
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39256486
70539
10.1038/s41598-024-70539-8
Article
A novel hybrid LFC scheme for multi-area interconnected power systems considering coupling attenuation
Wang Bing
Li Yinsheng 15150693623@163.com

Chen Yuquan
https://ror.org/01wd4xt90 grid.257065.3 0000 0004 1760 3465 College of Energy and Electrical Engineering, Hohai University, Nanjing, 210000 CO China
10 9 2024
10 9 2024
2024
14 2112913 5 2024
19 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In this paper, a hybrid load frequency control (LFC) scheme is proposed for multi-area interconnected power systems to decouple the intricate double control objectives, by dividing all subareas into the responsible areas and the free areas. The LFC in the responsible area has the function of regulating both the local frequency and the tie-line power, while the control objective of the LFC in the free area is thus simplified to regulate the local frequency only. Then, addressing the complex network coupling and uncertain dynamics, an integrated LFC controller is proposed for the free areas, which consists of two parts, namely, the coupling attenuation baseline controller and the disturbance compensation controller. The coupling attenuation baseline controller satisfying the predefined bounded L2-Gain condition is derived based on the solution to a multi-player zero-sum differential game. Additionally, a novel generalized integral observer is designed to estimate the system’s integrated disturbance, and the corresponding disturbance compensation controller is derived. After that, the ultimately uniformly bounded (UUB) stability of the integrated LFC controller combining baseline controller and disturbance compensation controller is proven rigorously. Finally, the performance superiority of the proposed hybrid LFC scheme is validated by the simulations in challenging operating modes.

Keywords

Load frequency control
Multi-area interconnected power system
Bounded L2-gain
Zero-sum differential game
Ultimately uniformly bounded
Coupling attenuation
Subject terms

Engineering
Mathematics and computing
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 51777058 62303158 Wang Bing Chen Yuquan http://dx.doi.org/10.13039/501100010014 Six Talent Peaks Project in Jiangsu Province XNY-010 Wang Bing issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

With the development of the smart grid and the interconnection of power grids in various large regions, the power system has increasingly developed into a super large artificial network. In the power system, frequency is an important index for evaluating the power quality. The dynamic load demand will cause frequency fluctuation and power quality decline. To ensure safe power transmission, the load frequency control (LFC)1–3 scheme is essential. In the current power systems, the PID-based LFC schemes are the mainstream due to their simple structure, easy implementation and wide stability margin4. Nevertheless, the PID method has its limitations including poor dynamic response, limited resistance against disturbance, and weak robustness5. Many advanced control theory-based LFC techniques have been proposed to replace the PID for better performances, such as sliding mode control6,7, model predictive control8,9, and intelligence algorithms-based control10–12. While the above methods improve the LFC performances in different aspects, some of them require accurate model information; some lead to high-order controllers; and some introduce complex computation.

Compared with the single-area power system, the multi-area interconnected power system contains intricate network coupling consisting of the AC tie-line power flows among subareas, which makes the LFC more challenging. Some distributed or decentralized LFC technologies are widely considered effective ways to handle network coupling13–17, which have the superiority of good scalability. In13, a multi-agent reinforcement learning approach is proposed for LFC in a decentralized way, the damping performance of frequency response is thus improved. A distributed online adaptive LFC scheme for multi-area power systems is proposed in14, which is combined with an internal model control (IMC) based PID controller15 and shows excellent robustness in the simulation. Some potential applications of distributed cooperative control methods in LFC are discussed and summarized in16. Nevertheless, the above methods also have the shortcoming of high communication and computing costs led by the interaction among power subareas. In18, an optimal control and differential game theory-based network coupling attenuation method is proposed for interconnected linear multi-agent systems, in which a feedback controller satisfying the bounded L2-gain condition for constraining coupling is created. This method can achieve predefined levels of network coupling attenuation with low communication and computing costs, and the results show that it can effectively restrain the diffusion of local faults in the communication network. Unfortunately, accurate model information is required in this method.

In the case that power systems suffer from uncertain dynamics and disturbances, the existing state feedback methods cannot enable steady frequency deviations to converge asymptotically. The disturbance observer (DO) based compensation methods19–21 are widely regarded as effective measures to deal with uncertain dynamics and disturbances. In19, a novel disturbance observer is presented to estimate the unmatched load disturbance of the power system and corresponding feed-forward compensation is designed to attenuate the disturbance effect. In20, a model-based unknown input observer (UIO) is designed to simultaneously estimate the system state and cyberattack disturbances, and the attack estimation is utilized to mitigate the attack impact. In21, a novel extended Kalman filter (EKF) based disturbance observer is designed to estimate inaccessible total disturbance, and the resultant disturbance compensation scheme is presented. However, the above-mentioned disturbance observers have a non-negligible drawback, that is, their designs are model-based, and accurate model information is required to ensure the tracking performance of the observer. In contrast, the active disturbance rejection control (ADRC) is a typical model-free DO-based compensation method, in which the extended state observer (ESO) is utilized to estimate the lumped disturbance. The ADRC and its linear form (LADRC) have been widely applied to LFC systems because of their excellent anti-disturbance performance and low dependency on model information22–25. In24, connections of the power systems and the effects of power contracts are treated as disturbances, the LADRC is investigated for the LFC in deregulated power systems. A LADRC LFC scheme based on Deep Q-Network (DQN) parameters tuning is designed in25, and excellent performances in terms of overshoot and settling time are obtained. However, in ADRC, the dynamics different from the standard integral cascade form is regarded as the lumped disturbance, while the design process is simplified, it also leads to the waste of available model information. On the other hand, only the bounded estimation errors can be guaranteed by ESO and it has poor tracking ability to dynamic disturbance, which will be explained later in the paper.

The LFC of multi-area interconnected power systems in the regulated environment considers two control objectives, namely, (a) regulating the local frequency deviation; (b) regulating the tie-line power flow26. Compared with single-area power systems only requiring control objective (a), extra objective (b) brings considerable complexity to multi-area power systems. This is also the reason why many advanced control methods cannot be directly applied.

Motivation of this paper

Inspired by the above discussion, we can locate the research gaps in three aspects, namely, (1) there is no reported method in attenuating the network coupling of multi-area interconnected power systems; (2) the existing DO-based LFC compensation methods addressing the system disturbance have the issues of high dependency on model information and poor tracking ability to dynamic disturbance; (3) Decoupling the intricate double LFC objectives of multi-area power systems remains open. These gaps in the existing research serve as the driving force for this paper.

In this paper, a hybrid LFC framework is proposed, in which the power areas at two sides of AC tie-lines are responsible for different control objectives and apply different control methods. One side of the AC tie-line follows the PID algorithm to regulate both the local frequency deviation and tie-line power, which is defined as the responsible area. While the other side only focuses on regulating the local frequency deviation, which is defined as the free area. In this way, the control objective of free area is simplified into the same as for single-area power systems. For the free areas, an integrated LFC method based on coupling attenuation and disturbance compensation is proposed. The proposed integrated LFC method is decentralized, and divided into two parts, namely, the coupling attenuation baseline controller and the disturbance compensation controller. Firstly, the full-actuated model of interconnected power systems is derived and formulated into the multi-player differential game. The bounded L2-gain problem for coupling attenuation is defined and its solution is transformed into the Nash equilibrium solution of a multi-player zero-sum game. The equilibrium solution of the multi-player zero-sum game is solved through an equivalent Algebraic Riccati equation (ARE) and the coupling attenuation baseline controller is derived. Then, a novel generalized integral observer is designed for estimating the dynamic disturbances and unmeasurable states, and the disturbance compensation controller is presented.

