Protocol-Based Inhomogeneous Steady-State Probability Power System Load Frequency Control Method

Abstract

The present invention discloses a method for load frequency control of a multi-region interconnected power system based on non-homogeneous dwell probabilities, including the steps of: 1) establishing a model of an interconnected multi-region power system with non-homogeneous dwell probabilities; 2) designing a novel event-triggered protocol based on dynamic quantization; 3) designing a proportional-integral controller based on Lyapunov theory to obtain sufficient conditions for the stability of the multi-region interconnected power system, ensuring the exponential mean-square stability of the system and meeting performance indicators. The proportional-integral controller gains are solved through the LMI toolbox of MATLAB, and the simulation results demonstrate the effectiveness of the proposed method. The control method of the proportional-integral controller proposed in the present invention under the event-triggered protocol can save network resources while ensuring the stochastic stability of the system model.

Description

Technical Field

[0001] The present invention relates to the field of network control of interconnected multi - area power systems, and specifically to a protocol - based non - homogeneous dwell - time probability power system load frequency control method. Background Art

[0002] In the past few decades, the rapid development of networked control systems has led to their widespread application, including multi - agent, power, and other systems. A typical national power system has long been regarded as an infrastructure crucial to national security in the world, and its stability affects the country's social economy. However, in recent years, the United States and Europe have experienced several large - scale power outages, causing direct losses of billions of dollars. The frequent occurrence of power outages has exposed potential problems in the current mathematical models and analysis methods of power systems, which also pose challenges to frequency stability. To protect the power system prone to sudden changes, frequency control is mainly used as an effective technique to maintain system performance. Therefore, load frequency control plays an indispensable role in power systems to adjust the frequency to the desired value. Following this trend, many control techniques have been applied to handle single / multi - area load frequency control in power systems.

[0003] In recent years, with the rapid development of computer networks and technology, on the one hand, it can promote information exchange; on the other hand, it has also led to the fact that power systems are vulnerable to attacks. Network attacks may cause instability in the interconnected multi - area power system, so it is urgent to study network security under network attacks.

[0004] In the load frequency control of multi - area interconnected power systems, data such as frequency deviations in each area and tie - line deviations between different areas are transmitted to the remote dispatching control center through an open network, serving as the basis for the control center to conduct frequency regulation. The open network is not only vulnerable to network attacks, but its network bandwidth is also limited. The influx of data such as weather and system electrical measurement parameters into the network will undoubtedly cause greater bandwidth pressure on the communication channel. Under this background, many effective protocols have been constructed in the prior art to standardize data transmission, such as sampled - data, cyclic, event - triggered, etc. Among them, the event - triggered protocol is the most widely used. In the framework of the event - triggered protocol, the transmitted data will be discarded unless the preset event - triggering condition is met, thus avoiding waste of resources. So far, many effective event - triggered protocols have emerged, including static, dynamic, memory - based, and distributed. Although some valuable results on event - triggered protocols have been reported, many network - induced phenomena, including quantization effects, have not been considered, which may lead to waste of resources in a certain context. Summary of the Invention

[0005] The object of the present invention is to provide a protocol-based non-homogeneous dwell probability power system load frequency control method in view of the deficiencies of the prior art. This method can save network resources while ensuring the stochastic stability of the system.

[0006] The technical solution for achieving the object of the present invention is as follows:

[0007] A protocol-based non-homogeneous dwell probability power system load frequency control method includes the following steps:

[0008] 1) Construct an interconnected multi-area power system: The load frequency regulation system of the interconnected multi-area power system involves loads, generators, prime movers, speed governors, and the LFC loop part. According to the operating characteristics of the power system, the interconnected multi-area power system is modeled as shown in Formula (1):

[0009]

[0010] Among them, Δf i (t) is the frequency deviation, is the tie-line power between area i and other control areas, ΔY i (t) is the valve position, is the mechanical output of the steam turbine, is the deviation of the load disturbance in the i-th control area, T g (r(t)) and T t (r(t)) are the governor and steam turbine time constants respectively, D i (r(t)) is the damping coefficient, M i (r(t)) is the generator inertia coefficient, R i (r(t)) is the droop coefficient of the i-th control area,

[0011] The generator model can be expressed as shown in Formula (2):

[0012]

[0013] The prime mover model can be expressed as shown in Formula (3):

[0014]

[0015] The governor model can be expressed as shown in Formula (4):

[0016]

