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Application of Nonlinear Differential Equation in Electric Automation Control System

Pubblicato online: 15 Jul 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 19 Jan 2022
Accettato: 20 Mar 2022
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License
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Abstract

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

MSC 2010

Introduction

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.

MPC-based voltage coordinated control model

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: min(m=0k(VrVm)TQ(VrVm)+ΔuTRΔu) \min \left( {\sum\limits_{m = 0}^k {{{\left( {{V_r} - {V_m}} \right)}^T}\,Q\left( {{V_r} - {V_m}} \right) + \Delta {u^T}R\Delta u} } \right) s.t. 0=f(x(t),y(t),zc(t),zd(t),u(t)) 0 = f\left( {x\left( t \right),\,y\left( t \right),\,{z_c}\left( t \right),\,{z_d}\left( t \right),\,u\left( t \right)} \right) 0=g(x(t),y(t),zc(t),zd(t),u(t)) 0 = g\left( {x\left( t \right),\,y\left( t \right),\,{z_c}\left( t \right),\,{z_d}\left( t \right),\,u\left( t \right)} \right) zc=hc(x(t),y(t),zc(t),zd(t),u(t)) {z_c} = {h_c}\left( {x\left( t \right),\,y\left( t \right),\,{z_c}\left( t \right),\,{z_d}\left( t \right),\,u\left( t \right)} \right) zd(t+)=hd(x(t),y(t),zc(t),zd(t),u(t)) {z_d}\left( {{t^ + }} \right) = {h_d}\left( {x\left( t \right),\,y\left( t \right),\,{z_c}\left( t \right),\,{z_d}\left( {{t^ - }} \right),\,u\left( t \right)} \right) yminy(t)ymaxt[tn,tn+tp] \matrix{ {{y_{\min }}\, \le y\left( t \right) \le {y_{\max }}} \hfill & {t \in \left[ {{t_n},\,{t_n} + {t_p}} \right]} \hfill \cr } uminu(t)umaxt[tn,tn+tc] \matrix{ {{u_{\min }}\, \le u\left( t \right) \le {u_{\max }}} \hfill & {t \in \left[ {{t_n},\,{t_n} + {t_c}} \right]} \hfill \cr } Δuminu(t)umaxt[tn,tn+tc] \Delta \matrix{ {{u_{\min }}\, \le u\left( t \right) \le {u_{\max }}} \hfill & {t \in \left[ {{t_n},\,{t_n} + {t_c}} \right]} \hfill \cr } zd(t+) and zd(t) are the vectors composed of discrete dynamic variables before and after switching, respectively. tn is the initial moment of the nth control cycle tn = t1 + (n − 1)ts. Where ts is the sampling period. tc is the control period. tp is the forecast period tp = Kts [6]. Equation (1) is the objective function. Equation (2)–Equation (5) is the QSS model. Equations (6), (7), and (8) are the upper and lower limits of the algebraic vector, control input, and single control variation, respectively. The total load has substantial uncertainty on the long-term scale and is difficult to model accurately. There may be a risk of control failure if there is no measured data during the optimization process [7]. Simplifying and rolling correction prediction models based on wide-area measurement information effectively solve the above problems.

Voltage coordinated control based on segmented correction model
Segmented correction model

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: z˙p=P0(VV0)asPdk {\dot z_p} = {P_0}{\left( {{V \over {{V_0}}}} \right)^{{a_s}}} - {{{P_d}} \over k} Pd=k(zpTp+P0(VV0)at) {P_d} = k\left( {{{{z_p}} \over {{T_p}}} + {P_0}{{\left( {{V \over {{V_0}}}} \right)}^{{a_t}}}} \right) z˙p=Q0(VV0)βsQdk {\dot z_p} = {Q_0}{\left( {{V \over {{V_0}}}} \right)^{{\beta _s}}} - {{{Q_d}} \over k} Qd=k(zpTp+Q0(VV0)βt) {Q_d} = k\left( {{{{z_p}} \over {{T_p}}} + {Q_0}{{\left( {{V \over {{V_0}}}} \right)}^{{\beta _t}}}} \right) zp and zq are the dynamic state variables of active and reactive loads, respectively. [zp = [zp, zq]T in the QSS model (Equation (2)–Equation (5)). P0, Q0, V0, is the steady-state value of active load power, reactive power, and voltage. V is the amplitude of the load voltage. Pd and Qd are the active and reactive power consumed by the load, respectively [8]. The transient load indexes at and βt are greater than the steady-state load indexes as and βs, respectively. Therefore, the transient characteristics are more sensitive to voltage, and the load dynamic state variables force the transient characteristics to transition to steady-state characteristics.

