Results of Signal Analysis

In addition to FIDF detection as discussed above, it is found from simulation that, when voltage collapse is about to happen for possible Q1, the residual signals appear to exhibit oscillation with growing amplitude and almost the same frequency. As such, it enables us to monitor the amplitude of such a frequency to help judge the occurrence of voltage collapse. An example is shown in Figure 3.6, where the load voltage and residual signal for Q1 ñ 11.1 are given in Figures 3.6(a) and (b), respectively. To avoid the influence of DC part, an averaged signal from the residual by the formula (3.22) below

sa(n) = s(n)à_L¹P

k=nàL+1

n s(k), L = 100 (3.22) and its spectrogram with sampling frequency fs = 100Hz are described in Figures 3.6(c) and (d), respectively. The oscillating frequency to be monitored is observed from Figure 3.6(d) to have f≈0.59Hz. The amplitude of the monitored frequency for the last 1024 points FFT before the occurrence of collapse versus Q1 is shown by the solid-line of Figure 3.7. To facilitate the detection using the monitored frequency, the threshold values for different Q1 are defined to be the amplitude of the monitored frequency 5 seconds ahead of voltage collapse, which are indicated by the dashed-line of Figure 3.7. Note that, the oscillating times before collapse are less than 5 seconds near the value of Q1 = 11.3. With the definitions of threshold values, the voltage collapse for Q1 = 11.1 is shown able to be successfully detected using both methods, as indicated in Figures 3.8(b) and (d). The alarm for the second method is fired around t = 55, which is near 5 seconds ahead of the collapse as desired. Finally, Figure 3.9 demonstrates the detection result for varying load using the threshold which is determined by the second method. Clearly, the alarm is also

３８

fired nearly 5 seconds ahead of the occurrence of voltage collapse. Finally, we consider the situation of a 5% load variation about the operating point. Figure 3.10 shows the detection result. It is clear that the alarm is fired nearly 5 seconds ahead of the occurrence of voltage collapse as our desire.

From these simulations, it is noted that the voltage collapse can be successfully detected before it occurs. By properly adjusting the threshold for generating the alarm signal, the FIDF may provide a precursor of avoiding undesirable effects of these unstable behaviors.

Figure 3.6: (a) Voltage response for Q1 ñ 11.1 (b) residual signal (c) residual signal after averaging (d) spectrogram

３９

Figure 3.7: Amplitude of the monitored frequency versus Q1

Figure 3.8: (a) residual signal for Q1ñ 11.1 (b) alarm by FIDF (c) amplitude of the monitored frequency (d) alarm by monitored frequency

４０

Figure 3.9: (a) load variation (b) residual signal (c) amplitude of the monitored frequency (d) alarm by monitored frequency

Figure 3.10: (a) load variation (b) residual signal (c) amplitude of the monitored frequency (d) alarm by monitored frequency

４１

CHAPTER 4

Voltage Regulation of the Electric Power Systems

In this chapter, we add an extra tap changer parallel to the nonlinear load to Dobson and Chiang's power system model for the purpose of voltage regulation. In Section 4.1, we derive the dynamic equations of the power system with tap changer. Then, we will apply Variable Structure Control design scheme to adjust the tap changer ratio to achieve voltage regulation for this model. In Section 4.2, we propose a parameter estimator as the load monitor to provide the load variation of the power system. In Section 4.3, we combine the designs of VSC voltage controller and load estimator to design an adaptive control system.

4.1 Variable Structure Controller Design

4.1.1 Controlled Power System Model

In this section, we add a voltage controller – tap changer to the original power

４２

system model. Here, we use the tap changer ratio as the control signal for the electric power system. We will utilize tap changer to regulate the voltage of the electric power system.

After adding a voltage controller – tap changer to the original power system model.

The controlled model is shown as in Figure 4.1.

