer robust DMPC controller is composed of a nominal DMPC and an ancillary DMPC controller. The coordinated nominal DMPC generates local control signals to regulate the frequency deviation through adjusting the power output of thermal/hydro generators, where the optimization objective is a strict convex combination of the local controller objectives. Under the wind power fluctuation, the ancillary DMPC tracks the state and control setpoints provided by the nominal controller, where a disturbance observer is initially designed to estimate the wind power signal.

2. Modeling of Power System LFC

A schematic diagram of the power system LFC is presented in Figure 1, in which the interconnected power system consists of thermal power plant, wind farm, and hydro power plant. The generation areas are fully connected, and the robust DMPC exchanges information through a highly reliable and high-speed communication network.

2.1. Model of Thermal Power Plant

The thermal power plant consists of four main components, which are described as follows [16]:

• Speed-governing subsystem (SG)

$$\mathrm{SG}: \Delta {\dot{X}}_{gi}=-\frac{1}{T_{Gi}{R}_i}\Delta f_i-\frac{1}{T_{Gi}}\Delta X_{gi}+\frac{1}{T_{Gi}}\Delta P_{ci}$$

(1)

where $\Delta X_{gi}$ denotes the governor valve position deviation, $\Delta f_i$ denotes the grid frequency deviation, $\Delta P_{ci}$ denotes the control signal coming from the local controller, $T_{Gi}$ denotes the thermal governor time constant, and $R_i$ denotes the speed droop.

2.

Steam turbine unit (STU)

$$\mathrm{STU}: \Delta {\dot{P}}_{gi}=-\frac{1}{T_{Ti}}\Delta P_{gi}+\frac{1}{T_{Ti}}\Delta P_{ri}$$

(2)

where $\Delta P_{gi}$ denotes the generator output power deviation, $\Delta P_{ri}$ denotes the reheat turbine output power deviation, and $T_{Ti}$ denotes the turbine time constant.

3.

Reheat time delay subsystem (RTD)

$$\mathrm{RTD}: \Delta {\dot{P}}_{ri}=-\frac{K_{ri}}{T_{Gi}{R}_i}\Delta f_i+\left(\frac{1}{T_{ri}}-\frac{K_{ri}}{T_{Gi}}\right)\Delta X_{gi}-\frac{1}{T_{ri}}\Delta P_{ri}$$

(3)

where $K_{ri}$ and $T_{ri}$ denote the reheat gain and time constant, respectively.

4.

Power system (PS)

$$\mathrm{PS}: \Delta {\dot{f}}_i=\frac{1}{T_{pi}}\left(-\Delta f_i+K_{pi}\left(\Delta P_{gi}-\Delta P_{tie,i}-\Delta P_{di}\right)\right)$$

(4)

where $K_{pi}$ and $T_{pi}$ denote the power system gain and time constant, respectively; $\Delta P_{di}$ denotes the load disturbance. $\Delta P_{tie,i}$ denotes the interchange power between generation areas, which can be expressed as:

$$\begin{array}{l}\Delta P_{tie,i}={\displaystyle \sum _{j\in \mathbb{M}}{K}_{sij}(\Delta f_i-\Delta f_j)}\\ \Delta P_{tie}^{ij}=-\Delta P_{tie}^{ji}\end{array}$$

(5)

where $\mathbb{M}=\{1,2,\cdots ,m\}$ is the number of generation areas, and $K_{ij}$ denotes the interconnection gain between the areas.

In power system LFC, the thermal power generation unit should satisfy the generation rate constraint:

$$\left|\Delta {\dot{P}}_{gi}\right|\le 0.0017$$

(6)

2.2. Model of Hydro Power Plant

The composition of a hydro power plant is similar to that of a thermal power plant, which are described as follows:

5.

Speed-governing subsystem (SG)

$$\mathrm{SG}: \Delta {\dot{X}}_{gi}=-\alpha \Delta f_i-\frac{1}{T_{2i}}\Delta X_{gi}-\beta \Delta X_{ghi}+\alpha R_i\Delta P_{ci}$$

(7)

where $\Delta X_{ghi}$ denotes the governor servo position deviation. $\alpha =\frac{T_{Ri}}{T_{1i}{T}_{2i}{R}_i}$ and $\beta =\frac{T_{Ri}-T_{1i}}{T_{1i}{T}_{2i}}$, where $T_{1i}$, $T_{2i}$, and $T_{Ri}$ denote hydro governor time constants.

6.

