This article uses fifth-order nonlinear differential equations to describe the dynamic process of electrical automation control systems. This method first derives the equivalent system of the nonlinear fuzzy global system and then uses the orthogonal polynomial series expansion technique and its integral operation matrix. The local manifold at the dominant unstable equilibrium point of a single-machine infinite-bus system after a failure described by a two-dimensional quadratic nonlinear differential equation is calculated, and the stability boundary of the power system is obtained. The research results show that the output frequency fluctuation of the electrical automation control system is small after the algorithm is adopted, and the intelligent control system can accurately diagnose and warn the electrical faults. The system can meet the requirements of online voltage coordinated control.

#### Keywords

- Nonlinear differential equation
- Electrical automation control system
- Voltage coordinated control
- Model predictive control

#### MSC 2010

- 34M03

The load level of the load center continues to increase, and the large-capacity long-distance transmission continues to increase. The problem of medium and long-term voltage stability driven by load recovery characteristics has become increasingly prominent, resulting in slow dynamic characteristics of related systems [1]. This requires timely voltage control measures to prevent the system from continuing to deteriorate and develop.

Some scholars use the variational method to convert the optimally coordinated voltage control problem into a linear programming solution. Some scholars have established the optimality conditions for coordinated voltage control and used the pull channel arrangement method to solve the two-point boundary value problem. Some scholars use the tree search method to solve the combined optimization problem predicted by the model. Some scholars use pseudo-gradient evolutionary programming to solve complex voltage coordination and optimization problems [2]. Based on the trajectory sensitivity method, some scholars use the linear model predictive control (MPC) for voltage control. The above methods can effectively solve coordinated voltage control, but they all involve time-domain simulation calculations of hybrid systems. Therefore, the above methods are difficult to meet the requirements of online applications.

The Wide Area Measurement System (WAMS) realizes the online synchronous measurement of the operation status of the wide-area power grid, combined with the data acquisition and monitoring (SCADA) system data to provide accurate voltage and node power injection information. The system can be applied to simplify and correct the predictive model of voltage coordinated control. The nonlinear model predictive control (NLMPC) method combined with the direct multiple shooting method and the measured information simplify and solve the voltage coordinated control problem [3]. However, this method ignores the influence of generator overexcitation limit on long-term discrete variables, and the Hessian matrix solution is time-consuming.

This paper proposes a segmented correction prediction model suitable for online applications. The load state variable trajectory is linearly processed in the forecast period, and the forecast model is calibrated on a rolling basis using the wide-area measurement information. This method can ensure the reliability of online control [4]. On this basis, the voltage response prediction method of the segmented correction model is proposed. Since neither the prediction nor the optimization process involves the processing of dynamic differential equations, it can greatly reduce the amount of calculation of the voltage coordinated control problem and meet the requirements of online control.

The goal of voltage coordinated control is to formulate reasonable voltage regulation measures according to the operating state of the power system, which prevents controller redundancy and disorderly regulation under the premise of ensuring voltage stability. MPC calculates the future optimal control strategy by minimizing the objective function in the finite time domain [5]. The voltage coordinated control model of the nth control cycle can be expressed as:
_{d}^{+}) and _{d}^{−}) are the vectors composed of discrete dynamic variables before and after switching, respectively. _{n}_{n}_{1} + (_{s}_{s}_{c}_{p}_{p}_{s}

In the QSS model, the load's dynamic recovery characteristics determine the system's continuous dynamic process, an important factor in voltage instability events. The article proposes an additive dynamic exponential recovery load model, which can be expressed as:
_{p}_{q}_{p}_{p}_{q}^{T} in the QSS model (Equation (2)–Equation (5)). _{0}, _{0}, _{0}, is the steady-state value of active load power, reactive power, and voltage. V is the amplitude of the load voltage. _{d}_{d}_{t}_{t}_{s}_{s}

The model parameters can be obtained in the existing system by field testing in the main high/medium voltage substation. The dynamic exponential recovery load model has been widely used to study medium and long-term voltage stability. Still, it ignores the time-varying and uncertainty of the load. Its simulation calculation is complex, so it isn't easy to apply to online voltage coordinated control [9]. This paper proposes a power system segmented correction model based on WAMS information based on the above situation.