Paper contribution

The main scientific contributions of this paper are summarized as follows:The proposed hybrid LFC scheme deals with the LFC objectives overlap and redundancy at both sides of AC tie-lines effectively. In this way, the intricate double control objectives are decoupled, and the LFC objective of the free areas is significantly simplified.

The coupling attenuation baseline controller satisfying the bounded L2-gain condition is derived for free areas. Only the nominal model-based equivalent ARE is needed to solve, with low computational cost.

The generalized integral observer is designed. Compared with the ESO24,25, the generalized integral observer has better tracking ability and more relaxed convergence conditions for dynamic disturbances.

An integrated LFC method based on the combination of coupling attenuation baseline controller and disturbance compensation controller is proposed for free areas. The UUB stability of the integrated LFC method is proved rigorously.

Paper structure

This paper is organized as follows: The multi-area power system model considering the uncertain modeling errors is investigated and the hybrid LFC framework is proposed in “Multi-area interconnected power system model” section. The coupling attenuation baseline controller for free areas is demonstrated in “Multi-player zero-sum game based coupling attenuation baseline controller” section. The generalized integral observer is designed and the resultant integrated LFC scheme is presented in “The disturbance compensation controller based on generalized integral observer” section. In “Numerical simulations and discussion” section, the performances of the proposed hybrid LFC scheme are verified through comparative simulations, and the simulation results are discussed in detail. Finally, “Conclusion” section presents the conclusion of the main findings.

Multi-area interconnected power system model

A linear model of a single-area power system involving an integral unit is introduced in27. This model is employed by various LFC schemes to test their feasibility. To analyze the influence of the uncertain dynamics rigorously, the modeling errors including nonlinear dynamics and parameter errors are considered and the full-actuated model of multi-area interconnected power systems is deduced in this section. The symbols used in the multi-area interconnected power systems are summarized in Table 1.Table 1 The symbols used in the multi-area interconnected power systems.

Nomenclature	
Ni	The set of power Areas connected with Area #i	
Δf{i}	Frequency deviation of Area #i (Hz)	
ΔPg{i}	Variation in generator’s output of Area #i (p.u.)	
ΔXg{i}	Variation in governor’s valve position of Area #i (p.u.)	
ΔPd{i}	Active load deviation of Area #i (p.u.)	
ΔPtie{i}	Tie-line power variation of Area #i (p.u.)	
u{i}	Control input signal of the Area #i governor	
TP{i}	Time constants of power system in Area #i(s)	
TT{i}	Time constants of turbine in Area #i(s)	
TG{i}	Time constants of governor in Area #i(s)	
Tij	Synchronization power coefficient between areas i and j	
KP{i}	Plant gain of Area #i	
Λf{i}	Modeling errors in frequency deviation dynamic of Area #i	
ΛP{i}	Modeling errors in generator’s output dynamic of Area #i	
ΛX{i}	Modeling errors in governor’s valve position dynamic of Area #i	
Λtie{i}	Modeling errors in tie-line power variation dynamic of Area #i	
R{i}	Speed regulation parameter of Area #i	

Hybrid load frequency control scheme

The multi-area interconnected power system can be regarded as a complex multi-node coupling network, which can be divided into several subsystems for analysis. The dynamic model of subarea is shown in Fig. 1, in which the exchanged tie-line power ΔPtiei is regarded as the coupling among subareas. The ΔPtiei should be kept to the planned value determined by the bilateral power supply contract. The power systems considered in this paper are in the regulated environment, and the planned value of ΔPtiei is 0. Nevertheless, as long as there is frequency discrepancy among the connected areas, the ΔPtiei would be driven away from 0. Therefore, these interactive systems can propagate and couple the frequency harmonics of every power area. The linear model27 is regarded as a nominal model, and the dynamical equations of each subarea are represented as follows by introducing the modeling errors:1 Δf˙{i}=-1TP{i}Δf{i}+KP{i}TP{i}ΔPg{i}-KP{i}TP{i}ΔPd{i}-KP{i}TP{i}ΔPtie{i}+Λf{i}ΔP˙g{i}=-1TT{i}Pg{i}+1TT{i}ΔXg{i}+ΛP{i}ΔX˙g{i}=-1R{i}TG{i}Δf{i}-1TG{i}ΔXg{i}+1TG{i}u{i}+ΛX{i}

where, subscript {i} denotes the power subsystem, Area #i.Fig. 1 Block diagram of the power system.

The dynamical equation of tie-line power variation is presented as2 ΔP˙tie{i}=∑j∈NiTijΔf{i}-Δf{j}+Λtie{i}

The above dynamic equation shows the energy exchanged among the connected subareas. The synchronization coefficients Tij,∀i,j act as the connected weights in the power network, which can be regarded as the coupling strength among subareas. This dynamical structure reveals how the frequency deviations of different subareas spread and grow among the coupled interconnected power network, in reaction to active load deviations from different subareas.

In the regulated environment, the LFC considers two objectives, namely, the frequency deviation Δf{i} and the exchanged tie-line power ΔPtiei. Nevertheless, there is only one control signal u{i}, which makes it difficult to find suitable control law to achieve simultaneous regulation of Δf{i} and ΔPtiei. In practical power systems, the area control error (ACE)28 based PID method is widely applied. The ACE is a linear combination of Δf{i} and ΔPtieiACE{i}=ΔPtie{i}+b{i}Δf{i}

in which, the weight of the two control objectives can be adjusted by the parameter bi.

The relative orders between Δf{i} and ΔPtiei are not equal to zero, thus, it is difficult to design state feedback control to adjust the closed-loop poles of (1) and (2) simultaneously. As can be seen from Eq. (2), the ΔPtiei can be controlled by both sides of the AC tie-line. Traditionally, the LFC controllers on both sides regard the exchanged tie-line power as one of the control objectives, which also leads to the function overlapping of the LFC controllers on both sides. Inspired by this, a hybrid LFC scheme is considered, in which all the power subareas are divided into two categories depending on the different LFC objectives, namely, the responsible areas and the free areas. The LFC controllers in the responsible areas consider two objectives, that is, stabilizing local frequency and the exchanged tie-line power to the expected value, while the LFC controllers in the free areas consider only one objective, i.e., stabilizing the local frequency to the expected value.

The principle of selecting responsible areas and free areas is summarized that we should select the maximum number of free areas with the premise that each AC tie-line exists at least one responsible area. Taking the 4-area interconnected power system in Fig. 2 as an example, Area #1 and Area #3 should be selected as responsible areas while Area #2 and Area #4 should be selected as free areas based on the proposed hybrid LFC scheme. In this way, both the control objectives on Δf{i} and ΔPtiei is achieved, meanwhile, the LFC in each free area is significantly simplified.Fig. 2 The 4-area interconnected power system topology.

In the hybrid LFC scheme, the IMC based PID method15 is adopted in responsible areas. Additionally, a novel integrated LFC method considering coupling attenuation and disturbance compensation is proposed for free areas. This will be introduced in the subsequent sections.

Full-actuated model of power system

Different from the responsible area in which the ACE is selected as output, the output defined in the free area is the local frequency deviation Δf{i}. This paper aims to develop a decentralized LFC method for the selected free areas to regulate the Δf{i}. The focus of LFC does not include the intermediate variables ΔPg{i} and ΔXg{i}, which is a typical output regulation problem. To control the output more intuitively, it is necessary to transform the dynamic Eq. (1) into the input–output full-actuated model.