Hydraulic turbine system (HT)

$$\mathrm{HT}: \Delta {\dot{X}}_{ghi}=-\frac{1}{T_{1i}{R}_i}\Delta f_i-\frac{1}{T_{1i}}\Delta X_{ghi}+\frac{1}{T_{1i}}\Delta P_{ci}$$

(8)

7.

Water hammer dynamic subsystem (WH)

$$\mathrm{WH}: \Delta {\dot{P}}_{gi}=2\alpha \Delta f_i-\frac{2}{T_{Wi}}\Delta P_{gi}+2\kappa \Delta X_{gi}+2\beta \Delta X_{ghi}-2\alpha R_i\Delta P_{ci}$$

(9)

where $T_{Wi}$ denotes the water starting time constant, $\kappa =\frac{T_{2i}+T_{Wi}}{T_{2i}{T}_{Wi}}$.

8.

Power system (PS)

$$\mathrm{PS}: \Delta {\dot{f}}_i=\frac{1}{T_{pi}}\left(-\Delta f_i+K_{pi}\left(\Delta P_{gi}-\Delta P_{tie,i}-\Delta P_{di}-\Delta P_{wi}\right)\right)$$

(10)

where $\Delta P_{wi}$ denotes the wind power fluctuation.

In power system LFC, the hydro power plant should satisfy the generation rate constraint:

$$\left|\Delta {\dot{P}}_{gi}\right|\le 0.045$$

(11)

2.3. Model of Wind Power Generation

The output power of a wind farm could be seriously affected since the wind speed varies randomly, and the grid frequency stability could be damaged. Traditionally, thermal and hydro power plants provide sufficient power reserves to compensate for wind power fluctuation and realize frequency regulation. Thus, in the process of LFC, wind power is not taken as a power energy source but is equivalent to a disturbance source [19].

Wind energy is converted as the aerodynamic power in the rotor, which can be expressed as:

$$P_t=\frac{\pi }{2}{\rho }_a{R}_r^2{v}_r^2{C}_p(\lambda ({\omega }_r,v_r),\beta )$$

(12)

where ${\rho }_a$ is the air density; $R_r$ is the radius of the rotor; $v_r$ is the effective wind speed; $\lambda$ and $\beta$ represent the tip speed ratio and the pitch angle of the blades, respectively; and ${\omega }_r$ is the angular velocity of the rotor. The power coefficient $C_p$ is generally expressed as:

$$C_p=0.5176\left(\frac{116}{{\lambda }_i({\omega }_r,v_r)}-0.4\beta -5\right)e^{-21/{\lambda }_i({\omega }_r,v_r)}+0.0068\lambda ({\omega }_r,v_r)$$

(13)

where $\frac{1}{{\lambda }_i({\omega }_r,v_r)}=\frac{1}{\lambda ({\omega }_r,v_r)+0.08\beta }-\frac{0.035}{{\beta }^3+1}$. The tip speed ratio $\lambda$ is defined as $\lambda ({\omega }_r,v_r)=R_r{\omega }_r/v_r$.

The output power of wind farm can be expressed as:

$$\Delta P_w=\frac{K_{WT}}{1+s{T}_{WT}}\Delta P_t$$

(14)

where $K_{WT}$ denotes the wind turbine gain, and $T_{WT}$ denotes the wind turbine time constant.

3. Two-Layer Robust Distributed Model Predictive Control

Following Section 2, the local discrete state-space model of the generation area can be uniformly described as:

$$\left\{\begin{array}{c}{x}_i(k+1)=A_{ii}{x}_i(k)+B_{ii}{u}_i(k)+F_{ii}{w}_i(k)+{\displaystyle \sum _{\begin{array}{c}j\ne i,\\ j=1,\cdots ,m\end{array}}{A}_{ij}{x}_j(k)}\\ y_i(k)=C_i{x}_i(k)\end{array}\right.$$