Figure 4.1: The power system model with tap changer

The original dynamical equations for the electric power system can be written as follow :

îç = ωm (4.1) Mωç = à d^mω + Pm+_n¹E_mYmV sin î( à î^mà ò^m)

+ E²_mYmsin òm (4.2) Kqwîç = à Kqv2V²à K^qvV + Q î( m, î, V)à Q⁰à Q¹ (4.3)

TKqwKpvVç = KpwKqv2V²+ K( pwKqvà KqwKpv) V + Kqw(P(îm, î, V)à P⁰à P¹)

à K^pw(Q(îm, î, V)à Q⁰à Q¹) (4.4) where

４３

Q(îm, î, V) = E⁰₀Y⁰₀V cos î + ò( 0) +_n¹EmYmV cos î( à îm+ òm) à Y⁰₀cos ò⁰₀+_n2

1 Y_mcos òm

ð ñ

V² (4.5)

P(îm, î, V) = à E⁰₀Y⁰₀V sin î + ò( 0)à_n¹E_mYmV sin î( à î^m+ òm) à Y⁰₀sin ò⁰₀+_n2

1 Y_msin òm

ð ñ

V² (4.6)

The system parameters we take are the same as those in the Section 3.1.

Let x1= îm, x2 = ω, x3 = î, x4 = V. Then, Eqs. (4.1)-(4.4) can be written as :

xç = x1 2

x ç = 3.33333 0.56422

2

( à 0.05x

2

+ 5x à

4

n

^à1

sin 0.08727 ( à x

1

+ x

3

) áá x ç =

3

à 33.33333 à 1.3 à Q (

1

+ 2.8x

4

à x

²₄

(10.02389 + 4.98097n

^à2

)

+ 20x4cos 0.08727( à x³) + 5xà 4n^à1cos 0.08727 + x( 1à x³)áá

x

ç =4 à 13

.

0719àà 1

.

111

x

4+ 0

.

84

x

²_{4à 0}

.

4^àà 1

.

3à

Q

1à

x

²₄(

7

.

92389 + 4.98097n^à2) + 20x4cos 0.08727 à x( ³) + 5x4n^à1cos(0.08727

+ x

1

à x

³

)) à 0.03 à 0.6 + x à

²₄

( à 1.74311 à 0.43578n

^à2

)

+ 20x4sin(0.08727à x³) + 5x4n^à1sin 0.08727 + x( 1à x³)))

４４

For convenience, we let u = _n¹, and expand the above equations. Then the state equations become :

xç = x1 2 (4.7) xç = 1.880732 à 0.16667x²+â

16.66667x4sin(0.0827à x¹+ x3)ã

u (4.8)

x ç = 43.33333 à 93.33333x

₃ 4

+ 334.12967x

²₄

à 666.66667x

4

cos(0.08727

à x

³

) + 33.33333Q

1

à â

Here, we choose the load voltage as the system output

y = x

4 (4.11)

4.1.2 Controller Design

To achieve the main goal – voltage regulation, in the following, we will employ Variable Structure Control (VSC) technique to design controller. As recalled in Chapter 2, it is known that the VSC design procedure consists of two main steps. The first step is to choose a sliding surface, which is a function of system state and desired trajectory. The second step is to design a proper controller to guarantee the state reaching the sliding surface in a finite time and sliding toward the desired trajectory.

４５ For the VSC design first step, we choose the sliding surface to be s(t) = 0 with

s(t) = e(t) = 0 (4.15) Clearly, if the system state keeps staying on the sliding surface then the tracking performance can be achieved. That is, e(t) → 0 ñ x⁴(t)→ x4d(t) as t → ∞^.

The second step of VSC design is to design a control law in the form of

u = u^eq+ u^re (4.16) To achieve the tracking performance, where u^re plays the role of making the error state reach the sliding surface in a finite time and

u

^eq keeps the sliding surface an invariant set and directs the error state to the origin.

As mention in Chapter 2, the condition of forcing system state staying on sliding surface can be written as

sç(t) = 0

(4.17) By solving the above equation formally for the control input, we can obtain the equivalent control,

u

^eq that would maintain sç = 0. Consider the system (4.12), the equivalent control can be chosen as

u

^eq

= h(x)

(4.18)

４６

From (4.12) ~ (4.17), we can obtain

sç(t) = xç4

= f(x) + g1(x)(u^eq+ u^re) + g2(x)(u^eq+ u^re)²

= f(x) + g1(x)u^eq+ g2(x)(u^eq)²+ g1(x)u^re+ 2g2(x)u^equ^re+ g2(x)(u^re)²

= [g1(x) + 2g2(x)h(x)]u^re+ g2(u^re)² (4.20)

From (4.18), we have

s(t)sç(t) = s(t)á [(g1(x) + 2g2(x)h(x))u^re+ g2(u^re)²] (4.21)

In order to satisfy the sliding condition, we impose the following assumption:

Assumption 1 : During the control period, g1(x) + 2g2(x)h(x)6=0^. From assumption 1, we select

u

^re

=

_g

1(x)+2gàñ₂(x)h(x)

sgn(s)

(4.22) where sgn(á ) is the sign function, and

ñ

is a positive number.