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 tn of the nth control cycle. After state estimation, the voltage amplitude vector Vcn and power injection vectors Pcn and Qcn of each load node are obtained. We substitute it into equation (10) and equation (12) to get the initial values zpcn and zqcn of the corresponding state vector. Assuming that the state variable grows linearly along its derivative direction at time tn, we can get the piecewise recovery model of the load. Equations (9) and (11) can be transformed into: {zp(m|n)=zpcn+mts(P0(VcnV0)asPcnk)zq(m|n)=zqcn+mts(Q0(VcnV0)βsQcnk) \left\{ {\matrix{ {{z_{p\left( {m|n} \right)}} = {z_{pcn}} + m{t_s}\left( {{P_0}{{\left( {{{{V_{cn}}} \over {{V_0}}}} \right)}^{{a_s}}} - {{{P_{cn}}} \over k}} \right)} \hfill \cr {{z_{q\left( {m|n} \right)}} = {z_{qcn}} + m{t_s}\left( {{Q_0}{{\left( {{{{V_{cn}}} \over {{V_0}}}} \right)}^{{\beta _s}}} - {{{Q_{cn}}} \over k}} \right)} \hfill \cr } } \right. m = 0,1,⋯, K; P0, Q0, V0 is the steady-state value of load active power vector, reactive power vector, and voltage vector. The subscript (m|n) is the m+1 sampling point in the nth control cycle. We replace the formula (4) in the QSS model with formula (13), then zc(m|n) = [zp(m|n), zq(m|n)]T, zc becomes a constant vector during the forecast period. The original differential equation is transformed into an algebraic equation, and the segmented correction model of the power system can be expressed as: {0=f(x(m|n),y(m|n),zc(m|n),zd(m|n),u(m|n))0=g(x(m|n),y(m|n),zc(m|n),zd(m|n),u(m|n))Cn=hc(x(m|n),y(m|n),zc(m|n),zd(m|n),u(m|n))zd(m|n)+=hd(x(m|n),y(m|n),zc(m|n),zd(m|n),u(m|n)) \left\{ {\matrix{ {0 = f\left( {{x_{\left( {m|n} \right)}},\,{y_{\left( {m|n} \right)}},{z_{c\left( {m|n} \right)}},{z_{d\left( {m|n} \right)}},{u_{\left( {m|n} \right)}}} \right)} \hfill \cr {0 = g\left( {{x_{\left( {m|n} \right)}},\,{y_{\left( {m|n} \right)}},{z_{c\left( {m|n} \right)}},{z_{d\left( {m|n} \right)}},{u_{\left( {m|n} \right)}}} \right)} \hfill \cr {{C_n} = {h_c}\left( {{x_{\left( {m|n} \right)}},\,{y_{\left( {m|n} \right)}},{z_{c\left( {m|n} \right)}},{z_{d\left( {m|n} \right)}},{u_{\left( {m|n} \right)}}} \right)} \hfill \cr {z_{d\left( {m|n} \right)}^ + = {h_d}\left( {{x_{\left( {m|n} \right)}},\,{y_{\left( {m|n} \right)}},{z_{c\left( {m|n} \right)}},z_{_{d\left( {m|n} \right)}}^ - ,{u_{\left( {m|n} \right)}}} \right)} \hfill \cr } } \right. m = 0, 1, ⋯ K; Cn is a constant vector, which represents the slope of the state vector to the time axis in the current forecast period. zd+ z_d^ + and zd z_d^ - are the vectors of discrete dynamic variables before and after the transformed model is switched at the sampling point, respectively.

If K = 3, tc = tp is taken, the restoration process of the active load state variable K = 3, tc = tp and active power Pd can be described in Figure 1. At time tn, the voltage and power data of each load node of the system are obtained through WAMS. Correct the state variable values zpcn and zqcn according to equation (10) and equation (12). The state variable trajectory is obtained by formula (13). Then the predicted value of the load node power and voltage at each sampling point in the prediction period is determined by the formula (14). The above process is repeated when the initial time tn+1 of the next control cycle arrives.