(15)

where $m$ represents the number of generation areas. The state vectors $x_i(k)\in X_i$, $x_j(k)\in X_j$, and $i,j\in \mathbb{M}$; the control input $u_i(k)\in U_i$; and the disturbance vector $w_i(k)\in W_i$. $X_i$, $X_j$, and $U_i$ are corresponding constraint sets. In the distributed state-space model, $x_i(k)$, $u_i(k)$, and $w_i(k)$ are different according to the different kinds of generation units. For the thermal power area $i$, $x_i={\left[\begin{array}{ccccc}\Delta f_i& \Delta P_{tie,i}& \Delta P_{gi}& \Delta X_{gi}& \Delta P_{ri}\end{array}\right]}^T$, $u_i=\Delta P_{ci}$, and $w_i=\Delta P_{di}$; for the hydro power area $i$, which incorporates the wind power, $x_i={\left[\begin{array}{ccccc}\Delta f_i& \Delta P_{tie,i}& \Delta P_{gi}& \Delta X_{gi}& \Delta X_{ghi}\end{array}\right]}^T$, $u_i=\Delta P_{ci}$, and $w_i=[\Delta P_{di}$$\Delta P_{wi}{]}^T$. $y_i=AC{E}_i$ $=K_{Bi}\Delta f_i+\Delta P_{tie,i}$ is the area control error (ACE), which is the output vector of each generation area. $A_{ii}$, $B_{ii}$, $C_i$, and $F_{ii}$ denote the appropriate system matrices in each area. The interaction matrices $A_{ij}$ are

$$A_{ij}(a,b)=\left\{\begin{array}{cc}-K_{sij}& (a,b)=(2,1)\\ 0& (a,b)\ne (2,1)\end{array}\right.$$

A general DMPC optimization problem for LFC can be denoted as:

$$\begin{gathered} \underset{u_i}{\min }{\displaystyle \sum _{p=0}^{N_p-1}\left[\left({‖x_i(k+p\left|k)\right.‖}_{{\overline{Q}}_i}^2+{‖u_i(k+p\left|k)\right.‖}_{{\overline{R}}_i}^2\right)+{\displaystyle \sum _{j=1,j\ne i}^m\left({‖x_j(k+p\left|k-1)\right.‖}_{{\overline{Q}}_j}^2+{‖u_j(k+p\left|k-1)\right.‖}_{{\overline{R}}_j}^2\right)}\right]} \\ s.t. \left(6\right), \left(11\right) \mathrm{and} \left(15\right). \end{gathered}$$

(16)

where $x_i(k+p\left|k)\right.$ and $u_i(k+p\left|k)\right.$ represent the predicted state and input at time instant $k+p$ based on the data at time $k$, respectively. ${\overline{Q}}_i$ and ${\overline{R}}_i$ are the symmetric weighting matrices of the state and input.

With the uncertain wind power disturbance $\Delta P_w$ propagating through power system dynamics (15), an explicit linear relation between $u_i$ and $x_i$ along the predictive horizon is not available. In this case, the DMPC optimization problem (16) can be nonconvex. Designing an observer for an uncertain LFC system is an effective method for providing a precise estimation of the states and disturbances. Then, the load change and the wind speed disturbance can be handled separately using the two-layer robust DMPC strategy.

3.1. Disturbance Observer Design

Based on the power system LFC model (15), the system disturbance observer can be designed as follows [22]:

$$\begin{array}{l}{\widehat{x}}_i(k+1)=A_{ii}{\widehat{x}}_i(k)+B_{ii}{u}_i(k)+{\tilde{F}}_{ii}\Delta P_{di}(k)+D_i\Delta {\widehat{P}}_{wi}(k)+{\displaystyle \sum _{j\ne i}{A}_{ij}{x}_j(k)}+L_i(y_i(k)-{\widehat{y}}_i(k))\\ \Delta {\widehat{P}}_{wi}(k+1)=\Delta {\widehat{P}}_{wi}(k)+H_i(y_i(k)-{\widehat{y}}_i(k))\\ {\widehat{y}}_i(k)=C_i{\widehat{x}}_i(k)\end{array}$$

(17)

where $L_i$ and $H_i$ are the observer gain matrices designed by the latter. ${\tilde{F}}_{ii}$ is the block in matrix $F_{ii}$ corresponding to load change $\Delta P_{di}(k)$. $D_i$ is the block in matrix $F_{ii}$ corresponding to wind power disturbance $\Delta P_{wi}(k)$. ${\widehat{x}}_i$ and ${\widehat{y}}_i$ denote the estimated system states and outputs. $\Delta {\widehat{P}}_{wi}$ denotes the estimation value of $\Delta P_{wi}$. The structure of the proposed disturbance observer is shown in Figure 2.