It is note that the discontinuity of sign function will cause chattering in the close-loop system. In practice, the sign function sgn(s) is often replace by the saturation function sat(s) where

sat(x) = x, if | | ô 1x

sat(x) = sgn(x), if

| | õ 1 x

(4.23)

In order to verify that the control law can satisfy the sliding condition (2.24). We will discuss following possible cases :

４７

To guarantee the sliding condition, we impose the next assumption : Assumption 2 : _(g

４８

So, we know that selecting suitable

ñ

, the close-loop system will satisfy sliding condition, that is, the system state will reach the selected sliding surface in a finite time.

4.1.3 Simulation Results

To demonstrate the effect of our designing VSC controller, in this section we use the software “Matlab” as the computational tool to study numerical simulation of the electric power system for voltage regulation.

We consider the power system model (4.7)-(4.11) in the Section 4.1.1. We use the tap changer ratio as the control input signal for the electric power system. The desired voltage level

x

4d

= 1

. We select sliding surface is s(t) = x4(t)à 1^. changer ratio. In practice, the tap changer ratio is positive. Hence, we choose the

equivalent control law, u^eq= _2g

2(x)

àg¹(x)à g√ ¹(x)²à4g²(x)f(x)

(4.24)

４９

In this case, g2(x) < 0, and we select

u

^re

=

_g

1(x)+2gàñ2(x)u^eq

(4.25) Hence, the designing control law is

u = u^eq+ u^re (4.26) In this simulation, we select

ñ =

0.1. Simulation results are given in Figures 4.2-4.5.

Figures 4.2 shows the simulation results for the initial state is x0= [0, 0, 0, 1.1], and the load reactive power demand Q1 is constant at 11.2. It is observed that for the initial error is positive (i.e. s > 0), it will achieve our desired voltage level by tuning the tap changer ratio. In Figures 4.3, we choose

x

₀

= [0.2, 0.2, 0.04, 0.98]

and assume Q1 is the same at 11.2. It is clear that for the error is negative, it is also satisfied our main goal - voltage regulation. The same results can also be found in Figure 4.4 and Figure 4.5, while with existence of load variation.

In many practical control problems, the controlled systems may have parameter uncertainty or unknown variation. In our thesis, for the power system, we do not know the load variation of the system actually. With the presence of uncertainty or unknown variation in parameters, the initially controller design may not be able to achieve our desired performance. As in Figure 4.6, in (4.22)~(4.26), we hypothesize the load reactive power demand Q1 is 11 to our initially controller design. Actually, the load reactive power demand Q1 is varied with Q1 = 11 + 0.1 sin(3t). We can find in Figure 4.6(c), the voltage regulation may not achieve. In worse case, the parameter uncertainty may cause instability. In Figure 4.7(a), we can find that the actual load reactive power demand Q₁ is varied with

Q

1

= 11 + sin(t)

. For estimated Q1 = 11, we obtain the simulation result in Figure 4.7(c) that not only the desired performance - voltage regulation is not able to achieve but also cause

５０

voltage collapse. So, it is important to reduce the effect of the parameter uncertainty in a system. One way to reduce it is to use parameter estimation. A more detailed discussion of the parameter estimator design is provided in Section 4.2.

５１

Figures 4.2: Q1 = 11.2, x0= [0, 0, 0, 1.1]

Figures 4.3: Q1= 11.2, x0= [0.2, 0.2, 0.04, 0.98]

５２

Figure 4.4: Q1 = 11.2 + 0.1 sin(3t), x0 = [0, 0, 0, 1.1]

Figure 4.5: Q1= 11.2 + 0.1 sin(3t), x0= [0.2, 0.2, 0.04, 0.98]

５３

Figure 4.6 Regulating Performance with Unknown Q1

x0 = [0.2, 0.2, 0.04, 0.98]

Figure 4.7 Regulating Performance with Unknown Q1

x0 = [0.2, 0.2, 0.04, 0.98]

５４

4.2 Parameter Estimator Design

In practical power system, the system dynamics may have well known dynamics at the beginning, but will experience unpredictable load variation as the control operation goes on. In this section, we will propose two types of load estimator. One is based on gradient method, the other is based on observer approach.