By taking the derivative of the first equation in Eq. (1) with the aid of Eqs. (1), (2), we have3 Δf¨{i}=-1TP{i}Δf˙{i}+KP{i}TP{i}-1TT{i}Pg{i}+1TT{i}ΔXg{i}+ΛP{i}-KP{i}TP{i}ΔP˙d{i}-KP{i}TP{i}∑j∈NiTijΔf{i}-Δf{j}+Λtie{i}+Λ˙f{i}

From Eq. (1), we can derive that4 ΔPg{i}=TP{i}KP{i}Δf˙{i}+1KP{i}Δf{i}+ΔPd{i}+ΔPtie{i}-TP{i}KP{i}Λf{i}

Then, combining Eqs. (4), (3), it can be obtained that5 ΔXg{i}=TP{i}TT{i}KP{i}Δf¨{i}+TP{i}+TT{i}KP{i}Δf˙{i}+1KP{i}+TT{i}∑j∈NiTijΔf{i}+ΔPd{i}+ΔPtie{i}+TT{i}ΔP˙d{i}-TT{i}∑j∈NiTijΔf{j}-TT{i}ΛP{i}+TP{i}KP{i}Λf{i}-TT{i}Λtie{i}+TP{i}TT{i}KP{i}Λ˙f{i}

Further, by taking the derivative of Eq. (3), we have6 Δf⃛{i}=-1TP{i}Δf¨{i}+KP{i}TT{i}2TP{i}Pg{i}-KP{i}TT{i}2TP{i}ΔXg{i}-KP{i}TT{i}TP{i}ΛP{i}-KP{i}R{i}TG{i}TT{i}TP{i}Δf{i}-KP{i}TG{i}TT{i}TP{i}ΔXg{i}+KP{i}TG{i}TT{i}TP{i}u{i}+KP{i}TT{i}TP{i}ΛX{i}+KP{i}TP{i}Λ˙P{i}-KP{i}TP{i}ΔP¨d{i}-KP{i}TP{i}∑j∈NiTijΔf˙{i}-Δf˙{j}+Λ˙tie{i}+Λ¨f{i}

Combining Eqs. (4)–(6), the input–output full-actuated dynamical equation is yielded7 Δf⃛{i}=KP{i}TG{i}TT{i}TP{i}u{i}-1TP{i}+1TG{i}+1TT{i}Δf¨{i}-TG{i}+TT{i}+TP{i}TG{i}TT{i}TP{i}+KP{i}∑j∈NiTijTP{i}Δf˙{i}-KP{i}TT{i}∑j∈NiTijTT{i}2TP{i}+R{i}+KP{i}+R{i}KP{i}TT{i}∑j∈NiTijR{i}TG{i}TT{i}TP{i}Δf{i}-KP{i}TP{i}ΔPd{i}+ΔPtie{i}+TG{i}+TT{i}ΔP˙d{i}TG{i}TT{i}+ΔP¨d{i}+Λ~{i}+∑j∈NiTijKP{i}TG{i}+KP{i}TT{i}TG{i}TT{i}TP{i}Δf{j}+KP{i}TP{i}Δf˙{j}

where, Λ~i represents the integrated modeling error as follows8 Λ~{i}=TP{i}TG{i}TT{i}KP{i}Λf{i}+1TG{i}ΛP{i}+1TT{i}ΛX{i}-1TT{i}+1TG{i}Λtie{i}+TP{i}TT{i}KP{i}+TP{i}TG{i}KP{i}Λ˙f{i}+Λ˙P{i}-Λ˙tie{i}+TP{i}KP{i}Λ¨f{i}

It can be seen from Eq. (7) that the output dynamic is affected by the frequency deviation coupling from the adjacent areas, the uncertain load, and the intricate integrated modeling error. Defining the following symbolic variables:

a1{i}=KP{i}TT{i}∑j∈NiTijTT{i}2TP{i}+1+KP{i}TT{i}∑j∈NiTijTG{i}TT{i}TP{i}+KP{i}R{i}TG{i}TT{i}TP{i},a2{i}=TG{i}+TT{i}+TP{i}TG{i}TT{i}TP{i}+KP{i}∑j∈NiTijTP{i},

a3{i}=1TP{i}+1TG{i}+1TT{i},w0{i}=1TG{i}TT{i}ΔPd{i}+1TG{i}TT{i}ΔPtie{i}+TG{i}+TT{i}TG{i}TT{i}ΔP˙d{i}+ΔP¨d{i}+Λ~{i},,wj{i}=KP{i}TG{i}+KP{i}TT{i}TG{i}TT{i}TP{i}Δf{j}+KP{i}TP{i}Δf˙{j}.

Then, the Eq. (7) is rewritten in the following standard form9 x˙{i}=A{i}x{i}+B{i}u{i}+B0{i}w0{i}+∑j∈NiBj{i}wj{i}

where,x{i}=x1{i}x2{i}x3{i}T=Δf{i}Δf˙{i}Δf¨{i}T represents the state vector. The system matrix A{i} and input matrix B{i},B0{i},Bj{i} are written asA{i}=010001-a1{i}-a2{i}-a3{i},B{i}=00KP{i}TG{i}TT{i}TP{i},B0{i}=00-KP{i}TP{i},Bj{i}=00Tij

In this paper, system (9) is regarded as the result of a multi-player game, in which the u{i},w0{i} and wj{i} are the players, and it will be demonstrated in the following subsection.

Remark 1:

The effects of parameter errors, unknown nonlinear dynamics, and the turbine generation rate constraint are summarized in the integrated modeling error Λ~i. The integrated modeling error Λ~i is incorporated into the signal w0{i} regarded as the main uncertainty. The signals wj{i},j∈Nj are regarded as the main coupling effects from the adjacent areas.

Remark 2:

Compared with model (1) including intermediate states ΔXg{i} and ΔPg{i}, model (9) can better reflect the control objective, that is the regulation of frequency deviation Δf{i}. In addition, as a full-actuated system, the model (9) makes the disturbances including uncertain modeling errors, load deviations, and network coupling match with the control input. Thus, the disturbances can be compensated equivalently through the same channel. All of these facilitate the subsequent design of the baseline controller and disturbance compensation controller.

Multi-player zero-sum game based coupling attenuation baseline controller

It can be seen from Eq. (9) that the output dynamics of free areas encounter complex network coupling which comes from the AC tie-line and accompanies the signals w0{i} and wj{i}. The network coupling presents a considerable challenge for LFC. Inspired by this, a coupling attenuation baseline controller for free areas will be designed in this section based on the nominal model information including system matrix A{i} and input matrix B{i},B0{i},Bj{i}.

The Bounded L2-gain problem for coupling attenuation

The bounded L2-gain problem for coupling attenuation is based on the quadratic performance index. For model (9), it is desired to design the baseline control u¯i satisfying the following bounded L2-gain condition with a given level γi>010 ∫0TxiTQixi+Riiu¯i2dt≤β(xi(0))+γi2∫0TRi0w0i2+∑j∈NiRijwji2dt

where,β(·) is a bounded function such that β(0)=0, Qi>0,Rii>0,Ri0>0,Rij>0. γi∗ is defined as the minimum value of γi while the coupling attenuation condition (10) is satisfied. Condition (10) is similar to the definition of H∞ control 29,30 except that it is based on the quadratic performance index and focuses on the attenuation of the multi-party network coupling and uncertain disturbances.

Multi-player zero-sum differential game

The following performance index is defined for each free areas11 Jixi0,u¯i,wi=12∫0∞xiTQixi+Riiu¯i2-γi2Ri0w0i2+∑j∈NiRijwji2dt

where, wi denotes the set of coupling signals w0{i} and wj{i},j∈Ni.