If we define the state estimation error $e_{xi}=x_i-{\widehat{x}}_i$ and the disturbance estimation error $e_{zi}=\Delta P_{wi}-\Delta {\widehat{P}}_{wi}$, using the actual LFC model (15) and the observer model (17), the dynamics of $e_{xi}$ and $e_{zi}$ can be written as:

$$e_{xi}(k+1)=(A_{ii}-L_i{C}_i)e_{xi}(k)+D_i{e}_{zi}(k)$$

(18)

$$\begin{array}{ll}{e}_{zi}(k+1)& =\Delta P_{wi}(k+1)-(\Delta {\widehat{P}}_{wi}(k)+H_i(y_i(k)-{\widehat{y}}_i(k)))\\ & =e_{zi}(k)-H_i{C}_i{e}_{xi}(k)+\left(\Delta P_{wi}(k+1)-\Delta P_{wi}(k)\right)\end{array}$$

(19)

Augmenting Equations (18) and (19), it can be obtained that

$$\begin{array}{ll}{e}_i(k+1)& =\left[\begin{array}{cc}{A}_{ii}-L_i{C}_i& D_i\\ -H_i{C}_i& I\end{array}\right]e_i(k)+\left[\begin{array}{c}\mathbf{0}\\ z_i(k+1)-z_i(k)\end{array}\right]\\ & =({\widehat{A}}_{ii}-{\widehat{L}}_i{\widehat{C}}_i)e_i(k)+\left[\begin{array}{c}\mathbf{0}\\ \Delta P_{wi}(k+1)-\Delta P_{wi}(k)\end{array}\right]\end{array}$$

(20)

where $e_i={\left[e_{xi}{}^T,e_{zi}{}^T\right]}^T$ and ${\widehat{A}}_{ii}=\left[\begin{array}{cc}{A}_{ii}& D_i\\ 0& I\end{array}\right]$, ${\widehat{L}}_i=\left[\begin{array}{c}{L}_i\\ H_i\end{array}\right]$, ${\widehat{C}}_i=\left[\begin{array}{cc}{C}_i& 0\end{array}\right]$.

To enable the estimation errors to vanish with time, the matrix ${\widehat{L}}_i$ should be found such that $({\widehat{A}}_{ii}-{\widehat{L}}_i{\widehat{C}}_i)$ is stable. The transposition of the matrix $({\widehat{A}}_{ii}-{\widehat{L}}_i{\widehat{C}}_i)$ is given as:

$${({\widehat{A}}_{ii}-{\widehat{L}}_i{\widehat{C}}_i)}^{\top }={\widehat{A}}_{ii}{}^{\top }-{\widehat{C}}_i{}^{\top }{\widehat{L}}_i{}^{\top }={\widehat{A}}_{ii}{}^{\top }+{\widehat{C}}_i{}^{\top }{(-{\widehat{L}}_i)}^{\top }={\widehat{A}}_{ii}{}^{\top }+{\widehat{C}}_i{}^{\top }{\widehat{K}}_i$$

(21)

Note that the transposition of the matrix $({\widehat{A}}_{ii}-{\widehat{L}}_i{\widehat{C}}_i)$ does not change the eigenvalues. Since the matrix $({\widehat{A}}_{ii}{}^{\top }+{\widehat{C}}_i{}^{\top }{\widehat{K}}_i)$ can be taken as a state feedback system matrix for the system $({\widehat{A}}_{ii}{}^{\top },{\widehat{C}}_i{}^{\top })$, the feedback gain ${\widehat{K}}_i$ for stabilizing the error dynamic system (20) can be calculated by solving the unconstrained LQR problem:

$${\hat{K}}_i={({\hat{R}}_i+{\widehat{C}}_i{\overline{P}}_i{\widehat{C}}_i{}^{\top })}^{-1}{\widehat{C}}_i{\overline{P}}_i{\widehat{A}}_{ii}{}^{\top }$$

(22)

where ${\widehat{Q}}_i\ge 0$, ${\widehat{R}}_i>0$, and ${\overline{P}}_i$ is the solution to the following Riccati equation:

$${\widehat{A}}_{ii}{\overline{P}}_i{}_i{\widehat{A}}_{ii}^{\top }+{\widehat{Q}}_i-{\widehat{A}}_{ii}{\overline{P}}_i{\widehat{C}}_i{}^{\top }{(R_i+{\widehat{C}}_i{\overline{P}}_i{\widehat{C}}_i{}^{\top })}^{-1}{\widehat{C}}_i{\overline{P}}_i{\widehat{A}}_{ii}^{\top }={\overline{P}}_i$$

(23)

3.2. Tube-Based Robust Distributed Model Predictive Controller

The two-layer tube-based DMPC scheme for power system LFC consists of two components: the nominal controller in the inner layer and the ancillary controller in the outer layer. The tube-based robust control can be depicted as in Figure 3. The ancillary controller generates a trajectory for the actual power system under wind power fluctuation, which is maintained within tubes centered around that of the nominal system without uncertainty, and the nominal controller provides a prediction of the system control input to alleviate the ACE in each area.