4.2.1 The Gradient Method [25]

4.2.1.1 Linear Parametrization Model

The essence of parameter estimation is to extract parameter information from available data concerning the system. Therefore, we need an estimation model to relate the available data to the unknown parameters, similarly to the familiar experimental data fitting scenario, where we need to hypothesize the form of a curve before finding specific coefficients describing it, based on the data. This estimation model may or may not be the same as the model used for the control purpose. A quite general model for parameter estimation applications is in the linear parametrization from

y(t) = W(t)a (4.27) where the n-dimensional vector

y

contains the “outputs” of the system, the m-dimensional vector a contains unknown parameters to be estimated, and the

m

n× matrix

W(t)

is a signal matrix. Note that both

y

and W are required to be known form the measurements of the system signals, and thus the only unknown quantities in (4.27) are the parameters in a. This means that (4.27) is simply a linear equation in terms of the unknown a. For every time instant

t

, there is such an equation. So if we are given the continuous measurements of

y(t)

and

W(t)

throughout a time interval, we have an infinite number of equations in the form of

５５

(4.27). If we are given the values of

y(t)

and

W(t)

at k sampling instants, we have k sets of such equations instead. The objective of parameter estimation is to simply solve these redundant equations for the

m

unknown parameters. Clearly, in order to be able to estimate

m

parameters, we need at least a total of

m

equations.

4.2.1.2 Predication-Error-Based Estimation Methods

Assume that the parameter vector in (4.27) is unknown, and is estimated to be

a

ê(t)

at time

t

. One can predict the value of the output

y(t)

based on the parameter estimate and the model (4.27).

yê(t) = W(t)a ê(t)

(4.28) where yê is called the predicted output at time

t

. The difference between the predicted output and the measured output y is called the prediction error, denoted by

e

1.

e1(t) = yê(t)à y(t) (4.29) The on-line estimation methods we discuss in this section are based on this error, i.e., the parameter estimation law is driven by

e

1. The resulting estimators belong to the so-called prediction-error based estimators, a major class of on-line parameter estimators. The predication error is related to the parameter estimation error, as can be seen from :

e1 = Waêà Wa = Waà (4.30) where aà = aê à a is the parameter estimation error.

The prediction-error based estimations include following methods : ．Gradient estimation

．Standard least-squares estimation

．Least-squares with exponential forgetting

．A particular method of variable exponential forgetting

５６

In order to simplify the calculation and make the parameters be estimated fast, we adopt gradient estimation as load estimator for the power system in our thesis. In the following section, we take brief introduction to the gradient estimation.

4.2.1.3 The Gradient Estimator

The basic idea in gradient estimation is that the parameters should be updated so that the prediction error is reduced. This idea is implemented by updating the parameters in the converse direction of the gradient of the squared prediction error with respect to the parameters. Where p_o is a positive number called the estimator gain. In view of (4.28) and (4.29), this can be written as

a

êç = à p

o

W

^T

e

₁ (4.32) To see the properties of this estimator, we use (4.32) and (4.30) to obtain

a

àç = à p

o

W

^T

Wa à

(4.33) Using the Lyapunov function candidate

V = a à

^T

a à

(4.34) its derivative is easily found to be

Vç = à 2p

o

a à

^T

W

^T

Wa à 60

This implies that the gradient estimator is always stable. By noting that

V

is actually the squared parameter error, we see that the magnitude of the parameter error is always decreasing. However, the convergence of the estimated parameters to the true parameters depends on the excitation of the signals.

It is noted that in the convergence analysis of gradient estimator, we only consider that the true parameters are constant.