The nominal model information is extracted from (9) and formulated into the following multi-player zero-sum differential game12 x˙{i}=A{i}x{i}+B{i}u¯{i}+B0{i}w^0{i}+∑j∈NiBj{i}w^j{i}

where, w^0{i} and w^j{i},j∈Ni denote the virtual players corresponding to signals w0{i} and wj{i},j∈Ni, which are the opponents against baseline control u¯i and are grouped in the set w^{i}.

According to Theorem 1 in18, the solution of the bounded L2-gain problem for coupling attenuation depicted in Sect. 3.1 is equivalent to the Nash equilibrium solution of the multi-player zero-sum differential game (12) based on the performance index function (11), which can be formulated as13 Vixit=Jixit,u¯i,w^i=12∫t∞xiTQixi+Riiu¯i2-γi2Ri0w^0i2+∑j∈NiRijw^ji2dtVi∗xi=minu¯imaxw^iJixi,u¯i,w^i

In this multi-player zero-sum game, the goal of the baseline controller u¯i is to minimize the value Vixit. On the contrary, the virtual opponents w^i are assumed to maximize the Vixit. This game has a unique solution if a game theoretic saddle point u¯i∗,w^i∗ exists, i.e.,14 Vi∗xi=minu¯imaxw^iJixi,u¯i,w^i=maxw^iminu¯iJixi,u¯i,w^i

Accordingly, the value Vi∗ in the above equation is the value of the zero-sum game and satisfies the following Nash equilibrium condition for all policies of players15 Jixi,u¯i∗,w^i≤Jixi,u¯i,w^i≤Jixi,u¯i,w^i∗

Taking the derivative of Eq. (13) along Eq. (12) with the selected policies u¯i,w^i, we can obtain the following differential equation16 0=12xiTQixi+12Riiu¯i2-γi22Ri0w^0i2+∑j∈NiRijw^ji2+∇ViTA{i}x{i}+B{i}u¯{i}+B0{i}w^0{i}+∑j∈NiBj{i}w^j{i},Vi0=0

where, ∇Vi=∂Vi∂xi∈R3 denotes the gradient vector. Defining the Hamiltonian function as follows17 Hixi,∇Vi,u¯i,w^i=12xiTQixi+12Riiu¯i2-γi22Ri0w^0i2+∑j∈NiRijw^ji2+∇ViTA{i}x{i}+B{i}u¯{i}+B0{i}w^0{i}+∑j∈NiBj{i}w^j{i}

Then, the principle of optimality18 gives18 ∂Hi∂u¯i=0⇒u¯i=-Rii-1BiT∇Vi∂Hi∂w^0i=0⇒w^0i=1γi2Ri0-1B0iT∇Vi∂Hi∂w^ji=0⇒w^ji=1γi2Rij-1BjiT∇Vi,j∈Ni

Defining the Nash equilibrium solution as Vi∗, the coupled Hamilton-Jacobi-Isaacs (HJI) equation31,32 is yielded by combining Eqs. (16), (18):19 Hi(xi,∇Vi∗,u¯i∗,w^i∗)=12xiTQixi+12Riiu¯i∗2-γi22Ri0w^0i∗2+∑j∈NiRijw^ji∗2+∇Vi∗TA{i}x{i}+B{i}u¯i∗+B0{i}w^0i∗+∑j∈NiBj{i}w^ji∗=12xiTQixi+∇Vi∗TA{i}x{i}-12∇Vi∗TBiRii-1BiT∇Vi∗+12γi2∇Vi∗TB0iRi0-1B0iT∇Vi∗+∑j∈Ni∇Vi∗TBjiRij-1BjiT∇Vi∗=0

where,20 u¯i∗=-Rii-1BiT∇Vi∗,w^0i∗=1γi2Ri0-1B0iT∇Vi∗,w^ji∗=1γi2Rij-1BjiT∇Vi∗,j∈Ni

And the Nash equilibrium solution Vi∗ can be obtained by solving the coupled HJI Eq. (19).

Lemma 1.

For any policies u¯i,w^i, the following equation holds.21 Hi(xi,∇Vi∗,u¯i,w^i)=Rii2(u¯i-u¯i∗)2-γi2Ri02(w^0i-w^0i∗)2-∑j∈Niγi2Rij2(w^ji-w^ji∗)2

Proof of Lemma 1.

Substituting ui,u-i for ui∗,u-i∗ in Eq. (19), it can be obtained that22 Hi(xi,∇Vi∗,u¯i,w^i)=12xiTQixi+12Riiu¯i2+12Riiu¯i∗2-γi22Ri0w^0i2+∑j∈NiRijw^ji2-γi22Ri0w^0i∗2+∑j∈NiRijw^ji∗2-12Riiu¯i∗2+γi22Ri0w^0i∗2+∑j∈NiRijw^ji∗2+∇Vi∗TA{i}x{i}+B{i}u¯i+B0{i}w^0i+∑j∈NiBj{i}w^ji+∇Vi∗TB{i}u¯i∗+B0{i}w^0i∗+∑j∈NiBj{i}w^ji∗+∇Vi∗T-B{i}u¯i∗-B0{i}w^0i∗-∑j∈NiBj{i}w^ji∗

Combining Eqs. (19), (22), it can be found that23 Hi(xi,∇Vi∗,u¯i,w^i)=12Riiu¯i2-12Riiu¯i∗2-γi22Ri0w^0i2+∑j∈NiRijw^ji2+γi22Ri0w^0i∗2+∑j∈NiRijw^ji∗2+∇Vi∗TB{i}u¯i+B0{i}w^0i+∑j∈NiBj{i}w^ji+∇Vi∗T-B{i}u¯i∗-B0{i}w^0i∗-∑j∈NiBj{i}w^ji∗

Then, completing the squares in (23) upon the relationship among u¯i∗,w^i∗ and ∇Vi∗ presented in (20) gives (21).

Remark 3:

The virtual players w^{i} corresponding to disturbances and the coupling effects from adjacent areas are assumed to maximize the value function. Nevertheless, the actual disturbances and coupling effects are not necessary to do this. In other words, the baseline control (20) derived from this zero-sum differential game is based on the worst scenario.

Solution of the bounded L2-gain problem for coupling attenuation and the equivalent algebraic Riccati equation

The following Theorem 1 shows that the solution of the coupled HJI Eq. (19) is equivalent to the solution of the bounded L2-gain problem defined in section III. A.

Theorem 1.

Selecting γi≥γi∗ and supposing that the coupled HJI Eq. (19) has a smooth positive definite solution Vi∗>0, the baseline control policy is selected as u¯i∗ given by the first equation in (20) Then, the bounded L2-gain condition (10) holds for all w^i∈L2[0,∞).

Proof of Theorem 1.

According to Lemma 124 Hi(xi,∇Vi∗,u¯i,w^i)=12xiTQixi+12Riiu¯i2-γi22Ri0w^0i2+∑j∈NiRijw^ji2+dVi∗dt=Rii2(u¯i-u¯i∗)2-γiRi02(w^0i-w^0i∗)2-∑j∈NiγiRij2(w^ji-w^ji∗)2

where, dVi∗dt is the derivative of Vi∗ along Eq. (12). Selecting the baseline control policy as u¯i=u¯i∗, we can obtain that25 12xiTQixi+12Riiu¯i2-γi22Ri0w^0i2+∑j∈NiRijw^ji2+dVi∗dt=-γiRi02(w^0i-w^0i∗)2-∑j∈NiγiRij2(w^ji-w^ji∗)2≤0

Integrating Eq. (25) on interval 0,T yields26 ∫0T12xiTQixi+12Riiu¯i2-γi22Ri0w^0i2+∑j∈NiRijw^ji2dt+Vi∗xiT-Vi∗xi0≤0

where, Vi∗ is smooth positive definite solution, i.e., Vi∗xiT≥0, one has27 ∫0TxiTQixi+Riiu¯i2dt≤γi2∫0TRi0w0i2+∑j∈NiRijwji2dt+12Vi∗xi0

Hence, the bounded L2-gain condition (10) for coupling attenuation is satisfied for all w^i∈L2[0,∞). The proof of Theorem 1 is completed.