[0017] The area control error signal for maintaining the steady-state error of the frequency deviation in the i-th area is defined as shown in Formula (5):

[0018]

[0019] The random variable r(t) is expressed as a switching variable belonging to a finite set of

[0020] Define

[0021] Then the interconnected multi - area power system in the i - th area is written as shown in Equation (6):

[0022]

[0023] where except for a 51 =-2πT ij for other a ls =0, and in addition, x i (t) is the state variable, u i (t) is the control input, y i (t) is the measurement output, is the disturbance input, and it satisfies:

[0024]

[0025] The proportional - integral controller in the i - th area is expressed as shown in Equation (7):

[0026]

[0027] where, represents the controller gain to be designed;

[0028] 2) Propose a quantization - based event - triggered scheduling protocol as shown in Equation (8):

[0029]

[0030] where,

[0031]

[0032] where ρ p >0, and h represents the sampling period, t k h represents the event - triggering moment, s represents the number of historical transmitted data packets, Ω ip is a positive definite matrix to be designed. Considering that there will be network - induced delays in the communication network, each time period of the zero - order hold is divided into the union of multiple small intervals, that is where Define the delay function Furthermore, we can obtain where

[0033] 3) Establish an interconnected multi - area power system with inhomogeneous dwell probability based on an event - triggered protocol under hybrid attacks: The process includes:

[0034] 3 - 1) Inhomogeneous dwell probability: In practical applications, subsystems of an interconnected multi - area power system always have various faults or trippings, which make the power system unable to operate normally. Assume that the accidental jump process is modeled using the mode of inhomogeneous dwell probability. This way represents the probability staying in the model. This is because obtaining the transition probability is very important, but it is very difficult to accurately obtain in practice. Using the method of dwell probability, it is easy to measure. Considering the inhomogeneity within the system, assume that there are m types of switching modes σ(t), where σ(t) ∈ M and M = {1, 2, …, m}. For any Define the indicative function as shown in formula (9):

[0035]

[0036] where and In some switching systems, the switching rules depend on time or the current state, but the switching rules based on dwell probability do not depend on time nor the current state. In this case, the probability that the system stays in a subsystem is known, that is, as shown in formula (10):

[0037]

[0038] 3 - 2) Dynamic quantizer: A dynamic quantizer can reduce the information exchange frequency and communication burden. The dynamic quantizer studied in this technical solution is defined as follows:

[0039] 3 - 2 - 1) If z ≤ M1, then |q(z) - z| ≤ Δ;

[0040] 3 - 2 - 2) If z ≥ M1, then |q(z)| ≥ M1 - Δ;

[0041] where z is the variable to be quantized, Δ represents the quantization error bound of the quantizer, and M1 represents the range of the quantizer. At the same time, the quantizer adopts the following dynamic adjustment rule, as shown in formula (11):

[0042]

[0043] where μ(t) represents the dynamic quantization parameter to be designed;

[0044] 3-3) Hybrid attack: Cyber attacks may occur in the power grid, leading to system instability. DOS attacks and spoofing attacks are the most destructive attacks. Therefore, hybrid attacks are considered by introducing a random variable α i (t) affected by the Bernoulli distribution to simulate DOS attacks. When the random variable α i (t) = 1, it means that no DOS attack has occurred; when α i (t) = 0, it means that the attacker disconnects the phasor measurement unit (PMU) and the load frequency control (LFC) to create an open-loop system. In addition, spoofing attacks are also considered. It is assumed that the aggression function satisfies the following bounded condition as shown in formula (12):

[0045]

[0046] The true output signal can be described as shown in formula (13):

[0047]

[0048] where α i (t) ∈ {0, 1} and the mathematical expectation is β i (t) ∈ {0, 1} and the mathematical expectation is

[0049] 3-4) Establish an interconnected multi-area power system with non-uniform dwell probability based on an event-triggered protocol under hybrid attacks:

[0050] For ease of subsequent processing, the model is transformed below. The matrix is increased by four columns to become a full-rank matrix in the i-th area system, that is Let Then the interconnected multi-area power system can be written as shown in formula (14):

[0051]

[0052] where

[0053] a(t) = diag N {a i (t)I}, β(t) = diag N {β i (t)I},

[0054] X(t) = colN {X i (t)}, y(t) = col N {y i (t)},

[0055] u(t) = col N {u i (t)},

[0056] 4) Based on Lyapunov theory, sufficient conditions for the exponential mean-square stability of the system are given and the H ∞ performance is satisfied. Solve the proportional-integral controller gains using the LMI toolbox of MTALAB: The process is as follows:

[0057] Based on Lyapunov theory, ensure that the mean-square exponential stability of the system is satisfied and the H ∞ performance sufficient condition is:

[0058] Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ω ip and an arbitrary matrix Z such that the following conditions hold, then the residual augmented system is stochastically stable and satisfies the H ∞ performance index:

[0059]

[0060] The average dwell time satisfies:

[0061]

[0062] where,

[0063]

[0064] Use linear matrix inequalities to solve the proportional-integral observer gains and controller gains:

[0065] Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ω ip and an arbitrary matrix Z such that the following conditions hold:

[0066]

[0067] where,

[0068]

[0069] In addition, the controller gain is where Z = diagN {Z i}, the i-th controller gain is as follows:

[0070]

[0071] This technical solution adopts a proportional-integral control method to resist some external interferences and enhance the system robustness. This is also an important load frequency control method, which can still make the system have strong robustness when there are interferences and cyber attacks in the system.

[0072] Compared with the prior art, the advantages of this technical solution are as follows:

[0073] 1. Model the power system as a system based on non-homogeneous sojourn probability: In practical applications, the subsystems of the interconnected multi-area power system always have various faults or trippings, which make the power system unable to operate normally. This accidental jump process is modeled by this technical solution using the non-homogeneous sojourn probability mode. This is because obtaining the transition probability is very important, but it is very difficult to accurately obtain it in practice. Using the sojourn probability method, it is easy to measure, and this technical solution is closer to the actual environment;

[0074] 2. To adjust the transmission frequency, this technical solution proposes a new event-triggered protocol, which considers the influence of dynamic quantization on the latest trigger signal, and ensures better control performance while saving network resources.

[0075] This method can save network resources and ensure the stochastic stability of the system. Brief Description of the Drawings

[0076] Figure 1 It is a schematic flow chart of the control method for the embodiment;

[0077] Figure 2 It is a schematic diagram of a three-area power system with multiple attacks in the embodiment;

[0078] Figure 3 It is a block diagram of the event-triggered LFC system under hybrid cyber attacks in the embodiment;

[0079] Figure 4 It is a state input trajectory diagram of the interconnected multi-area power system based on the improved event-triggered protocol in the embodiment;

[0080] Figure 5 It is a state output trajectory diagram of the interconnected multi-area power system based on the improved event-triggered protocol in the embodiment;

[0081] Figure 6 It is a system mode change diagram in the embodiment;

[0082] Figure 7 State input trajectory diagram of the interconnected multi - area power system based on the traditional event - triggered protocol in the embodiment;

[0083] Figure 8 State output trajectory diagram of the interconnected multi - area power system based on the traditional event - triggered protocol in the embodiment;

[0084] Figure 9 Schematic diagram of the event - trigger release time and release interval simulation results of the interconnected multi - area power system based on the improved event - triggered protocol in the embodiment;

[0085] Figure 10 Schematic diagram of the event - trigger release time and release interval simulation results of the interconnected multi - area power system based on the traditional event - triggered protocol in the embodiment. Detailed implementation manners

[0086] The following further elaborates on the content of the present invention in conjunction with the accompanying drawings and embodiments, but does not limit the present invention.

[0087] Embodiment:

[0088] Refer to Figure 1 , a non - homogeneous dwell - probability power system load - frequency control method based on a protocol, comprising the following steps:

[0089] 1) Construct an interconnected multi - area power system: As Figure 2 shown, the load - frequency regulation system of the multi - area power system involves loads, generators, prime movers, speed - regulation systems, and LFC loop parts. According to the operating characteristics of the power system, the interconnected multi - area power system is modeled as shown in formula (1):

[0090]

[0091] where, Δf i (t) is the frequency deviation, is the tie - line power between area i and other control areas, ΔY i (t) is the valve position, is the mechanical output of the steam turbine, is the deviation of the load disturbance in the i - th control area, T g (r(t)) and T t (r(t)) are the governor and steam - turbine time constants respectively, D i (r(t)) is the damping coefficient, M i (r(t)) is the generator inertia coefficient, R i (r(t)) is the droop coefficient of the i - th control area,

[0092] The generator model can be expressed as shown in formula (2):

[0093]

[0094] The prime mover model can be expressed as shown in Equation (3):

[0095]