Assume that WAMS can summarize the system measurement information at the initial time _{n}_{cn}_{cn}_{cn}_{pcn}_{qcn}_{0}, _{0}, _{0} is the steady-state value of load active power vector, reactive power vector, and voltage vector. The subscript (_{c}_{(m|n)} = [_{p}_{(m|n)}, _{q}_{(m|n)}]^{T}, _{c}_{n}

If _{c}_{p}_{c}_{p}_{d}_{n}_{pcn}_{qcn}

We use the ESPLO method to calculate the QSS model load node voltage response to the control Δ^{*}, the value of the response track Δ

Since the state variables in the segmented correction model increase linearly during the prediction period, the sensitivity of the control input to the slope of the state variable at a time ^{*} cannot be calculated, so this method cannot calculate the voltage response of the segmented correction model [10]. This article proposes a voltage response prediction method for the segmented correction model by assuming that the system is dynamically combined with ESPLO.

The linearized segmented correction model at a time ^{*} can obtain the sensitivity vector of the relative control of the load node power and voltage. After the implementation of the new control, the predicted instantaneous values of the load node active power, reactive power vector, and voltage vector are:

Taking the load node power and voltage values of the segmented correction model at a time ^{*} into equations (9) and (11), the derivative
^{*} can be expressed as:

Assuming
_{c}

The voltage prediction method at the sampling point is shown in Figure 2.

Assuming no new control adjustments are applied in the current control period, the voltage trajectory (_{(1/n)}, Δ_{(2/n)},⋯, Δ_{(k|n)} to obtain the voltage response value Δ_{(m|n)} at each sampling point in the control period. After the system is controlled, the predicted voltage amplitude vector of each target node can be expressed as:

We substitute formula (20) into formula (1), and the voltage coordination optimization model (formula (1)–formula (8)) is transformed into a mixed-integer programming problem with the control input adjustment variable Δ

The calculation process of the online voltage coordinated control based on the segmented correction model is shown in Figure 3.

At the initial moment of control, we correct the initial value of the system state according to the measurement information. We obtain the output trajectory of the system in the prediction period according to equation (14). According to equation (18) and equation (19), the output response in the forecast period is calculated. The voltage coordinated control model is transformed by formula (20)–formula (22). At the beginning of the next control cycle, _{n}_{+1} = _{n}_{s}

To highlight the voltage stability problem, we reduced the output of No. 6 and No. 7 generators to 0.87 and 0.75, respectively. The generators all consider the role of OLE, so the maximum excitation current is selected to be 1.05 times of its rated value. We approximate the QSS model as the existing system, the load adopts a dynamic exponential recovery model, and the recovery parameters are selected as: _{s}_{s}_{1} = _{t}_{p}_{q}

The predictive controller parameter chooses _{s}_{c}_{p}

When t = 10s, No. 3 generator and line 10–11 trip due to a fault. With the recovery of load power, the OEL of the No. 5 generator will be activated at t=59.7s. If emergency control measures are not taken, the system will experience voltage collapse in 291s. The voltage response curves of nodes 4, 7, and 8 are shown in Figure 4.

Figure 5 shows the voltage curves of nodes 4, 7, and 8 under the online voltage coordinated control. _{ref, G}_{1} is the AVR reference voltage value of the No. 1 generator. _{t}_{(12–11)} is the transformation ratio of OLTC connecting nodes 12 and 11. _{load}_{20} is the load shedding coefficient at node 20.

Control the initial moment _{1} = 40

In a control period, the average calculation time of the system's predicted trajectory is 0.61s, and the average optimization time (including optimization model formation and data import time) is 0.32s. If the state estimation and measurement information transmission time are not considered, the total calculation time of the online voltage coordinated control in a single control cycle is only 0.93s. Applying the existing system after the sampling period _{s}_{offset}

_{r}_{i}_{1} = 40_{offset}

If the system QSS model is not linearized, we use the NLMPC method to directly use the system's hybrid differential-algebraic equations for predictive control. The voltage response curve obtained is shown in Figure 7. The genetic algorithm is used for optimization.

The simulation results show that the three control methods can maintain the system voltage stability and the load shedding conditions. The average voltage offset index Δ_{offset}

This paper proposes an online voltage coordination control strategy based on a segmented correction model, which can effectively coordinate control measures of different locations and types and gradually stabilize the system voltage after a fault. This method mainly has the following advantages.

Simplify wide-area measurement information and use roll-correct the prediction model during the control cycle, which keeps the predictive model consistent with the actual operating state of the system and ensures the reliability of control.

The segmented correction model approximates the continuous dynamic characteristics of the system, considers the influence of the generator excitation limit, avoids the time domain simulation process and greatly improves the prediction time of system trajectory.

We propose a voltage response prediction method for the segmented correction model. This method transforms the complex optimal coordinated voltage control model into a mixed integer programming problem whose independent variable is the control regulation.

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