Robust Distributed Model Predictive Load Frequency Control of Interconnected Power System

Abstract

Considering the load frequency control (LFC) of large-scale power system, a robust distributed model predictive control (RDMPC) is presented. The system uncertainty according to power system parameter variation alone with the generation rate constraints (GRC) is included in the synthesis procedure. The entire power system is composed of several control areas, and the problem is formulated as convex optimization problem with linear matrix inequalities (LMI) that can be solved efficiently. It minimizes an upper bound on a robust performance objective for each subsystem. Simulation results show good dynamic response and robustness in the presence of power system dynamic uncertainties.

1. Introduction

The load frequency control (LFC) has long been a much concerned research interest for power system engineers over the past forty years [1]. In modern power system, undesirable frequency and scheduled tie-line power changes in multiarea power system are a direct result of the imbalance between generated power and system demand plus associated system losses. The main objectives of the LFC are to keep the system frequency at the scheduled value and regulate the generator units to make the area control error tend to zero under the continuous adjustment of active power, so that the generation of the entire system and the load power well match.

In a practical power system, there exist different kinds of uncertainties, such as changes in parameter. And each control area contains various disturbances due to increased complexity, system modeling errors, and changing power system structure. Thus the robustness must be taken into theoretical consideration in the LFC design procedure to promise high power quality. A fixed controller based on classical theory is not very suitable for the LFC problem. It is necessary that a flexible controller should be developed [2–4]. Robust LFC was early designed based on the Riccati equation approach [5], which is simple and effective and can ensure the overall system to be asymptotically stable for all admissible uncertainties. Motivated by the large uncertainty in dynamic models of power system components and their interconnections, paper [6] proposes a physically motivated passivity objective as a means to achieve effective closed-loop control. Recently, robust LFC can be realized using linear matrix inequalities [7], fuzzy logic [8], neural networks [9], and genetic algorithms [10].

Model predictive control (MPC) has been an attracting method for power system LFC, which can perform an optimization procedure to calculate optimal control actions within the realistic power system constrains. In LFC research, there is the practical limit on the rate of change in the generating power, called the generation rate constraints (GRC), which can result in the LFC to be a constraint optimal problem. Traditional MPC is unable to explicitly incorporate plant uncertainty. Thus, robust MPC has been well developed recently [11, 12].

Most existing MPCs assume that all subsystems are identical, which is not the case of actual power systems. Subsequently, a number of decentralized/distributed load frequency controllers were developed to eliminate the above drawback. In [13], the distributed model predictive control (DMPC) is used in LFC, which offers an effective means of achieving the desired controller coordination and performance improvements. A decentralized MPC framework for multiarea power system has been presented in [14]. Accordingly, the robustness of DMPC strategies to model errors has been identified as a key factor for the successful application of DMPC [15].

In this paper, a robust distributed MPC (RDMPC) strategy for load frequency control in interconnected power system is presented. The entire power system is composed of several subareas and the problem is formulated as convex optimization problem with linear matrix inequalities (LMI) that can be solved efficiently using the algorithm. The method shows good dynamic response and robustness in the presence of power system model dynamic uncertainties.

2. Mathematical Model of Power System

The interconnected power system consists of at least two control areas connected by tie lines. Usually the subsystem contains thermal power system, hydro power system, nuclear power system, and renewable power system, in which thermal power system and hydro power system generally participate in load frequency control. Figures 1 and 2 show, respectively, the structures of thermal power plant and hydro power plant in power system LFC. The original model has been given in [16]. Comparing to [16], the model in this article contains the reheater part, which is quite common in modern thermal power plant. Each control area has its own controller. The variables and parameters are given in Table 1.

Table 1 

Variables and parameters used in thermal and hydro power plant.

Figure 1 

The block diagram of thermal power plant in area 𝑖.

Figure 2 

The block diagram of hydro power plant in area 𝑖.

When load change happens in one area, all the interconnected areas will be affected, and the controllers act to adjust the frequency deviation and tie-line active power to return to steady state. The LFC using RDMPC will be applied to the whole control areas.