[0096] The governor model can be expressed as shown in Equation (4):

[0097]

[0098] The area control error signal for maintaining the steady-state error of the frequency deviation in the \(i\)th area is defined as shown in Equation (5):

[0099]

[0100] The random variable \(r(t)\) is expressed as a switching variable belonging to a finite set of

[0101] Define

[0102] Then the interconnected multi-area power system in the \(i\)th area is written as shown in Equation (6):

[0103]

[0104] where except for \(a\) 51 =-2πT ij the other \(a\) ls =0, and in addition \(x\) i (t) is the state variable, \(u\) i (t) is the control input, \(y\) i (t) is the measured output, is the disturbance input, and satisfies:

[0105]

[0106] The proportional-integral controller in the \(i\)th area is expressed as shown in Equation (7):

[0107]

[0108] where represents the controller gain to be designed;

[0109] 2) A quantization-based event-triggered scheduling protocol is proposed as shown in Equation (8):

[0110]

[0111] Among them,

[0112]

[0113] Among them, ρ p > 0, and h represents the sampling period, t k h represents the event triggering moment, s represents the number of historical transmitted data packets, and Ω ip is a positive definite matrix to be designed. Considering that there will be network-induced delays in the communication network, each time period of the zero-order hold is divided into the union of multiple small intervals, that is Among them Define the delay function Furthermore, it can be obtained that Among them

[0114] 3) Establish an interconnected multi-area power system with non-uniform dwell probability based on an event-triggered protocol under hybrid attacks: The process includes:

[0115] 3-1) Non-uniform dwell probability: In practical applications, the subsystems of an interconnected multi-area power system always have various faults or trips, making the power system unable to operate normally. Assume that the accidental jump process is modeled using the non-uniform dwell probability mode. This means that the probability stays in the model. This is because obtaining the transition probability is very important, but it is very difficult to accurately obtain it in practice. Using the dwell probability method, it is easy to measure. Considering the non-uniformity that appears inside the system, assume that there are m types of switching modes σ(t), where σ(t) ∈ M, M = {1, 2,..., m}. For any Define the indicative function as shown in formula (9):

[0116]

[0117] Among them and In some switched systems, the switching rules depend on time or the current state, but the switching rules based on dwell probability do not depend on time nor on the current state. In this case, the probability that the system stays in the subsystem is known, that is, as shown in formula (10):

[0118]

[0119] 3-2) Dynamic quantizer: The dynamic quantizer can reduce the information exchange frequency and communication burden. The dynamic quantizer in this example is defined as follows:

[0120] 3-2-1) If z ≤ M1, then |q(z) - z| ≤ Δ;

[0121] 3-2-2) If \(z\geq M_1\), then \(|q(z)|\geq M_1 - \Delta\);

[0122] where \(z\) is the variable to be quantized, \(\Delta\) represents the quantization error bound of the quantizer, \(M_1\) represents the range of the quantizer. Meanwhile, the quantizer adopts the following dynamic adjustment rule, as shown in formula (11):

[0123]

[0124] where \(\mu(t)\) represents the dynamic quantization parameter to be designed;

[0125] 3-3) Hybrid attack: Network attacks may occur in the power grid, leading to system instability. DOS attack and spoofing attack are the most destructive attacks. Therefore, hybrid attack is considered. By introducing a random variable \(\alpha(t)\) affected by Bernoulli distribution i to simulate the DOS attack. When the random variable \(\alpha(t)\) i = 1, it means that no DOS attack occurs; when \(\alpha(t)\) i = 0, it means that the attacker disconnects the connection between the phasor measurement unit PMU and the load frequency control LFC to create an open-loop system. In addition, spoofing attack is also considered. It is assumed that the aggression function satisfies the following bounded condition as shown in formula (12):

[0126]

[0127] The true output signal can be described as shown in formula (13):

[0128]

[0129] where \(\alpha(t)\) i \(\in\{0,1\}\) and the mathematical expectation is \(\beta(t)\) i \(\in\{0,1\}\) and the mathematical expectation is

[0130] 3-4) Establish an interconnected multi-area power system with non-homogeneous dwell probability based on event-triggered protocol under hybrid attack:

[0131] For the convenience of subsequent processing, the model is transformed as follows. The matrix is increased by four columns to become a full-rank matrix in the \(i\)-th area system, that is Let As Figure 3 shown, the interconnected multi-area power system can be written as shown in formula (14):