It can be seen from the above results that the coupling attenuation baseline controller satisfying the condition (10) can be derived by solving the coupled HJI Eqs. (19), (20). It will be shown that the coupled HJI Eq. (19) is equivalent to an ARE.

The solution of Eq. (19) is defined as the quadratic value function Vi∗=xiTPixi, where Pi is the symmetric positive definite matrix to be solved. Combining with Eq. (20), we can obtain that28 u¯i∗=-2Rii-1BiTPixiw^0i∗=2γi2Ri0-1B0iTPixiw^ji∗=2γi2Rij-1BjiTPixi,j∈Ni

Then, substituting Eq. (28) and Vi∗=xiTPixi into (19) yields29 2γi2xiTPiB0iRi0-1B0iTPixi+∑j∈NixiTPiBjiRij-1BjiTPixi+xiTPiAi+AiTPixi+12xiTQixi-2xiTPiBiRii-1BiTPixi=0

The above equation is equivalent to30 PiAi+AiTPi+12Qi-2PiBiRii-1BiTPi+2γi2PiB0iRi0-1B0iTPi+∑j∈NiPiBjiRij-1BjiTPi=0

Define the following extended matrix‵Ri=diagRii2,-γi2Ri02,-γi2Rij12,-γi2Rij22,⋯,-γi2RijDi2‵Bi=BiB0iBj1iBj2i⋯BjDii,j1,j2,⋯,jDi∈Ni

where, Di represents the number of power areas connected with Area #i. Then, Eq. (30) is rewritten as the following ARE31 PiAi+AiTPi+12Qi-Pi‵Bi‵Ri-1‵BiTPi=0

After solving Eq. (31), the coupling attenuation baseline controller u¯i∗ can be obtained through the first equation in Eq. (28). It can be seen from Eq. (31) that only the nominal model information is required in the design process.

Refer to Theorem 2 in18, the system (9) could achieve ultimately uniformly asymptotic stability only if the coupling signals set wi ultimately stabilizes at the origin. However, the coupling signals set wi is generated by modeling errors, load deviation, and frequency deviations of adjacent areas and it is dynamic and complex. In addition, the baseline controller requires the first and second derivatives of Δfi as state feedback, nevertheless, they are not available by the sensor directly. In summary, it is necessary to design a specific observer for estimating the unavailable states and dynamic disturbance, which will be investigated in the next section.

Remark 4:

In the design process of coupling attenuation baseline controller for practical LFC, Qi, Rii, Ri0,Rij and γi should be selected based on detailed engineering performance requirements. If a high convergence speed is required for frequency deviation, the Qi with large eigenvalues should be selected; if low control energy consumption is emphasized, the large Rii should be selected; the coupling attenuation level can be adjusted by Rij and γi. Additionally, Qi, Rii, Ri0,Rij and γi must satisfy that the coupled HJI Eq. (19) has a positive definite solution Vi∗>0.

The disturbance compensation controller based on generalized integral observer

To eliminate the steady-state error, the DO-based disturbance compensation method is investigated in this section. According to the separation principle, the disturbance compensation controller is designed independently of the baseline controller. It can be seen from Eqs. (8), (9) that the effects including modeling errors, load deviation, and coupling of adjacent areas are dynamic and complex. The most common disturbance observer is the linear extended state observer (LESO), which can only ensure that the estimation errors are ultimately bounded rather than asymptotically convergent for dynamic disturbance. For the dynamic disturbance of interconnected power systems, in this section, the generalized integral observer is developed based on LESO, and the resultant disturbance compensation controller is designed.

For the system (9), the following integrated disturbance including effects of modeling errors, load deviation and the power coupling from adjacent areas is defined32 d{i}=-KP{i}TP{i}w0{i}+∑j∈NiTijwj{i}

By expanding the integrated disturbance and its derivative into the new states, the following extended system is obtained based on Eq. (9)33 x¯˙{i}=A¯{i}x¯{i}+B¯{i}u{i}+F¯{i}y{i}=C¯x¯{i}

where, x¯{i}=Δf{i}Δf˙{i}Δf¨{i}d{i}d˙{i}T and y{i}=Δf{i} represent the extended state vector and output of system (33). The A¯{i}, B¯{i}, F¯{i} and C¯{i} are written as

A¯{i}=0100000100-a1{i}-a2{i}-a3{i}100000100000,B¯{i}=00KP{i}TG{i}TT{i}TP{i}00T,F¯{i}=0000d¯¨{i}T,C¯{i}=10000T

Then, the following state observer is designed34 x¯^˙{i}=A¯{i}x¯^{i}+B¯{i}u{i}+L{i}y{i}-y^{i}y^{i}=C¯x¯^{i}

where, x¯^{i}=Δf^{i}Δf^˙{i}Δf^¨{i}d^{i}d^˙{i}T represents the estimation of the extended state vector. L{i}=β1{i}β2{i}β3{i}β4{i}β5{i}T represents the gain matrix to be designed. All the poles of the observer are placed on -ω{i} and we can derive that35 β1=5ω{i}-a3{i},β2=10ω{i}2-a2{i}-5a3{i}ω{i}+a3{i}2β3=10ω{i}3-a1{i}-5a2{i}ω{i}+a2{i}a3{i}-10a3{i}ω{i}2+a2{i}a3{i}+5a3{i}2ω{i}-a3{i}3β4=5ω{i}4,β5=ω{i}5

The above observer is called the generalized integral observer. Its main difference from the LESO is the integral chain introduced at the end, and the steady-state estimation errors are effectively eliminated in this way.

The estimation errors vector of the generalized integral observer is defined as x¯~{i}=x¯{i}-x¯^{i} and the following errors dynamic equation is yielded by subtracting Eq. (33) from Eq. (35)36 x¯~˙{i}=A¯{i}-L{i}C¯x¯~{i}+F¯{i}

Lemma 2.33

Let Φt∈C1 is a continuous differentiable positive function with a bounded initial value Φ0. If Φ˙t≤-αΦt+β is held, where α and β are positive constants, then, there exists the following result:37 Φt≤e-αtΦ0+βα1-e-αt

Considering the engineering environment of practical power systems, the following assumption is made.

Assumption:

The second derivative of the integrated disturbance is bounded and satisfies that38 F¯{i}TF¯{i}=d¨{i}2≤ξ{i}

Theorem 2.

The poles of the generalized integral observer (34) are selected such that ω{i}>12. If the above Assumption is true, the estimation errors vector of the generalized integral observer would be UUB and satisfy that39 x¯~{i}Tx¯~{i}≤2e-2ω{i}-1tx¯~{i}Tx¯~{i}0+ξ{i}2ω{i}-11-e-2ω{i}-1t

Proof of Theorem 2.

The following Lyapunov candidate is selected40 VGIO{i}=12x¯~{i}Tx¯~{i}

The derivative of the above Lyapunov candidate along (36) is computed as follows41 V˙GIO{i}=x¯~{i}Tx¯~˙{i}=x¯~{i}TA¯{i}-L{i}C¯x¯~{i}+F¯{i}≤-ω{i}x¯~{i}Tx¯~˙{i}+12x¯~{i}Tx¯~˙{i}+12F¯{i}TF¯{i}≤-2ω{i}-12VGIO{i}+ξ{i}2

The proof is then completed according to Lemma 2.