The time-varying linearized mathematical model of thermal and hydro plant used in interconnected power system can be described as 𝐱̇_𝑖(𝑡)=𝐀_𝑖𝑖(𝑡)𝐱_𝑖(𝑡)+_𝑗𝐀_𝑖𝑗(𝑡)𝐱_𝑗(𝑡)+𝐁_𝑖𝑖(𝑡)𝐮_𝑖(𝑡)+𝐅_𝑖𝑖(𝑡)𝐝_𝑖(𝑡), (1) where 𝑖 represents the control area; 𝐱_𝑖∈ℝ^𝑛𝑖, 𝐮_𝑖∈ℝ^𝑚𝑖, and 𝐝_𝑖∈ℝ^𝑧𝑖 represent the state, input, and disturbance vector in the 𝑖’s subsystem, respectively. 𝐱_𝑗∈ℝ^𝑛𝑗 is a state vector of the neighbor system.

Define the area control error (ACE) to be ACE_𝑖=𝐲_𝑖(𝑡)=𝐂_𝑖𝑖𝐱_𝑖(𝑡)=𝐵_𝑖Δ𝑓_𝑖(𝑡)+Δ𝑃_tie𝑖(𝑡), (2) where 𝐲_𝑖∈ℝ^𝑣𝑖 represents system output signal. Matrices in (1) and (2) have dimensions 𝐀_𝑖𝑖∈ℝ^{𝑛𝑖×𝑛𝑖},𝐀_𝑖𝑗∈ℝ^{𝑛𝑖×𝑛𝑗},𝐁_𝑖𝑖∈ℝ^{𝑛𝑖×𝑚𝑖},𝐅_𝑖𝑖∈ℝ^{𝑛𝑖×𝑧𝑖},𝐂_𝑖𝑖∈ℝ^{𝑣𝑖×𝑛𝑖}. (3)

In Figure 1, the state variable in the thermal power system is 𝐱_𝑖(𝑡)=Δ𝑓_𝑖(𝑡)Δ𝑃_tie𝑖(𝑡)Δ𝑃_𝑔𝑖(𝑡)Δ𝑋_𝑔𝑖(𝑡)Δ𝑃_𝑟𝑖(𝑡)^𝑇; (4)

while the state variable in the hydro power system is 𝐱_𝑖(𝑡)=Δ𝑓_𝑖(𝑡)Δ𝑃_tie𝑖(𝑡)Δ𝑃_𝑔𝑖(𝑡)Δ𝑋_𝑔𝑖(𝑡)Δ𝑋_𝑔ℎ𝑖(𝑡)^𝑇. (5)

The control signal and disturbance in both the thermal power system and the hydro power system are as follows: 𝐮_𝑖(𝑡)=Δ𝑃_𝑐𝑖(𝑡),𝐝_𝑖(𝑡)=Δ𝑃_𝑑𝑖(𝑡). (6)

The state, control, and disturbance matrix in thermal power system are 𝐀_𝑖𝑖=−1𝑇_𝑝𝑖−𝐾_𝑝𝑖𝑇_𝑝𝑖𝐾_𝑝𝑖𝑇_𝑝𝑖00_𝑗≠𝑖𝑇_𝑠𝑖𝑗1000000−𝑇_𝑇𝑖01𝑇_𝑇𝑖−1𝑇_𝐺𝑖𝑅_𝑖100−𝑇_𝐺𝑖0−𝐾_𝑟𝑖𝑅_𝑖𝑇_𝐺𝑖100𝑇_𝑟𝑖−𝐾_𝑟𝑖𝑇_𝐺𝑖−1𝑇_𝑟𝑖,𝐁_𝑖𝑖=0001𝑇_𝐺𝑖0,𝐅_𝑖𝑖=−𝐾_𝑝𝑖𝑇_𝑝𝑖0000, (7)

while the state, control, and disturbance matrix in hydro power system are 𝐀_𝑖𝑖=−1𝑇_𝑃𝑖−𝐾_𝑃𝑖𝑇_𝑃𝑖𝐾_𝑃𝑖𝑇_𝑃𝑖00_𝑗≠𝑖𝑇_𝑠𝑖𝑗200002𝛼0−𝑇_𝑊𝑖12𝜅2𝛽−𝛼00−𝑇_2𝑖−1−𝛽𝑇_1𝑖𝑅_𝑖1000−𝑇_1𝑖,𝐁_𝑖𝑖=00−2𝑅_𝑖𝛽𝑅_𝑖𝛽1𝑇_1𝑖,𝐅_𝑖𝑖=−𝐾_𝑃𝑖𝑇_𝑃𝑖0000, (8) where 𝛼=𝑇_𝑅𝑖/𝑇_1𝑖𝑇_2𝑖𝑅_𝑖, 𝛽=(𝑇_𝑅𝑖−𝑇_1𝑖)/𝑇_1𝑖𝑇_2𝑖, 𝜅=(𝑇_2𝑖+𝑇_𝑊𝑖)/𝑇_2𝑖𝑇_𝑊𝑖.