The proposed control structure for each generation area is shown in Figure 4. In the LFC problem, neglecting the wind power disturbance in (17), the nominal system can be defined as:

$$\left\{\begin{array}{c}{\tilde{x}}_i(k+1)=A_{ii}{\tilde{x}}_i(k)+B_{ii}{\tilde{u}}_i(k)+{\tilde{F}}_{ii}\Delta P_{di}(k)+{\displaystyle \sum _{j\ne i}{A}_{ij}{\tilde{x}}_j(k)}\\ {\tilde{y}}_i(k)=C_i{\tilde{x}}_i(k)\end{array}\right.$$

(24)

where ${\tilde{x}}_i(k)$, ${\tilde{u}}_i(k)$, and ${\tilde{y}}_i(k)$ denote the state, input, and output variables for the nominal system, respectively. The dynamic model for the states ${\tilde{x}}_i(k)$ is the same as (15) given for the actual states $x_i(k)$. The cost function of each local area can be defined in terms of the nominal state ${\tilde{x}}_i(k)$ and control input ${\tilde{u}}_i(k)$:

$$\begin{array}{ll}{\overline{J}}_i\left(k\right)& =\sum _{p=1}^{N_p}\left[\begin{array}{c}{\left({\tilde{x}}_i(k+p\left|k\right.)-x_{i,ref}(k+p)\right)}^{\top }{\tilde{Q}}_i\left({\tilde{x}}_i(k+p\left|k\right.)-x_{i,ref}(k+p)\right)\hfill \\ +\sum _{j=1,j\ne i}^m{\left({\tilde{x}}_j(k+p\left|k-1\right.)-x_{j,ref}(k+p)\right)}^{\top }{\tilde{Q}}_j\left({\tilde{x}}_j(k+p\left|k-1\right.)-x_{j,ref}(k+p)\right)\hfill \end{array}\right]\\ & +\sum _{q=0}^{N_c-1}\left[{\tilde{u}}_i^{\top }(k+q\left|k\right.){\tilde{R}}_i{\tilde{u}}_i(k+q\left|k\right.)+\sum _{j=1,j\ne i}^m{\tilde{u}}_j^{\top }(k+q\left|k-1\right.){\tilde{R}}_j{\tilde{u}}_j(k+q\left|k-1\right.)\right]\hfill \end{array}$$

(25)

We define the predicted nominal input trajectory vector as

$${\tilde{\mathit{u}}}_i={\left[{\tilde{u}}_i{(k\left|k\right.)}^{\top }\hspace{1em}{\tilde{u}}_i{(k+1\left|k\right.)}^{\top }\hspace{1em}\cdots \hspace{1em}{\tilde{u}}_i{(k+N_c-1\left|k\right.)}^{\top }\right]}^{\top }$$

(26)

The local nominal optimization problem which computes the optimal input sequence ${\tilde{\mathit{u}}}_i$ is:

$$\underset{{\tilde{u}}_i}{\min }{\overline{J}}_i(k)$$

(27)

$$s.t.{\tilde{x}}_i(k)=x_i(k)$$

(28)

$${\tilde{x}}_i(k+1)=A_{ii}{\tilde{x}}_i(k)+B_{ii}{\tilde{u}}_i(k)+{\tilde{F}}_{ii}\Delta P_{di}(k)+{\displaystyle \sum _{j\ne i}{A}_{ij}{\tilde{x}}_j(k)}$$

(29)

$${\tilde{u}}_i(k+q\left|k\right.)\in {\overline{U}}_i,\forall q\in \left\{0,1,\dots ,N_c\right\}$$

(30)

$${\tilde{x}}_i(k+p\left|k\right.)\in {\overline{X}}_i,\forall p\in \left\{1,\dots ,N_p\right\}$$

(31)

where $N_p$ is the prediction horizon and $N_c$ is the control horizon. The constraints (30) and (31) are tightened to ensure the satisfaction of the original constraints for the actual system, which are specified in an offline mode [21].