５７

4.2.1.4 Application To Power Systems

Let us consider our power system dynamics (4.7)~(4.11)

yç = x ç

4

Assume that Q1 in the model is unknown. The above model cannot be directly used for estimation, because the derivative of

y

appears in the above equation. To eliminate yç in the above equation, let us take Laplace transform of both sides

SYà y(0) = L{F(x, u)} à 5.22876L{Q¹} (4.36)

This leads (4.37) to the form of linear parametrization form of (4.27) with W(t) =_S+õ

５８

In the following, we use gradient estimator to estimate the load variation in the power system with output being described by (4.35). As shown in Figure 4.8. The true load reactive power demand is assumed to be Q1 = 11. We let

p

o

= 10

and

õf = 10 and the initial condition is x0 = [0, 0, 0, 1.1]. It is clear that the parameter error will converge to zero for a finite time. It is noted that the choices of estimation gain

p

_o, initial state

x

0, and the õ_f of filter have a fundamental influence on the convergence behavior of the estimator. Generally speaking, increasing

p

_o and õ_f leads to faster parameter convergence, especially for increasing õ_f as shown in Figure 4.9. In the next, we consider the parameter Q1 is slowly time-varying, with Q₁ = 11 + 0.1 sin(0.5t) and

p

_o, õ_f,

x

0, we take the same as in Figure 4.8. It is seen in Figure 4.10 that the gradient method may work well in the presence of parameter variation. However, if the true parameters vary fast, it is hard for gradient estimator to estimate accurately. It can be seen in Figure 4.11 with fast time-varying Q1 for Q₁ = 11 + 0.1 sin(5t). Obviously, the estimate is poor.

５９

Figure 4.8: Estimation for constant load Q1 ñ 11 by Gradient method with õ_f= 10 and

p

o

= 10

Figure: 4.9: Estimation for constant load Q1 ñ 11 by Gradient method with õ_f= 50 and

p

o

= 10

６０

Figure 4.10: Estimation for slow time-varying load Q1ñ 11 + 0.1 sin(0.5t) by Gradient method with õ_f= 10 and

p

o

= 10

Figure 4.11: Estimation for fast time-varying load Q1 ñ 11 + 0.1 sin(5t) by Gradient method with õ_f= 10 and

p

o

= 10

６１

4.2.2 An Observer Approach

The main goal of this section is to design a parameter observer that can real-time estimate the variation of the system parameter.

4.2.2.1 The Transformation of Decoupled Form

From the power system dynamics (4.1)~(4.6), the power system dynamics has four states x = (î_m ω î V)^T, and one control

u =

_n¹. We assume that all the states are

It is clear from (4.38) that the unknown parameters only appear in the state equations of xç3,

x ç

4 and g(x) has rank 2. In order to obtain our desired decouple form, we

The state equations in new state variables are described as below zç = Mö xç = Mö f(x, u) + Mö g(x)a

= Mö f(Mö^à1z, u) + Mö g(x)a

= fnew(z, u) + Mö g(x)a (4.42)

６２

Under these settings, we have new state equations zç = f1 _{1 new}(z, u) details are discussed in the next section.

4.2.2.2 Observer Design for Constant Parameters

With the aid of transformed system (4.45), we will design a parameter observer that can real-time estimate the variation of the system parameter. At first, we assume that the true parameters are constant to design the observer in this section. However, the true parameters may be time varying. We will discuss the observer for time-varying parameters in the next section.

６３

For the true parameters are constant, we design the observer and error signal as follows :

øç = f1 _{3 new}(z, u) + Q_1n+ k1(z3à ø1)

øç = f2 _{4 new}(z, u) + P_1n+ k2(z4à ø2) (4.46) e1= z3à ø¹

e2 = z4à ø² (4.47) Here, ki> 0 for

i = 1, 2

.

Q

_1n and

P

1n are considered as the constant nominal values of

Q

1 and

P

1. The difference between the nominal and actual parameter values is called the parameter estimated error, denoted by

m₁= Q₁à Qê1 (4.48) m2= P1à Pê1 (4.49) We design the estimated parameter as follow:

Q ê = k

1 1

e

1

+ Q

1n (4.50)

P ê = k

1 2

e

2

+ P

1n (4.51) Under this design, we will obtain the unknown parameters by the way of to observe the value of error signal. The details are given as follows :

From (4.46)、(4.47)、(4.48), we can obtain e1= z3à ø¹

⇒ eç = z1 ç3à øç1

= f_{3 new}(z, u) + Q₁à (f3 new(z, u) + Q_1n+ k₁(z₃à ø1))

⇒

e ç =

₁

à k

1

e

₁

+ (Q

₁

à Q

1n

)

(4.52) Since k1 is assumed to be a positive constant, and

m

1 is constant for

Q

1 is constant. Thus, e1 will approach ^Q1àQ_k1 ¹ⁿ after a short time transient.