Here 𝐴_𝑖𝑗 have 5×5 dimensions. All their elements are equal to zero, except for the element at position (1,2), which is equal to −𝑇_𝑠𝑖𝑗.

For the whole power system, the state-space equation is as follows: 𝐱̇(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑑(𝑡),𝐲(𝑡)(𝑡)𝑥(𝑡),=𝐀+𝐁+𝐅=𝐂 (9) where 𝐱(𝑡)=𝐱₁(𝑡)𝐱₂(𝑡)⋮𝐱_𝑀(𝑡)𝐮(𝑡)=𝐮₁(𝑡)𝐮₂(𝑡)⋮𝐮_𝑀(𝑡),𝐝(𝑡)=𝐝₁(𝑡)𝐝₂(𝑡)⋮𝐝_𝑀(𝑡)𝐲(𝑡)=𝐲₁(𝑡)𝐲₂(𝑡)⋮𝐲_𝑀(𝑡). (10)

This is a general continuous-time linear system with added disturbance. 𝑀 is the number of control areas of the interconnected power system. After using the zero-order hold (ZOH) discretization method, each control area’s distributed discrete-time linear model is expressed as follows: 𝑥_𝑖(𝑘+1)=𝐴_𝑖𝑖(𝑘)𝑥_𝑖(𝑘)+𝐵_𝑖𝑖(𝑘)𝑢_𝑖(𝑘)+^𝑀_𝑗≠𝑖𝐴_𝑖𝑗(𝑘)𝑥_𝑗(𝑘)+𝐵_𝑖𝑗(𝑘)𝑢_𝑗(𝑘),𝑦_𝑖(𝑘)=𝐶_𝑖𝑖𝑥_𝑖(𝑘). (11)

From (11), the polytopic model of each control area is 𝐴_𝑖𝑖(𝑘)𝐵_𝑖𝑖(𝑘)⋯𝐴_𝑖𝑗(𝑘)𝐵_𝑖𝑗(𝑘)⋯=^𝐿_ℓ=1𝛽_ℓ𝐴^(ℓ)_𝑖𝑖𝐵^(ℓ)_𝑖𝑖⋯𝐴^(ℓ)_𝑖𝑗𝐵^(ℓ)_𝑖𝑗⋯{}∈Ω∀𝑗∈1,…,𝑀,𝑗≠𝑖. (12)

𝐴_𝑖𝑖(𝑘), 𝐵_𝑖𝑖(𝑘), 𝐴_𝑖𝑗(𝑘), 𝐵_𝑖𝑗(𝑘), and 𝐶_𝑖𝑖 are the relative matrices in the discrete-time model (11). Ω is the model parameter uncertainty set. 𝛽_ℓ’s are used to represent a convex combination of the model vertices since the convex hull (the polytope) is the extreme model vertices. Each vertex ℓ corresponds to a linear model. The states are assumed to be available.

3. Robust Distributed Model Predictive Control Algorithm

Considering the distributed discrete-time power system model (11), the min-max problem to be solved for each subsystem is expressed as min_{𝑢𝑖()𝑘+𝑙∣𝑘}max𝐴_{𝑖𝑖()𝑘+𝑙}𝐵_{𝑖𝑖()⋯𝑘+𝑙}𝐴_{𝑖𝑗()𝑘+𝑙}𝐵_{𝑖𝑗()⋯𝑘+𝑙,𝑙≥0}𝐽_𝑖(𝑘)s.t.𝑢_𝑖(𝑘+𝑙∣𝑘)≤𝑢_𝑖^max,𝑙≥0, (13)(14) where 𝐽_𝑖(𝑘) is an object function for subsystem 𝑖 to guarantee the cooperation of subsystem controllers, defined as 𝐽_𝑖(𝑘)=^∞_𝑙=0𝑥^′_𝑖(𝑘+𝑙∣𝑘)𝑆_𝑖𝑥_𝑖(𝑘+𝑙∣𝑘)+𝑢^′_𝑖(𝑘+𝑙∣𝑘)𝑅_𝑖𝑢_𝑖(+𝑘+𝑙∣𝑘)_{𝑀∞𝑗≠𝑖𝑙=0}𝑥^′_𝑗(𝑘+𝑙∣𝑘)𝑆_𝑗𝑥_𝑗(𝑘+𝑙∣𝑘)+𝑢^′_𝑗(𝑘+𝑙∣𝑘)𝑅_𝑗𝑢_𝑗(𝑘+𝑙∣𝑘), (15) where 𝑥_𝑖(𝑘+𝑙∣𝑘) and 𝑢_𝑖(𝑘+𝑙∣𝑘) are the predicted state and input variables for the 𝑖th subsystem at time instant 𝑘+𝑙, 𝑙≥0, based on data at time 𝑘. 𝑆_𝑖, 𝑅_𝑖, 𝑆_𝑗, and 𝑅_𝑗 are the weighting matrices.