[0132]

[0133] where

[0134] a(t) = diag N {a i (t)I}, β(t) = diag N {β i (t)I},

[0135] X(t) = col N {X i (t)}, y(t) = col N {y i (t)},

[0136] u(t) = col N {u i (t)},

[0137] 4) Based on the Lyapunov theory, sufficient conditions for the exponential mean-square stability of the system are given and the H ∞ performance is satisfied. The proportional-integral controller gains are solved using the LMI toolbox of MTALAB. The process is as follows:

[0138] Based on the Lyapunov theory, the sufficient condition to ensure the mean-square exponential stability of the system and satisfy the H ∞ performance is:

[0139] Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ω ip and an arbitrary matrix Z such that the following conditions hold, then the residual augmented system is stochastically stable and satisfies the H ∞ performance index:

[0140]

[0141] The average dwell time satisfies:

[0142]

[0143] where,

[0144]

[0145] Using linear matrix inequalities, the proportional-integral observer gains and controller gains are solved:

[0146] Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ωip and any matrix Z such that the following conditions hold:

[0147]

[0148] where,

[0149]

[0150]

[0151] In addition, the controller gain is where Z = diag N {Z i}, and the i-th controller gain is as follows:

[0152]

[0153] To verify the effectiveness of the method in this example, a simulation example of an interconnected multi-area power system is given, and its system parameters are shown in Table 1:

[0154] Table 1

[0155]

[0156] Other parameters are as follows:

[0157] T 12 (r(t)) = 0.2, T 13 (r(t)) = 0.25, T 23 (r(t)) = 0.12, γ = 0.5,

[0158] η2 = 0.0069, b = 0.2, θ = 36, M1 = 3, ρ1 = 0.1, ρ2 = 0.5,

[0159] Initial parameter selection:

[0160] X 10 = [-0.043 -0.06 -0.02 0.005 0.4] T ,

[0161] X 20 = [0.039 -0.08 0.01 0.018 0.07] T ,

[0162] X 30 = [-0.0233 -0.002 -0.001 -0.03 -0.46] T ,

[0163] The proportional-integral controller gains and event-triggering matrices are obtained using the LMI toolbox in MATLAB as follows:

[0164] K 11 = [0.0533 -0.4668], K 12 = [-0.8562 -0.0023]

[0165] K 21 = [0.0313 0.0259], K 22 = [-0.6208 -0.2583]

[0166] K 31 = [0.0259 0.0314], K 32 = [-0.5943 -0.2570]

[0167]

[0168] The simulation results are as follows Figures 4 - 10 shown. Selecting the sampling period h = 0.1 and the simulation time as 35 s, Table 2 shows the average transmission volume affected by hybrid attacks in each region during 350 simulations under the traditional event-triggering protocol and the improved event-triggering protocol. It can be clearly seen that the average transmission volume based on the improved event-triggering mechanism is less than that of the traditional event-triggering mechanism, which indicates that the method in this example occupies less communication bandwidth.

[0169] Table 2

[0170] Improved Event Triggering Protocol Traditional Event Triggering Protocol Region 1 11.4 32.56 Region 2 21.07 22.06 Region 3 23.90 26.83 ,

[0171] Comparing Figures 4 - 5 and Figures 7 - 8 , it can be seen that both event-triggering protocols can make the system quickly tend to be stable. However, from the perspective of saving bandwidth resources, from Table 2 Figure 9 and Figure 10 it can be seen that under the condition of maintaining the same system performance, the method in this example results in fewer triggering times. This indicates that under the same parameter conditions, compared with the traditional method, the method in this example is more general and also has the advantage of a short settling time. Therefore, the method in this example can achieve better control effects.