Figure 1

Assumption of linear recovery of load state variables

Voltage response prediction of segmented correction model

We use the ESPLO method to calculate the QSS model load node voltage response to the control Δu. If the control is applied at the sampling time t*, the value of the response track Δy at the sampling point is: Δzc(t*+mts,u)mts(zc(t*)+hcuΔu) \Delta {z_c}\left( {{t^*} + m{t_s},u} \right) \approx m{t_s}\left( {{z_c}\left( {{t^*}} \right) + {{\partial {h_c}} \over {\partial u}}\Delta u} \right) Δy(t*+mts,u)yzcΔzc(t*+mts,u)+yuΔu {\Delta _y}\left( {{t^*} + m{t_s},u} \right) \approx {{\partial y} \over {\partial {z_c}}}\Delta {z_c}\left( {{t^*} + m{t_s},u} \right) + {{\partial y} \over {\partial u}}\Delta u

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 t* 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 t* 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: {Pd(t*)+ΔPdPd(t*)+PduΔuQd(t*)+ΔQdQd(t*)+QduΔuV(t*)+ΔVV(t*)+VuΔu \left\{ {\matrix{ {{P_d}\left( {{t^*}} \right) + \Delta {P_d} \approx {P_d}\left( {{t^*}} \right) + {{\partial {P_d}} \over {\partial u}}\Delta u} \hfill \cr {{Q_d}\left( {{t^*}} \right) + \Delta {Q_d} \approx {Q_d}\left( {{t^*}} \right) + {{\partial {Q_d}} \over {\partial u}}\Delta u} \hfill \cr {V\left( {{t^*}} \right) + \Delta V \approx V\left( {{t^*}} \right) + {{\partial V} \over {\partial u}}\Delta u} \hfill \cr } } \right.

Taking the load node power and voltage values of the segmented correction model at a time t* into equations (9) and (11), the derivative (Cn*) \left( {C_n^*} \right) of the state variable to time at that moment can be obtained. Assuming that after control is applied, the model is recalibrated according to the instantaneous value of the system predicted by equation (17), the sensitivity of the control input to Cn* C_n^* at time t* can be expressed as: B[P0as(V(t*))qs1V0saVuPdkuQ0βs(V(t*))βs1V0sβVuQdku] B \approx \left[ {\matrix{ {{{{P_0}{a_s}{{\left( {V\left( {{t^*}} \right)} \right)}^{{q_s} - 1}}} \over {V_{{0^s}}^a}}{{\partial V} \over {\partial u}} - {{\partial {P_d}} \over {k\partial u}}} \cr {{{{Q_0}{\beta _s}{{\left( {V\left( {{t^*}} \right)} \right)}^{{\beta _s} - 1}}} \over {V_{{0^s}}^\beta }}{{\partial V} \over {\partial u}} - {{\partial {Q_d}} \over {k\partial u}}} \cr } } \right]

Assuming z(t*)=Cn* z\left( {{t^*}} \right) = C_n^* , ∂hc / ∂u = B in formula (15), the voltage response calculation method for the segmented correction model during the prediction period is: {Δzc(t*+mts,u)mts(Cn*+BΔu)Model6 \left\{ {\matrix{ {\Delta {z_c}\left( {{t^*} + m{t_s},u} \right) \approx m{t_s}\left( {C_n^* + B\Delta u} \right)} \hfill \cr {{\rm{Model}}6} \hfill \cr } } \right.

Online voltage coordinated control based on segmented correction model

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

Figure 2

Voltage prediction method at the sampling point

Assuming no new control adjustments are applied in the current control period, the voltage trajectory (V*) in the predicted period can be quickly obtained from equation (14). The voltage change of the load node caused by the control adjustment can be obtained by formula (19). According to the nature of the linear system, we can linearly superimpose the voltage changes produced by the control sequence (Δu(1/n), Δu(2/n),⋯, Δu(k|n) to obtain the voltage response value ΔV(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: V(m|n)=V(m|n)*+ΔV(m|n) {V_{\left( {m|n} \right)}} = V_{\left( {m|n} \right)}^* + \Delta {V_{\left( {m|n} \right)}}

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 Δu as an independent variable: min(m=0k(VrV(m|n))TQ(VrV(m|n))+ΔuTRΔu) \min \left( {\sum\limits_{m = 0}^k {{{\left( {{V_r} - {V_{\left( {m|n} \right)}}} \right)}^T}\,Q\left( {{V_r} - {V_{\left( {m|n} \right)}}} \right) + \Delta {u^T}R\Delta u} } \right) s.t. {u(m|n)=u((m1)|n)+Δu(m|n)yminy(m|n)ymaxuminu(m|n)umaxΔuminΔu(m|n)Δumax \left\{ {\matrix{ {\,{u_{\left( {m|n} \right)}} = {u_{\left( {\left( {m - 1} \right)|n} \right)}} + \Delta {u_{\left( {m|n} \right)}}} \cr {{y_{\min }} \le {y_{\left( {m|n} \right)}} \le {y_{\max }}} \cr {{u_{\min }} \le {u_{\left( {m|n} \right)}} \le {u_{\max }}} \cr {\Delta {u_{\min }} \le \,\Delta {u_{\left( {m|n} \right)}} \le \Delta {u_{\max }}} \cr } } \right.