The maximization is to choose time-varying model [𝐴_𝑖𝑖(𝑘+𝑙)𝐵_𝑖𝑖(𝑘+𝑙)⋯𝐴_𝑖𝑗(𝑘+𝑙)𝐵_𝑖𝑗(𝑘+𝑙)⋯] in the uncertainty set Ω to get the worst situation of 𝐽_𝑖(𝑘), and this worst situation will be minimized on the current and the future horizons.

To solve the optimal problem (13), it is necessary to find an upper bound of the object function (15). Considering the quadratic function 𝑉_𝑖(𝑥)=𝑥^𝑇𝑃_𝑖𝑥,𝑃_𝑖>0, (16) where 𝑥=[𝑥^′₁𝑥^′₂⋯𝑥^′_𝑀]^′. For all the subsystem 𝑖, 𝑉_𝑖(𝑥) should satisfy the following stability constraint: 𝑉_𝑖(𝑥(𝑘+𝑙+1∣𝑘))−𝑉_𝑖(𝑥(𝑥𝑘+𝑙∣𝑘))≤−^′_𝑖(𝑘+𝑙∣𝑘)𝑆_𝑖𝑥_𝑖(𝑘+𝑙∣𝑘)+𝑢^′_𝑖(𝑘+𝑙∣𝑘)𝑅_𝑖𝑢_𝑖()+𝑘+𝑙∣𝑘_{𝑀𝑗≠𝑖}𝑥^′_𝑗(𝑘+𝑙∣𝑘)𝑆_𝑗𝑥_𝑗(𝑘+𝑙∣𝑘)+𝑢^′_𝑗(𝑘+𝑙∣𝑘)𝑅_𝑗𝑢_𝑗(𝑘+𝑙∣𝑘). (17)

For 𝑙=0,1,…,∞, the accumulation of (17) is 𝑉_𝑖(𝑥(𝑘∣𝑘))≥𝐽_𝑖(𝑘). (18)

So the upper bound of object function can be proved to be max𝐴_𝑖𝑖()^𝑘+𝑙𝐵_𝑖𝑖()^⋯𝑘+𝑙𝐴_𝑖𝑗()^𝑘+𝑙𝐵_𝑖𝑗()^{⋯𝑘+𝑙,𝑙≥0}𝐽_𝑖(𝑘)≤𝑉_𝑖(𝑥(𝑘∣𝑘)). (19)

A state-feedback law is sought for each subsystem 𝑖 as follows: 𝑢_𝑖(𝑘+𝑙∣𝑘)=𝑓_𝑖𝑖𝑥_𝑖(𝑘+𝑙∣𝑘)+^𝑀_𝑗≠𝑖𝑓_𝑖𝑗𝑥_𝑗(𝑘+𝑙∣𝑘)=𝑓_𝑖𝑥(𝑘+𝑙∣𝑘), (20) where 𝑓_𝑖=𝑓_𝑖1𝑓_𝑖2⋯𝑓_𝑖𝑀.

When solving optimization problem of the subsystem 𝑖, the state-feedback law of the neighboring subsystem 𝑗(𝑗≠𝑖) is expressed as 𝑢_𝑗(𝑘+𝑙∣𝑘)=𝑓^∗_𝑗𝑗𝑥_𝑗(𝑘+𝑙∣𝑘)+^𝑀_𝑗≠𝑠𝑓^∗_𝑗𝑠𝑥_𝑠(𝑘+𝑙∣𝑘)=𝑓^∗_𝑗𝑥(𝑘+𝑙∣𝑘), (21) where 𝑓^∗_𝑗=[𝑓^∗_𝑗1𝑓^∗_𝑗2⋯𝑓^∗_𝑗𝑀].