To keep each generation area’s ACE minimal, the different located MPCs are required to cooperate with each other in achieving systemwide objectives. Thus, the objective function (25) measures the systemwide impact of local control actions. Since the cost function can be transformed to a quadratic form with respect to ${\tilde{\mathit{u}}}_i$, the quadratic programming procedure can then be utilized to solve the optimization problem.

It should be noted that only the local control sequence ${\tilde{\mathit{u}}}_i$ is optimized and updated, while all the other areas’ predicted state and input sequences are substituted by the estimated value ${\tilde{x}}_j(k+p\left|k-1\right.)$ and ${\tilde{u}}_j(k+q\left|k-1\right.)$ based on the information at time $k-1$ [23].

With the nominal optimal solutions ${\tilde{x}}_{i,opt}$ and ${\tilde{u}}_{i,opt}$ obtained from (25), the ancillary MPC is intended to track the optimal solutions while alleviating the effect of wind power. The cost function of the ancillary controller can be defined as:

$$\begin{array}{ll}{J}_i(k)& ={\displaystyle \sum _{p=1}^{N_p}{\left(x_i(k+p\left|k\right.)-{\tilde{x}}_{i,opt}(k+p)\right)}^{\top }{Q}_i\left(x_i(k+p\left|k\right.)-{\tilde{x}}_{i,opt}(k+p)\right)}\\ & +{\displaystyle \sum _{q=0}^{N_c}{\left(u_i(k+p\left|k\right.)-{\tilde{u}}_{i,opt}(k+p)\right)}^{\top }{R}_i\left(u_i(k+p\left|k\right.)-{\tilde{u}}_{i,opt}(k+p)\right)}\end{array}$$

(32)

We define the predicted input trajectory vector as

$${\mathit{u}}_i={\left[u_i{(k\left|k\right.)}^{\top }\hspace{1em}{u}_i{(k+1\left|k\right.)}^{\top }\hspace{1em}\cdots \hspace{1em}{u}_i{(k+N_c-1\left|k\right.)}^{\top }\right]}^{\top }$$

(33)

The corresponding local optimization problem which computes the optimal input sequence ${\mathit{u}}_i$ is:

$$\underset{u_i}{\min }{J}_i(k)$$

(34)

$$s.t.x_i(k+1)=A_{ii}{x}_i(k)+B_{ii}{u}_i(k)+F_{ii}{w}_i(k)+{\displaystyle \sum _{j\ne i}{A}_{ij}{x}_j(k)}$$

(35)

$$u_i(k+q\left|k\right.)\in U_i,\forall q\in \left\{0,1,\dots ,N_c\right\}$$

(36)

$$x_i(k+p\left|k\right.)\in X_i,\forall p\in \left\{1,\dots ,N_p\right\}$$

(37)

Solving (34)–(37), the actual control sequence is obtained and the first sample in the sequence is applied to every generation area in each sampling instant.

The two-layer robust DMPC algorithm can be summarized as follows:

(1)

Initialization: Set $k=0$, and for all areas $i$, obtain an initial feasible solution ${\tilde{u}}_i(0)$ with the current state ${\tilde{x}}_i(0)$. Each generation area sends the state and input information to all other areas. Go to Step (3).

(2)

At sampling instant $k$, each area $i$ makes the estimation of the solutions as follows: ${\tilde{\mathit{u}}}_i^{esti}={\left[\begin{array}{cccc}{\tilde{u}}_i{(k\left|k-1\right.)}^{\top }& \hspace{1em}{\tilde{u}}_i{(k+1\left|k-1\right.)}^{\top }& \hspace{1em}\cdots & \hspace{1em}{\tilde{u}}_i{(k+N_c-1\left|k-1\right.)}^{\top }\end{array}\right]}^{\top }$. Each area exchanges ${\tilde{\mathit{u}}}_i^{esti}$ and ${\tilde{x}}_i(k)$ with other areas.

(3)

For $i=1$ to $m$, the area $i$:

• Receives the estimated trajectories ${\tilde{\mathit{u}}}_j^{esti}$ and ${\tilde{x}}_j(k)$ from all other areas.
• Solves the nominal problem (27)–(31) and obtains the nominal optimal solutions ${\tilde{x}}_{i,opt}$ and ${\tilde{u}}_{i,opt}$.
• Transmits ${\tilde{x}}_{i,opt}$, ${\tilde{u}}_{i,opt}$, and the estimated wind power signal ${\widehat{w}}_i(k)$ produced by the disturbance observer (if there is wind power incorporated in the area) to the ancillary controller.
• Solves the optimization problem (34)–(37) and obtains the actual input trajectory ${\mathit{u}}_i^{\ast }$. Go to Step 4.