Remark 5.

It can be seen from Eq. (39) that the error vector can be ultimately forced to the following bounded neighborhood of the origin

ΩGIO{i}=x¯~{i}x¯~{i}Tx¯~{i}≤ξ{i}2ω{i}-1.

This means that the estimation accuracy can be improved by increasing ω{i}. However, due to the unavoidable measurement noise, ω{i} cannot be increased without limit. In addition, the error vector would ultimately converge, with the premise that the second derivative of the integrated disturbance stables at the origin ultimately. Compared with the LESO which requires the ultimate stability of the integrated disturbance’s first derivative at origin (it is impossible for dynamic integrated disturbance), the inclusiveness of the proposed generalized integral observer is better.

The state estimation vector x^{i}=Δf^{i}Δf^˙{i}Δf^¨{i}T is applied to the coupling attenuation baseline controller, it can be obtained that42 u¯^i∗=-2Rii-1BiTPix^i

Based on the disturbance estimation x¯^{i}4=d^{i} and system (9), the disturbance compensation controller is designed as43 u~{i}=-TG{i}TT{i}TP{i}KP{i}d^{i}

Combining the baseline controller (42) and the disturbance compensation controller (43) yields the integrated LFC controller (44) for the free areas,44 ui=u¯^i∗+u~i=-2Rii-1BiTPi-TG{i}TT{i}TP{i}KP{i}0x¯^i

and its block diagram is shown in Fig. 3.Fig. 3 Block diagram of the integrated LFC controller for the free areas.

Theorem 3.

The integrated LFC controller (44) can guarantee that the frequency deviation state vector x{i}=Δf{i}Δf˙{i}Δf¨{i}T of system (9) is UUB and satisfies that45 x{i}TP{i}x{i}≤e-λminηi-1λmaxPitx{i}TP{i}x{i}0+λmaxPiϑiλminηi-11-e-λminηi-1λmaxPit

where,ηi=12Qi+2γi21Ri0PiB0iB0iTPi+∑j∈Ni1RijPiBjiBjiTPiϑi=max2λmaxPiBiBiTPiRii,λmax2Pix¯~iTx¯~imax

x¯~iTx¯~imax represents the maximum value of x¯~iTx¯~i. λmax(·) and λmin(·) represent the maximum and minimum eigenvalue of the matrix, respectively.

Proof of Theorem 3.

The solution Vi∗=xiTPixi of Eq. (31) is selected as the Lyapunov candidate. According to Lemma 1 and Eq. (17), the derivative of the Vi∗ along system (9) with the LFC control scheme (44) is46 V˙i∗=∇Vi∗TA{i}x{i}+B{i}u¯^i∗+u~i+B0{i}w0i+∑j∈NiBj{i}wji=∇Vi∗TA{i}x{i}+B{i}u¯^i∗-d^{i}+B0{i}w0i+∑j∈NiBj{i}wji=∇Vi∗TA{i}x{i}+B{i}u¯^i∗+B0{i}w~0i+∑j∈NiBj{i}w~ji

where, w~0i=w0i-w⌢0i,w~ji=wji-w⌢ji,j∈Ni. w⌢0i,w⌢ji,j∈Ni is any combination satisfying the following equation47 -KP{i}TP{i}w⌢0{i}+∑j∈NiTijw⌢j{i}=d^{i}

Following the Lemma 1, we can obtain that48 V˙i∗=∇Vi∗TA{i}x{i}+B{i}u¯^i∗+B0{i}w~0i+∑j∈NiBj{i}w~ji=Hi(xi,∇Vi∗,u¯^i∗,w~0i,w~ji)-12xiTQixi-12Riiu¯^i∗2+γi22Ri0w~0i2+∑j∈NiRijw~ji2=-12xiTQixi-12Riiu¯^i∗2+Rii2(u¯^i∗-u¯i∗)2-γiRi02(w~0i-w^0i∗)2-∑j∈NiγiRij2(w~ji-w^ji∗)2+γi22Ri0w~0i2+∑j∈NiRijw~ji2

Combining Eqs. (28), (42) yields that49 V˙i∗≤-12xiTQixi-γi22Ri0w^0i∗2+∑j∈NiRijw^ji∗2+2Riixi-x^iTPiBiBiTPixi-x^i+γi2Ri0w~0iw^0i∗+∑j∈NiRijw~jiw^ji∗

Equation (49) is equivalent to50 V˙i∗≤2Riixi-x^iTPiBiBiTPixi-x^i+2w~0iB0iT+∑j∈Niw~jiBjiTPixi-xiTQi2+2γi21Ri0PiB0iB0iTPi+∑j∈Ni1RijPiBjiBjiTPixi

Following the Eq. (47), we can obtain that51 V˙i∗≤2Riixi-x^iTPiBiBiTPixi-x^i-xiTQi2+2γi21Ri0PiB0iB0iTPi+∑j∈Ni1RijPiBjiBjiTPixi+200di-d^iPixi

Further, employing the Young’s inequality, we can obtain that52 V˙i∗≤2Riixi-x^iTPiBiBiTPixi-x^i+λmax2Pidi-d^i2+xiTxi-xiTQi2+2γi21Ri0PiB0iB0iTPi+∑j∈Ni1RijPiBjiBjiTPixi≤-λminηi-1xiTxi+max2λmaxPiBiBiTPiRii,λmax2Pix¯~iTx¯~i

From Theorem 2, the generalized integral observer errors vector is UUB, thus53 V˙i∗≤-λminηi-1xiTxi+max2λmaxPiBiBiTPiRii,λmax2Pix¯~iTx¯~imax

we can derive from Eq. (53) that54 V˙i∗≤-λminηi-1λmaxPixiTPixi+ϑi

One can deduce the result of Eq. (45) by combining Eq. (54) and Lemma 2, and the proof of Theorem 3 is completed.

From Theorem 2 and Theorem 3, under the control of the proposed integrated LFC controller (44), the system (9) will be ultimately forced to the following bounded neighborhood of the origin55 Ω{i}=x{i}x{i}TPix{i}≤ciλmaxPiξ{i}λminηi-12ω{i}-1

where, ci=max2λmaxPiBiBiTPiRii,λmax2Pi.

The steady-state control accuracy is closely related to the generalized integral observer. The system (9) would converge if the error vector of the generalized integral observer is ultimately stable at the origin, whose premise is d¨{i}→0,ast→∞. This condition is reasonable in practical multi-area interconnected power systems. Therefore, the integrated LFC controller (44) can achieve the LFC objective of free area.

Numerical simulations and discussion

The proposed hybrid LFC scheme is applied to multi-area interconnected power systems configured with challenging operating conditions, in which the responsible areas adopt the IMC based PID controller15 and the free areas adopt the proposed integrated LFC controller (44). The four-area power system illustrated in Fig. 2 is considered in the numerical simulation, in which Area #1 and Area #3 are selected as responsible areas while Area #2 and Area #4 are selected as free areas. Two LFC schemes are used as comparative cases to validate the performance of the proposed hybrid LFC scheme, namely,

Comparative Case 1: the IMC based PID LFC controller is utilized in all the power areas, which is represented by “IMC based PID”;

Comparative Case 2: the generalized integral observer is replaced by the model-free LESO on the basis of the proposed hybrid LFC scheme, which is represented by “LESO based hybrid LFC scheme”.