The RDMPC algorithm will be redefined using state-feedback law (20) to minimize the upper bound min_{𝑢𝑖()𝑘+𝑙∣𝑘}𝑉_𝑖(𝑥(𝑘∣𝑘))=min_𝑓𝑖𝑥^′(𝑘∣𝑘)𝑃_𝑖𝑥(𝑘∣𝑘),𝑃_𝑖>0. (22)

For the whole power system, the expression of 𝑥 is 𝑥()=𝑥𝑘+1¹()𝑥𝑘+1₂()⋮𝑥𝑘+1_𝑀()=𝐴𝑘+1₁₁(𝑘)𝐴₁₂(𝑘)⋯𝐴_1𝑀(𝑘)𝐴₂₁(𝑘)𝐴₂₂(𝑘)⋯𝐴_2𝑀(𝑘)𝐴⋮⋮⋮_𝑀1(𝑘)𝐴_𝑀2(𝑘)⋯𝐴_𝑀𝑀(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)⋮𝑥_𝑀(𝑘)+𝐵₁₁(𝑘)𝐵₂₁(𝑘)⋮𝐵_𝑀1(𝑘)𝑢₁(𝑘)+𝐵₁₂(𝑘)𝐵₂₂(𝑘)⋮𝐵_𝑀2(𝑘)𝑢₂(𝑘)𝐵+⋯+^1𝑀(𝑘)𝐵_2𝑀(𝑘)⋮𝐵_𝑀𝑀(𝑘)𝑢_𝑀(𝑘). (23)

Define 𝐴(𝑘)=𝐴₁₁(𝑘)𝐴₁₂(𝑘)⋯𝐴_1𝑀(𝑘)𝐴₂₁(𝑘)𝐴₂₂(𝑘)⋯𝐴_2𝑀(𝑘)𝐴⋮⋮⋮_𝑀1(𝑘)𝐴_𝑀2(𝑘)⋯𝐴_𝑀𝑀(𝑘)𝐵_𝑖,(𝑘)=𝐵_1𝑖(𝑘)𝐵_2𝑖(𝑘)⋮𝐵_𝑀𝑖(𝑘). (24)(25)

Using (20) and (21), the state (23) can be simplified as 𝑥(𝑘+1)=𝐴(𝑘)+𝐵_𝑖(𝑘)𝑓_𝑖𝑥(𝑘) (26) in which 𝐴(𝑘)=𝐴(𝑘)+∑^𝑀_𝑗≠𝑖𝐵_𝑗(𝑘)𝑓^∗_𝑗.

The robust stability constraint in (17) becomes 𝐴^(ℓ)(𝑘+𝑙)+𝐵_𝑖^(ℓ)(𝑘+𝑙)𝑓_𝑖^′×𝑃_𝑖𝐴^(ℓ)(𝑘+𝑙)+𝐵_𝑖^(ℓ)(𝑘+𝑙)𝑓_𝑖−𝑃_𝑖≤−𝑆_𝑖+^𝑀_𝑗≠𝑖𝑓^∗′_𝑗𝑅_𝑗𝑓^∗_𝑗+𝑓^′_𝑖𝑅_𝑖𝑓_𝑖, (27) where 𝑆_𝑖=𝑆₁𝑆₂⋱𝑆_𝑀. (28)

By defining an upper bound, 𝐽_𝑖(𝑘)≤𝑉_𝑖(𝑥(𝑘∣𝑘))≤𝛾_𝑖. (29)

The optimal problem (22) is equivalent to min_{𝛾𝑖,𝑃𝑖}𝛾_𝑖s.t.𝑥^′(𝑘∣𝑘)𝑃_𝑖𝑥(𝑘∣𝑘)≤𝛾_𝑖. (30)