(4)

Each area applies $u_i^{\ast }(k\left|k\right.)$ to the generation unit.

(5)

Let $k=k+1$ and then go to Step (2).

4. Simulation Results

To evaluate the control performance of the proposed two-layer robust DMPC strategy, the four-area power system LFC was analyzed. Area 1 and area 4 included thermal power plants; area 2 and area 3 included hydro power plants, as shown in Figure 5. Wind power was integrated into area 2. The specific settings of the parameters are shown in

Table 1. Parameters of the interconnected power system.

$K_{p1}=K_{p4}=120\mathrm{Hz}/\mathrm{p}.\mathrm{u}.\mathrm{MW}$, $K_{p2}=K_{p3}=115\mathrm{Hz}/\mathrm{p}.\mathrm{u}.\mathrm{MW}$, $T_{p1}=T_{p4}=20 \mathrm{s}$, $K_{r1}=K_{r4}=0.5 \mathrm{Hz}/\mathrm{p}.\mathrm{u}.\mathrm{MW}$,

$R_1=R_4=2.4 \mathrm{Hz}/\mathrm{p}.\mathrm{u}.\mathrm{MW}$, $R_2=R_3=2.5 \mathrm{Hz}/\mathrm{p}.\mathrm{u}.\mathrm{MW}$, $K_{B1}=K_{B2}=K_{B3}=K_{B4}=0.425 \mathrm{p}.\mathrm{u}.\mathrm{MW}/\mathrm{Hz}$,

$T_{R2}=T_{R3}=0.6 \mathrm{s}$, $T_{12}=T_{13}=48.7 \mathrm{s}$, $T_{22}=T_{23}=5 \mathrm{s}$, $T_{W2}=T_{W3}=1 \mathrm{s}$, $T_{G1}=T_{G4}=0.08 \mathrm{s}$, $T_{T1}=T_{T4}=0.3 \mathrm{s}$,

$T_{r1}=T_{r4}=10 \mathrm{s}$, $K_{sij}=0.545 \mathrm{p}.\mathrm{u}.\mathrm{MW}$

.

For the parameters related to the two-layer robust DMPC design, the prediction and control horizons were 10 and 3, respectively, and the sampling time was 1 s in both nominal and ancillary DMPCs. The tightened constraint sets were calculated through the MPT toolbox in MATLAB [24]. The weighting matrices in (25) and (32) were chosen using the trial-and-error method:

$$Q_1=Q_2=Q_3=Q_4=diag\left(\begin{array}{ccccc}10& 10& 10& 10& 10\end{array}\right)$$

$${\tilde{Q}}_1={\tilde{Q}}_2={\tilde{Q}}_3={\tilde{Q}}_4=diag\left(\begin{array}{ccccc}10& 100& 0& 0& 0\end{array}\right)$$

$$R_i={\tilde{R}}_i=0.01\left(i=1,2,3,4\right)$$

For the disturbance observer in area 2, the gain matrices were calculated based on Equations (22) and (23) as $L_2={\left[\begin{array}{ccccc}0.7291& 0.0775& -0.0107& -0.0106& -0.009\end{array}\right]}^{\top }$ and $H_2=0.2322$.

For comparison purposes, the conventional DMPC was also applied to the four-area power system. The DMPC strategy solved the same coordinated optimization problem as Equation (25), except that the actual power system model (15) was used and the constraints were not tightened. The prediction and control horizon of DMPC were also chosen to be the same as those of the proposed robust DMPC.

The wind power was impacted by fluctuations in wind speed, making it subject to uncertainty as shown in Figure 6a. A step load decrease of 0.01 p.u. was applied in area 2 at $t=50 \mathrm{s}$, and a load increase of 0.02 p.u. was applied in area 4 at $t=300 \mathrm{s}$, as shown in Figure 6b.

4.1. LFC under Fluctuation in Wind Power

Figure 7 depicts the frequency responses of each generation area during the contingency, showing the results associated with the nominal DMPC, the two-layer robust DMPC, and the conventional DMPC. As can be seen, the wind power fluctuation propagated to all generation areas through the tie-line link. In the proposed two-layer robust DMPC scheme, a nominal response trajectory of the frequency deviation with no wind uncertainty was initially generated, and then the actual grid frequency was regulated to lie close to the nominal trajectory. When the load demand increased or decreased, the frequency deviation of the proposed strategy resulted in an in obvious reduction compared to that of the conventional DMPC. In other words, the proposed two-layer robust DMPC controller was less impacted by the wind power fluctuation.