Claims

1. A protocol-based load frequency control method for non-homogeneous residence probability in a power system, characterized in that, It includes the following steps: 1) Construct an interconnected multi - area power system: The load - frequency regulation system of the interconnected multi - area power system involves loads, generators, prime movers, speed governors, and the LFC loop part. The interconnected multi - area power system is modeled as shown in Equation (1) as follows: Among them, Δf i (t) is the frequency deviation, is the tie-line power between area i and other control areas, ΔY i (t) is the valve position, is the mechanical output of the steam turbine, is the deviation of the load disturbance in the i-th control area, T g (r(t)) and T t (r(t)) are the governor and turbine time constants respectively, D i (r(t)) is the damping coefficient, M i (r(t)) is the generator inertia coefficient, R i (r(t)) is the droop coefficient of the i-th control area, The generator model is expressed as shown in Equation (2): The prime mover model is expressed as shown in Equation (3): The speed governor model is expressed as shown in Equation (4): The area control error signal used to maintain the steady - state error of the frequency deviation in the \(i\) - th area is defined as shown in Equation (5): The random variable r(t) is represented as a switching variable belonging to a finite set of Definition Then, the interconnected multi - area power system of the $i$-th area is written as shown in Equation (6): Among them except for a 51 = -2πT ij for other a ls = 0, and in addition x i (t) is a state variable, u i (t) is a control input, y i (t) is a measured output, is a disturbance input, and satisfies: The proportional - integral controller in the \(i\) - th area is expressed as shown in Equation (7): Among them, represents the controller gain to be designed; 2) Propose a quantization - based event - triggered scheduling protocol as shown in Equation (8): where where ρ p > 0, and h represents the sampling period, t k h represents the event triggering moment, s represents the number of historical transmitted data packets, and Ω ip is a positive definite matrix to be designed. There will be network-induced delay in the communication network. Each time period of the zero-order hold is divided into the union of multiple small intervals, that is where Define the delay function Furthermore, we get where 3) Establish an interconnected multi - area power system with non - homogeneous dwell - time probability based on the event - triggered protocol under hybrid attacks: The process includes: 3-1) Non-homogeneous sojourn probability: Assume that the accidental jump process is modeled with a non-homogeneous sojourn probability pattern, which represents the non-uniformity within the system that occurs when the probability stays in the model. Assume that there are m types of switching patterns σ(t), where σ(t) ∈ M and M = {1, 2, …, m}. For any Define the indicative function as shown in Equation (9): Among them and In the switching system, the switching rule depends on time or the current state, but the switching rule based on the residence probability does not depend on time nor on the current state. In this case, the probability that the system stays in the subsystem is known, as shown in formula (10): 3 - 2) Dynamic quantizer: The dynamic quantizer is defined as follows: 3 - 2 - 1) If \(z\leq M_1\), then \(|q(z)-z|\leq\Delta\); 3 - 2 - 2) If \(z\geq M_1\), then \(|q(z)|\geq M_1-\Delta\); where \(z\) is the variable to be quantized, \(\Delta\) represents the quantization error bound of the quantizer, and \(M_1\) represents the range of the quantizer. Meanwhile, the quantizer adopts the following dynamic adjustment rule, as shown in Equation (11): where \(\mu(t)\) represents the dynamic quantization parameter to be designed; 3-3) Hybrid attack: Introduce a random variable α affected by the Bernoulli distribution i α(t) is used to simulate the DOS attack. When the random variable α i α(t) = 1, it means that no DOS attack occurs; when α i α(t) = 0 means that the attacker disconnects the PMU from the load frequency control (LFC) to create an open-loop system, and the aggression function satisfies the following bounded condition as shown in Equation (12): The true output signal is described as shown in Equation (13): where α i (t) ∈ {0, 1} and the mathematical expectation is β i (t) ∈ {0, 1} and the mathematical expectation is 3 - 4) Establish an interconnected multi - area power system with non - homogeneous dwell - time probability based on the event - triggered protocol under hybrid attacks: Add four columns to the matrix to form a full-rank matrix in the \(i\)-th regional system, i.e., Let Then the interconnected multi-regional power system can be written as shown in Equation (14): where a(t) = diag N {a i (t)I}, β(t) = diag N {β i (t)I}, X(t) = col N {X i (t)}, y(t) = col N {y i (t)}, u(t) = col N {u i (t)}, 4) Give the sufficient conditions for the exponential mean-square stability of the system and satisfy the H ∞ performance, and solve the proportional-integral controller gain: The process is as follows: Based on the Lyapunov theory, ensure that the system is mean-square exponentially stable and satisfies H ∞ The sufficient condition for performance is as follows: Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ω ip and an arbitrary matrix Z such that the following conditions hold, then the residual augmented system is stochastically stable and satisfies the H ∞ performance index: The average dwell - time satisfies: where Use linear matrix inequalities to solve the proportional - integral observer gain and controller gain: Given scalars η2 > 0, ρ p > 0, κ > 1, if there exist positive definite matrices M, R, W, Ω ip and any matrix Z such that the following conditions hold: where The controller gain is where Z = diag N {Z i}, and the i-th controller gain is as follows:

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