Substituting 𝑃_𝑖=𝛾_𝑖𝑄_𝑖⁻¹>0, 𝑌_𝑖=𝑓_𝑖𝑄_𝑖, with the input constraints given in (13) and the stability constraint (27), followed by a Schur complement decomposition, the minimization of 𝐽_𝑖(𝑘) can be replaced by the minimization problem (30) as in the following linear minimization problem with LMI constraints: min_{𝛾𝑖,𝑃𝑖,𝑄𝑖}𝛾_𝑖1s.t.𝑥^′(𝑘∣𝑘)𝑥(𝑘∣𝑘)𝑄_𝑖𝑄≥0,_𝑖𝑄_𝑖𝐴^′(ℓ)+𝑌^′_𝑖𝐵^′_𝑖^(ℓ)𝑄_𝑖𝑆_𝑖^1/2𝑌_𝑖𝑅_𝑖^1/2𝐴^(ℓ)𝑄_𝑖+𝐵_𝑖^(ℓ)𝑌_𝑖𝑄_𝑖00𝑆_𝑖^1/2𝑄_𝑖0𝛾_𝑖𝑅𝐼0_𝑖1/2𝑌_𝑖00𝛾_𝑖𝐼𝑢≥0ℓ=1,2,…,𝐿,_𝑖max2𝐼𝑌_𝑖𝑌^′_𝑖𝑄_𝑖≥0. (31)

For the constraints on power system state max𝐴_𝑖𝑖()^𝑘+𝑙𝐵_𝑖𝑖()^⋯𝑘+𝑙𝐴_𝑖𝑗()^𝑘+𝑙𝐵_𝑖𝑗()^{⋯𝑘+𝑙∈Ω,𝑙≥0}𝑦_𝑖(𝑘+𝑙∣𝑘)₂≤𝑦_𝑖,max. (32)

Transform it to LMI form as 𝑄_𝑖𝐴^(ℓ)𝑄_𝑖+𝐵_𝑖^(ℓ)𝑌_𝑖^′𝐶^′_𝑖𝑖𝐶_𝑖𝑖𝐴^(ℓ)𝑄_𝑖+𝐵_𝑖^(ℓ)𝑌_𝑖𝑦²_𝑖,max𝐼≥0ℓ=1,2,…,𝐿. (33)

4. The Simulation

Two examples are considered to demonstrate the effectiveness of the proposed RDMPC. In the first one, the RDMPC is utilized in a two-control area thermal power system, while in the second one, a three-area thermal-hydro power system is considered.

Case 1 (a two-control area thermal power system). A two-control area thermal power system is shown in Figure 3. The parameters used in the simulation is as follows: 𝐾_𝑃1=120Hz/p.u.MW;𝐾_𝑃2𝑇=75Hz/p.u.MW;_𝑃1=20s;𝑇_𝑃2=15s;𝐾_𝑟1=𝐾_𝑟2𝑅=0.5Hz/p.u.MW;₁=2.4Hz/p.u.MW;𝑅₂𝐵=3Hz/p.u.MW;₁=0.425p.u.MW/Hz;𝐵₂𝑇=0.347p.u.MW/Hz;_𝐺1=0.08s;𝑇_𝐺2=0.2s;𝑇_𝑇1=𝑇_𝑇2𝑇=0.3s;_𝑠12=0.545p.u.MW;𝑇_𝑟1=𝑇_𝑟2=10s. (34)
The power system model in Figure 1 with included GRC is shown in Figure 4. In simulations, GRC was set to |Δ𝑃̇_𝑔𝑖|≤𝑟=0.0017 p.u.MW/s.
In real time power system LFC, the power system time constant 𝑇_𝑃 and turbine time constant 𝑇_𝑇 can change frequently. Thus the robustness study is performed by applying intentional changes in these two parameters. The maximum range of parameter variation is chosen to be 40%. The polytopic of uncertain LFC system has four vertices, which are 𝐴_𝑖⁽¹⁾0.6𝑇_𝑃𝑖,0.6𝑇_𝑇𝑖;𝐴_𝑖⁽²⁾1.4𝑇_𝑃𝑖,1.4𝑇_𝑇𝑖;𝐴_𝑖⁽³⁾0.6𝑇_𝑃𝑖,1.4𝑇_𝑇𝑖;𝐴_𝑖⁽⁴⁾1.4𝑇_𝑃𝑖,0.6𝑇_𝑇𝑖. (35)
Under the parameter changes, the performance of the RDMPC is assessed by applying load disturbance. At 𝑡=2 s, a step load disturbance on control area is added to be Δ𝑃_𝑑1=0.01 p.u. Choose the sample time to be 𝑇_𝑠=0.1 s, 𝑆_𝑖=1, and 𝑅_𝑖=0.05.
The proposed RDMPC is compared with two other schemes, for example, the conventional robust centralized MPC, which solves the min-max optimization problem using the centralized model by the formulation of a linear matrix inequality and also with the communicated-based MPC, which utilizes the objective function for local subsystem only. Figures 5, 6, and 7 show the comparison results of the ACE signals, the frequency deviations, and the tie-line power flow, respectively. It is clear that the proposed RDMPC has the best performance, since the MPC controllers cooperate with each other in achieving system-wide objectives. The performance of the robust centralized MPC is quite close to that of the RDMPC, since it is also robust to parameter changes. The only shortcoming of the centralized MPC is its high computation burden. The performance of the communicated-based MPC is the worst, since it can neither realize the cooperation of the subsystems nor adapt to parameter changes.