Figure 8 depicts the control variables for the proposed robust DMPC and the conventional DMPC. It can be clearly seen that the conventional DMPC demonstrated more significant fluctuations in its control variables in comparison to the two-layer robust DMPC, indicating a higher degree of variability and instability in its control outputs. This is because DMPC is required to simultaneously realize two objectives: coordinated load frequency control and wind power disturbance rejection. As a result, the frequency regulation performance of LFC was affected by the wind disturbance. The two-layer robust DMPC appropriately discriminated the effect of the wind power disturbance from that of LFC. Specifically, the nominal DMPC controller operated in a coordinated fashion to track load variations and ensure frequency stability with no wind power disturbance, while the ancillary DMPC controller focused on tracking the optimal setpoint produced by the nominal controller. The nominal and ancillary DMPC controllers were executed in turn at each sampling instant so that load change and wind power variation are separately handled. In this way, the ancillary DMPC generated the actual system trajectory in the tube centered around the nominal trajectory so that the effect of wind power disturbance was alleviated. For further comparison, the performance of the proposed algorithm and the conventional DMPC was evaluated with the root mean square error (RMSE) index:

$$\mathrm{RMSE}(\Delta f)={\displaystyle \sum _{i=1}^4\sqrt{{\displaystyle \sum _{k=1}^{Tstop}{\left(\Delta f_i(k)\right)}^2}/Tstop}}, \mathrm{RMSE}(\Delta P_{tie})={\displaystyle \sum _{i=1}^4\sqrt{{\displaystyle \sum _{k=1}^{Tstop}{\left(\Delta P_{tie,i}(k)\right)}^2}/Tstop}}$$

where $Tstop$ denotes the simulation steps.

Table 2. The performance comparison under different strategies.

Strategy	$\mathbf{RMSE}({\Delta }\mathit{f})$	$\mathbf{RMSE}({\Delta }{\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}})$	Average Computation Time
Two-layer robust DMPC	0.5892	0.8404	0.283
Conventional DMPC	0.8857	0.9568	0.171

lists the comparison results. The proposed two-layer robust DMPC had lower RMSE indices than the conventional DMPC in both frequency deviation and tie-line power flow. However, since the two-layer robust DMPC requires solving two optimization problems sequentially at each sampling instant, it causes a heavier burden than DMPC.

4.2. Performance of Disturbance Observer

Since the uncertain wind power was integrated in area 2, the disturbance observer for area 2 was incorporated in the control scheme for estimating the disturbance. Figure 9 depicts the estimation error of frequency deviation. Based on Equations (22) and (23), the observer gain matrices $L_2$ and $H_2$ for area 2 could be obtained offline, which kept the error dynamic system (20) stable. It was observed that the estimation error of frequency deviation exhibited a rapid convergence within 10 s, enabling a precise estimation of the system states when the load demand changes. Figure 10 depicts the measured and estimated wind power fluctuation. The estimated disturbance approached the real-time value of the wind power disturbance within 5 s. By incorporating the estimated value of wind power fluctuation into the optimization problem of the ancillary DMPC controller, the controller is able to accurately track the optimized setpoint, effectively minimizing the influence of uncertain disturbance. The frequency responses are compared in Figure 11 for applying the two-layer robust DMPC controller with and without the disturbance observer. It can be seen that the frequency response using the disturbance observer had a lower maximum value when the load demand changed.

5. Conclusions

A two-layer robust distributed model predictive control strategy was presented for the LFC of an interconnected power system under wind power fluctuation. The local nominal DMPC controllers worked cooperatively towards satisfying systemwide LFC objectives. The optimal setpoints obtained from nominal controllers were transmitted to the ancillary DMPC, and the ancillary DMPC controllers produced a sequence of control signals while minimizing the difference between the trajectory of the actual system and that of the nominal system. The load change and wind power fluctuation in the power system were handled separately in the proposed control scheme. In such a way, the trajectory of the actual power system was kept within tubes centered around that of the nominal power system. Simulations on the four-area interconnected power system demonstrated that the coordination between the hydro and thermal plants has been achieved. Based on the precise estimation of the real-time wind power variation, the two-layer robust DMPC strategy presented in this paper demonstrates the ability to achieve smoother changes in control variables while effectively mitigating the impact of wind power fluctuations on frequency deviation.