Remark 6:

Comparative Case 1 is designed to verify the performance of the proposed hybrid LFC framework and the coupling attenuation baseline controller, the main distinction between the proposed hybrid LFC scheme and comparative Case 1 lies in the LFC controller of free areas, Area #2 and Area #4; Comparative Case 2 is designed to verify the performance of the proposed generalized integral observer, the main distinction between the proposed hybrid LFC scheme and comparative Case 2 lies in the equipped disturbance observer. Additionally, to further validate the performance superiority of the proposed LFC scheme, we will introduce extra comparative simulation with other advanced robust LFC schemes, such as the model-free LADRC scheme and the improved LADRC based on DQN parameters tuning, on the basis of IEEE 39-Bus testing system.

To test the proposed hybrid LFC scheme more comprehensively, the following operating modes are investigated: (1) dynamic load variations; (2) dynamic load variations, with parameter mismatch; (3) dynamic load variations, with generation rate constraint on turbine; (4) frequency failure occurs in Area#1, with lossy connection topology. Simulation parameters of the four-area power system are listed in Table. 2 and parameters of the proposed hybrid LFC scheme are listed in Table. 3. Notably, the parameters tuning rule of the proposed integrated LFC controller is available in Remark 4 and Remark 5. The parameters of the compared IMC based PID scheme refer to15, while the parameters of the second compared LFC scheme are all consistent with the proposed hybrid LFC scheme except for the LESO.Table 2 Simulation parameters of the four-area power system.

Parameter	Area #1	Area #2	Area #3	Area #4	
TP(s)	20	20.5	19.7	19.5	
TT(s)	0.3	0.33	0.32	0.27	
TG(s)	0.08	0.08	0.08	0.08	
KP(s)	120	120	119.5	121	
R	2.4	2.4	2.4	2.4	
Tij	T12=0.545,T13=0.7,T14=0.35,T23=0.9,T34=0.25	

Table 3 Simulation parameters of the proposed hybrid LFC scheme.

Responsible Area #1	Free Area #2	Responsible Area #3	Free Area #4	
IMC based PID with parameters referred to15	Q2=diag(30,10,10)R22=1

R20=1

R21=1

R23=1

γ2=0.0244

ω{2}=30

	IMC based PID with parameters referred to15	Q4=diag(30,10,10)R44=1

R40=1

R41=1

R43=1

γ4=0.0241

ω{4}=30

	

Scenario 1. dynamic load variations

In this scenario, the closed-loop performance is tested in the presence of dynamic load deviations. To reflect the dynamic convergence ability, the initial frequency deviations of the four power regions are set asΔf{1}0=0.01,Δf{2}0=0.02,Δf{3}0=0.03,Δf{4}0=0.04Hz

Then the asymmetric changes in the active load demands ofΔPd{1}=0.01,ΔPd{2}=0.02,ΔPd{3}=0.03,ΔPd{4}=0.04p.u.

are considered. The frequency deviations of the four areas are shown in Figs. 4, 5, the tie-line exchange power flows are shown in Figs. 6, 7, in which the results are decomposed into the initial stage and the stage encountering dynamic load variations. Whether in the initial stage or the stage encountering dynamic load variations, the frequency deviations and tie-line exchange power flows of the four areas are quickly driven back to zero. It is observed that the proposed hybrid LFC scheme can ensure the tie-line exchange powers of the free areas converge to the planned value, even if the LFC controllers in free areas do not have the tie-line exchange power regulsation function. The proposed hybrid LFC scheme achieves the best damping for frequency and tie-line exchange power in all four areas. In terms of settling times and fluctuation ranges in the frequency deviations, the performance advantages of Area #2 and Area #4 are particularly evident. This also demonstrates that the proposed integrated LFC method for free areas has excellent perturbation suppression ability. As can be seen from Figs. 5, 7, it is difficult to eliminate the steady-state errors quickly by traditional LESO.Fig. 4 Frequency deviation responses of the four-area power system in the initial stage ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Fig. 5 Frequency deviation responses of the four-area power system in the stage encountering dynamic load variations ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Fig. 6 Tie-line exchange power flows of the four-area power system in the initial stage ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Fig. 7 Tie-line exchange power flows of the four-area power system in the stage encountering dynamic load variations ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

To verify the effect of γi, the compared case with γ2=0.2 in Area #2 is introduced. The frequency deviation responses encountering dynamic load variations are shown in Fig. 8. We can find that smaller γi reflects stronger attenuation ability for load perturbations. In practical engineering, we can gradually search for smaller γi to improve the performance under the premise that there exists a positive definite solution for ARE (31).Fig. 8 Frequency deviation responses of Area #2 with different γi.

Scenario 2. dynamic load variations, with parameter mismatch

To test the dependence on the power system parameters information, all parameters of Area #2 are increased by 20%, while all parameters of Area #4 are reduced by 20%. In this scenario, the serious load deviations ΔPd{2}=0.1,ΔPd{4}=0.1p.u. in free areas are investigated.

The frequency deviations and tie-line exchange power flows are shown in Figs. 9, 10. First, the proposed hybrid LFC scheme exhibits the best performances in terms of settling times and fluctuation range, compared with the IMC based PID scheme and LESO based hybrid LFC scheme. This proves that the proposed hybrid LFC scheme has a high tolerance for parameter mismatch. Second, similar to scenario 2, the LESO based hybrid LFC scheme shows the longest settling time, although its frequency deviations and tie-line exchange power flows can converge to an arbitrarily small neighborhood of origin by increasing the simulation time. This reflects the value of the proposed generalized integral observer, that is the stronger tracking performance than LESO to dynamic disturbances. Considering the shortcomings of LESO based hybrid LFC scheme, which have been proved in scenario 1 and scenario 2, subsequent simulations will not introduce it for comparison.Fig. 9 Frequency deviation responses of the four-area power system with inaccurate parameters ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Fig. 10 Tie-line exchange power flows of the four-area power system with inaccurate parameters ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Scenario 3. dynamic load variations, with generation rate constraint on turbine

The generation rate constraint of 0.1 p.u. for each turbine is considered in this scenario. Then, the load deviations ΔPd{2}=0.01,ΔPd{4}=-0.01p.u. are introduced. The frequency deviations and tie-line exchange power flows are shown in Figs. 11, 12. The results show that a serious overshoot caused by the generation rate constraint occurs in the IMC based PID scheme, and it is oscillatory in Area #1 and Area #3. In contrast, the proposed hybrid LFC scheme is less affected by the generation rate constraint, and good damping performance is achieved. Even though the deterioration of the PID controllers in the responsible areas extends the settling times of the tie-line exchange power in free areas, the proposed hybrid LFC scheme still guarantees convergence accuracy at a level of 10−6.Fig. 11 Frequency deviation responses of the four-area power system with generation rate constraint on each turbine ((a) Area #1; (b) Area #2; (c) Area #3; (d) Area #4).

Fig. 12 Tie-line exchange power flows of the four-area power system with generation rate constraint on each turbine. ((a) Area #1; (b) Area #2; (c) Area #3;(d) Area #4).

Scenario 4. Frequency failure occurs in power Area#1, with lossy connection topology

An aggressive scenario is introduced to test the robustness in this section refer to14. This scenario considers that the AC tie-line between Area#1 and Area#3 is disconnected, as shown in Fig. 13. Moreover, a frequency fault occurs in Area#1, in which its frequency deviation is fixed at 0.1 Hz.Fig. 13 The four-area interconnected power system with lossy connection topology.