Figure 3 

The two-control area interconnected thermal power system.

Figure 4 

GRC in power system LFC.

(a)

(b)

Figure 5 

ACE signals in the two subsystems.

(a)

(b)

Figure 6 

The frequency deviations.

Figure 7 

The tie-line power flow between two control areas.

Case 2 (a three-area thermal-hydro power system). The three-control area interconnected power system containing thermal and hydro power plant is showed in Figure 8.
The power system model in Figures 1 and 2 with included GRC in hydro power plant is shown in Figure 9, where |Δ𝑃̇_𝑔𝑖|≤𝑟=0.045 p.u.MW/s.
The parameters used in the simulation are as follows: 𝐾_𝑃1=120Hz/p.u.MW;𝐾_𝑃3𝐾=115Hz/p.u.MW;_𝑃3=75Hz/p.u.MW;𝑇_𝑃1=20s;𝑇_𝑃2𝑇=20s;_𝑃3=15s;𝐾_𝑟1=𝐾_𝑟3𝑅=0.5Hz/p.u.MW;₁=2.4Hz/p.u.MW;𝑅₂𝑅=2.5Hz/p.u.MW;₃=3Hz/p.u.MW;𝐵₁𝐵=0.425p.u.MW/Hz;₂=0.494p.u.MW/Hz;𝐵₃𝑇=0.347p.u.MW/Hz;_𝑅2=0.6s;𝑇₁₍₂₎=48.7s;𝑇₂₍₂₎𝑇=5s;_𝑊2=1s;𝑇_𝐺1=0.08s;𝑇_𝐺3𝑇=0.2s;_𝑇1=𝑇_𝑇3=0.3s;𝑇_𝑟1=𝑇_𝑟3𝑇=10s;_𝑠12=0.545p.u.MW;𝑇_𝑠23=0.545p.u.MW. (36)
Since the maximum range of parameter variation is also chosen to be 40% for hydro power system, the polytope is 𝐴₂⁽¹⁾0.6𝑇_𝑃2;𝐴₂⁽²⁾1.4𝑇_𝑃2. (37)
At 𝑡=1 s, a step load disturbance on control area 1 is added as Δ𝑃_𝑑1=0.01 p.u., and at 𝑡=10 s, a step load disturbance on control area 3 is added as Δ𝑃_𝑑3=−0.01 p.u. Figures 10 and 11 show the comparison results of the ACE signals and the frequency deviations, demonstrating clearly the advantage of the proposed RDMPC.

Figure 8 

The three-control area interconnected thermal-hydro power system.

Figure 9 

GRC in hydro power system.

(a)

(b)

(c)

Figure 10 

ACE signals in three-area thermal-hydro power system.

(a)

(b)

(c)

Figure 11 

The frequency deviation in the three-area thermal-hydro power system.

5. Conclusion

In this paper, a robust distributed MPC scheme for load frequency control of interconnected power system is presented. The overall system consisted of at least two control areas, which either can be thermal-thermal power system or thermal-hydro power system. Each control area has its own polytopic distributed model in order to consider the uncertainty because of parameter variation. A min-max cost function is used for the optimization problem, and the LMI method is involved to solve this problem. The simulation results illustrate the advantage of the proposed RDMPC, due to its cooperative function. Thus it is suitable for LFC of power system, which is large-scale complex system and subject to parameter uncertainty.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grants 60974051, 61273144, and 61203041, Natural Science Foundation of Beijing under Grant 4122071, Chinese National Postdoctoral Science Foundation under Grants 2011M500217 and 2012T50036, and the Doctoral Fund of Ministry of Education of China under Grant 20120036120013.