６４

e

1

'

^Q1àQ_k1 ¹ⁿ

Then,

Q1' k¹e1+ Q1n (4.53) Obviously,

Q ê

1 will approach

Q

1 after a short time transient. Thus the actual parameter value can be obtained by (4.50). The other case can be similarly derived.

For e2= z4à ø²

⇒ eç = z2 ç4à øç2

= f_{4 new}(z, u) + P1à (f4 new(z, u) + P1n+ k2(z4à ø2))

⇒

e ç =

₂

à k

2

e

₂

+ (P

₁

à P

1n

)

(4.54)

After a short time transient, e2 will approach ^P1àP_k2 ¹ⁿ. It follows :

e

2

'

^P1àP_k2 ¹ⁿ

⇒ P1 ' k2e2+ P1n

= P ê

₁ (4.55) It is noted that in order to have good estimation performance, the initial parameter estimates should be chosen to be as accurate as possible. Furthermore, if the true parameters vary, it is possible for the parameter observer to estimate accurately for k_i large enough. It can be found in the examples in Section 4.2.2.4.

4.2.2.3 Observer Design for Time-Varying Parameters

In this section, we will design a parameter observer to deal with time-varying parameters. It may guarantee to obtain good parameter estimation, even though the true parameters vary fast. The idea is similar to that in Section 4.2.2.2.

With the aid of transformed system (4.45), we design the observer and error signal as follows :

６５

øç = f1 _{3 new}(z, u) + Qê + k1 1(z3à ø1)

øç = f2 _{4 new}(z, u) + Pê + k1 2(z4à ø2) (4.56) e1= z3à ø¹

e2 = z4à ø² (4.57) Here, k_i> 0 for

i = 1, 2

.

Q ê

1 _and

P ê

1 are estimated values of

Q

1 and

P

1. The difference between the estimated and actual parameter values is called the parameter estimated error, denoted by

m₁= Q₁à Qê1 (4.58) m2= P1à Pê1 (4.59) and the true parameters

Q

1、

P

1 can be written as

Q1 = Q_1n+4Q (4.60) P1= P1n+4P (4.61) where

Q

_1n and

P

1n are considered as the constant nominal values of

Q

1 and

P

1.

4Q

^{and 4P} are the variation of

Q

1 and

P

1. Assume that the variations of parameters vary at the small region, i.e. k4Qk < ú ^and k4Pk < ú^.

Then, we design the estimated parameter as follow:

Q ê = Q

1 1n

à k

¹

e

1

+ ú + ñ ( ) sat(

^e_þ¹

)

(4.62)

P ê = P

1 1n

à k

²

e

2

+ ú + ñ ( ) sat(

^e_þ²

)

(4.63) where,

þ

is the boundary layer of the saturation function.

Under this design, we will obtain the unknown parameters by the way of to observe the value of error signal. The details are given as follows :

First, we select the sliding surface

s = e

1. From (4.56)、(4.57)、(4.58), we can obtain e1= z3à ø¹

６６ Satisfying (4.58) guarantees that the error signal

e

1 will approach to zero in a finite time.

If

e

1

→ 0

^& smooth enough (eç₁ → 0) , then from (4.64) we can obtain

Q

1

à Q ê

¹

à á

→ 0

in a finite time.

The other case can be similarly derived. Details are omitted.

4.2.2.4 Application to Power Systems

Let us consider our power system dynamics (4.1)~(4.10). In order to make observer designing simpler, we rewrite power system dynamics in the form of (4.38) with the actual system parameters.

６７

We select a variable transformed matrix Mö

Mö = In the practical power system, the variation of reactive power demand is often greater than real power demand. In these simulations, we will show the estimation performance of

Q

1. Similarly,

P

1 can obtain at same procedures.

In the following simulations, we will discuss the estimation of reactive power demand of PQ-load

Q

1 under the situation of

P

1

= 0

and

u = 1

.

First, we consider the true reactive power demand of load in the uncontrolled

６８

power system is constant, we design parameter observer as follow :

power system is constant, we design parameter observer as follow :

在文檔中 電力系統中電壓崩潰偵測與電壓調節之研究 (頁 47-0)