The frequency deviations of the other 3 areas are described in Fig. 14. It is shown that, in the IMC based PID scheme, the frequency deviations of the other three areas are severely affected by the faulted Area#1 and eventually deviate from the origin. In contrast, in the proposed hybrid LFC scheme, the other 3 areas are less affected by the frequency fault in Area#1, and their frequency deviations are eventually stabilized at the origin. The selected free areas guarantee the stability of local frequency, which effectively blocks the propagation of frequency fault in Area#1. From this result, the proposed hybrid LFC scheme shows considerable robustness to frequency faults and lossy connections in local areas. The proposed hybrid LFC scheme can prevent the diffusion of local frequency faults and provide more time to remove the faults, which is significant for practical power systems.Fig. 14 Frequency deviation responses of the other 3 areas when frequency fault occurs in Area#1. ((a) Area #2; (b) Area #3; (c) Area #4).

To test the robustness of the coupling attenuation baseline controller to local faults, the performance comparison with different γi is introduced to this scenario. In this simulation, the Qi of the coupling attenuation controller is modified as Q2=Q4=diag30,1,1 to increase the weight of the frequency deviation. The two sets of simulations adopt γ2=0.0108,γ4=0.0109 and γ2=0.2,γ4=0.2, respectively. The comparative results are described in Fig. 15, which shows that the lower γi corresponds to the stronger damping performance against the frequency fault.Fig. 15 Frequency deviation responses of the other 3 areas when frequency fault occurs in Area#1, with different γi. ((a) Area #2; (b) Area #3; (c) Area #4).

IEEE 39-bus test system

To further verify the effectiveness of the proposed hybrid LFC scheme, a case study based on the IEEE 39-Bus testing system is implemented. The single-line 39-Bus test system is depicted in Fig. 16, which consists of 34 transmission lines, 12 transformers, 19 loads and 10 generators. The testing system is divided into three areas according to20, and all the generators participate in the LFC of the local areas based on the participation factor ci,k of the generator Gk in Area #i, where the sum of ci,k in Area #i equals 1. The LFC parameters of the testing systems are shown in Table. 4, and Area#1 and #3 are selected as the responsible areas while Area#2 is selected as the free area. In the design of LFC scheme, the generators in each area are equivalent to a single one according to the simplification method in34. To highlight the superiority of the proposed generalized integral observer, the Deep Q-Network (DQN)-ESO based LADRC scheme (DQN-ESO based LADRC) and the fixed parameter ESO based LADRC scheme (FESO based LADRC) are introduced for comparison. Additionally, the integral of deviation absolute value (IDA) is employed as the criteria for evaluating the LFC performances, which are shown in (56) and (57).56 IDAΔf{i}=∫tIEΔf{i}dt

57 IDAΔPtie{i}=∫tIEΔPtie{i}dt

where, tI and tE represent the initial and end times of simulation, respectively.Fig. 16 The IEEE 39-Bus testing system.

Table 4 The LFC parameters of the testing systems.

	Area#1	Area#2	Area#3	
	G1	G2	G3	G4	G5	G6	G7	G8	G9	G10	
TT	0.3	0.3	0.3	0.35	0.35	0.35	0.35	0.4	0.4	0.4	
TG	0.1	0.1	0.1	0.17	0.17	0.17	0.17	0.2	0.2	0.2	
R	0.8	0.8	0.8	0.8	0.8	0.8	0.8	0.8	0.8	0.8	
KP	120	120	120	128	128	128	128	120	120	120	
TP	20	20	20	21.5	21.5	21.5	21.5	20.5	20.5	20.5	
ci,k	0.33	0.33	0.33	0.25	0.25	0.25	0.25	0.33	0.33	0.33	
Tij	T12=0.4166,T13=1.3272,T14=0.2959	

Area#1 and Area#2 suffer from the stepping load disturbances ΔPd{1}=0.02,ΔPd{2}=0.1p.u. at Bus− < 31 > and Bus− < 20 > , respectively. The frequency deviations and the tie-line exchange power flows are shown in Figs. 17, 18, and the resultant IDA is listed in Table. 5. One can find that the LFC schemes equipped with corresponding DO and disturbance compensations measure possess superior dynamic response performance by comparing with the IMC based PID scheme. Additionally, among the three DO-based LFC schemes, the proposed hybrid LFC scheme based on generalized integral observer shows the best damping characteristic to the load disturbances, which can be seen from the settling times and fluctuation ranges of frequency and tie-line power. A similar conclusion is drawn based on the results of Table.5, in which, the lower IDA corresponds to the better dynamic performance for regulating local frequency and tie-line power.Fig. 17 The frequency deviations of IEEE 39-Bus testing system ((a) Area #1; (b) Area #2; (c) Area #3).

Fig. 18 The tie-line power of IEEE 39-Bus testing system ((a) Area #1; (b) Area #2; (c) Area #3).

Table 5 The IDA of compsarative case studies.

	The proposed hybrid LFC scheme	DQN - ESO based LADRC scheme	FESO based LADRC scheme	IMC based PID scheme	
	IDAΔf{i}	IDAΔPtie{i}	IDAΔf{i}	IDAΔPtie{i}	IDAΔf{i}	IDAΔPtie{i}	IDAΔf{i}	IDAΔPtie{i}	
Area#1	7.75 × 10−3	3.38 × 10−3	1.42 × 10−2	5.87 × 10−3	2.74 × 10−2	9.43 × 10−3	2.55 × 10−2	1.11 × 10−2	
Area#2	2.95 × 10−3	3.53 × 10−3	1.55 × 10−2	7.88 × 10−3	2.80 × 10−2	2.38 × 10−2	5.45 × 10−2	2.64 × 10−2	
Area#3	7.53 × 10−3	2.30 × 10−3	1.32 × 10−2	8.56 × 10−3	2.75 × 10−2	1.65 × 10−2	2.55 × 10−2	1.53 × 10−2	

Conclusion

A hybrid LFC scheme is proposed for multi-area power systems in a regulated environment, where the subareas are divided into free areas and responsible areas. The selected responsible areas are responsible for regulating the local frequency and tie-line exchange power, while the free areas only focus on regulating the local frequency. On this basis, an integrated LFC controller combining the coupling attenuation baseline controller with a generalized integral observer is designed for the free areas. The UUB stability of the proposed integrated LFC controller is proven. The case studies are implemented on a four-area power system and the IEEE 39-bus testing system. The simulation results show that the proposed hybrid LFC scheme can stabilize the local frequency and tie-line exchange power of each area at the planned value effectively, and excellent damping performance against aggressive conditions is achieved. Additionally, the hybrid LFC scheme can block the transmission of local frequency faults effectively, which is significant in improving the risk resistance ability.

Future research will be devoted to the improvement of the interconnected power systems topology with the proposed hybrid LFC scheme.

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No.51777058 and No.62303158, and the Six Talent Peak Projects in Jiangsu Province under Grant XNY-010.

Author contributions

Conceptualization: Yinsheng Li, Bing Wang; Methodology: Yinsheng Li, Bing Wang; Validation: Yinsheng Li, Yuquan Chen; Software: Yinsheng Li, Bing Wang; Formal analysis and investigation: Bing Wang; Writing—original draft preparation: Yinsheng Li; Writing—review and editing: Yinsheng Li, Bing Wang; Funding acquisition: Bing Wang, Yuquan Chen; Supervision: Bing Wang, Yuquan Chen. All authors have read and agreed to the published version of the manuscript.

Funding

It is declared that the particular research has been funded by the National Natural Science Foundation of China under Grant No.51777058 and No.62303158, and the Six Talent Peak Projects in Jiangsu Province under Grant XNY-010.

Data availability

The data that support the findings of this paper are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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