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

Figure 3

Online voltage coordinated control flow

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, tn+1 = tn + ts. Repeat the above steps.

Example analysis
Example system

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: as = βs = 0, a1 = βt = 2, Tp = Tq = 60. This can be used to provide the measurement data at the initial moment of the control cycle.

The predictive controller parameter chooses K= 3, ts = 10s, tc = tp = 30s, and the voltage offset weight is 1. The model prediction is realized under Matlab7.9/Simulink. The optimization problem is solved by the mini lip method of GAMS. The computer condition is Pentium Dual-Corre E58003.20GHz, and the memory is 2.00GB.

Example simulation

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 4

Nodal voltage response curve without control

Figure 5 shows the voltage curves of nodes 4, 7, and 8 under the online voltage coordinated control. Vref, G1 is the AVR reference voltage value of the No. 1 generator. nt(12–11) is the transformation ratio of OLTC connecting nodes 12 and 11. kload20 is the load shedding coefficient at node 20.

Figure 5

Voltage response curve after applying control

Control the initial moment t1 = 40s. Through the segmented correction model, the system trajectory in the prediction period t ∈ [40, 70]s can be quickly obtained. At the predicted sampling time t = 60 of the first control cycle, the excitation current of the No. 5 generator obtained by the segmented correction model is greater than the upper limit. At this sampling point, it is determined that the existing system OLE has acted, and the excitation current of the No. 5 generator returns to the upper limit [11]. We can approximately reflect the effect of OEL in the forecast period. At the initial moment of each control cycle, the prediction model will be calibrated with the WAMS information. It can be seen from the optimization results that the optimization decision only removes 10% of the load at node 20, where the load shedding prediction response is the most significant. It takes into account both safety and economic requirements. The control strategy proposed in this paper can gradually stabilize the system voltage after a fault by coordinating different locations and types of control measures.

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 ts = 10s can effectively avoid control failure caused by the calculation time delay. This paper uses the power system QSS model to optimize the calculation examples and compare the results. Introduce the control performance index ΔVoffset to describe the average voltage offset of the 19 load nodes in the system in a certain period after the control is applied: ΔVoffset=119i=119(1Δtt1t1+Δt|VrVi(t)|dt) \Delta {V_{offset}} = {1 \over {19}}\sum\limits_{i = 1}^{19} {\left( {{1 \over {\Delta t}}\int_{{t_1}}^{{t_1} + \Delta t} {\left| {{V_r} - {V_i}\left( t \right)} \right|dt} } \right)}

Vr is the node reference voltage. Vi is the predicted voltage at node i. Δt is selected as 360s. The integration start time is the initial control time t1 = 40s; ΔVoffset. The smaller the value, the better the global optimization effect of the control strategy. When the system parameters and optimization tools are unchanged, the proposed LMPC method realizes the prediction, which does not perform sub-approximation processing of load state variables. After implementing the optimization results, the voltage traces of nodes 4, 7, and 8 are shown in Figure 6.

Figure 6

Voltage response curve obtained by using LMPC method

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.

Figure 7

Voltage response curve obtained by using NLMPC method

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 ΔVoffset of the control method based on NLMPC is the smallest. We have achieved the best global optimization effect in this calculation, but the average calculation time of the optimal control sequence in the control cycle is as long as 1217.31s. This isn't easy to apply online. Online control tends to adopt fast and reliable control decisions, and it is acceptable to sacrifice part of the optimality of the solution to simplify the optimization problem. The control method based on LMPC and based on the segmented correction model uses a linearized predictive model, which significantly reduces the calculation time of the voltage coordinated control problem. This method can track the actual operating state of the system. Combined with WAMS information rolling correction prediction model to ensure the reliability of online control.

Conclusion

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.

Figure 1

Assumption of linear recovery of load state variables
Assumption of linear recovery of load state variables

Figure 2

Voltage prediction method at the sampling point
Voltage prediction method at the sampling point

Figure 3

Online voltage coordinated control flow
Online voltage coordinated control flow

Figure 4

Nodal voltage response curve without control
Nodal voltage response curve without control

Figure 5

Voltage response curve after applying control
Voltage response curve after applying control

Figure 6

Voltage response curve obtained by using LMPC method
Voltage response curve obtained by using LMPC method

Figure 7

Voltage response curve obtained by using NLMPC method
Voltage response curve obtained by using NLMPC method

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