1
00:00:17,350 --> 00:00:26,050
Based on the understanding of how linear systems
behave during transient conditions, we were
2
00:00:26,050 --> 00:00:33,710
able to understand the simple linear circuit
consisting of capacitors and inductors in
3
00:00:33,710 --> 00:00:39,250
the previous lecture. In fact, there was one
capacitor and two inductors in that circuit.
4
00:00:39,250 --> 00:00:43,970
We could take out a time response. And very
importantly, we could take out a few characteristics
5
00:00:43,970 --> 00:00:49,180
of the system. In fact, the characteristics
of the system depend on the model parameters,
6
00:00:49,180 --> 00:00:53,390
in addition to the circuit itself.
Now, one of the most important things which
7
00:00:53,390 --> 00:00:58,930
came out of the study was that, there is a
combination of fast and slow transients. Now,
8
00:00:58,930 --> 00:01:04,750
this is a very fundamental, you know kind
of modeling issues which usually encounter
9
00:01:04,750 --> 00:01:11,611
in our studies of power systems. So, I would
recommend that a great deal of attention to
10
00:01:11,611 --> 00:01:16,299
be paid on these; the modeling principles
as well as systems in which there are mixture
11
00:01:16,299 --> 00:01:21,340
of fast and slow transients. A mixture of
fast and slow transients also brings into
12
00:01:21,340 --> 00:01:28,890
play several effects, when we try to numerically
simulate these systems. That is, when we are
13
00:01:28,890 --> 00:01:35,409
trying to understand the system by numerically
integrating them. A stiff system or a stiff
14
00:01:35,409 --> 00:01:43,689
system in which there are fast and slow transients
coming into the picture require some specific
15
00:01:43,689 --> 00:01:46,570
tools, numerical tools to solve.
16
00:01:46,570 --> 00:01:51,440
We shall see that, we shall be able to make
some modeling simplifications based on the
17
00:01:51,440 --> 00:01:57,619
fact that there is a clear time scale difference
between the nature of transients, which are
18
00:01:57,619 --> 00:02:07,189
seen in certain circuits. Now, so this lecture,
we will try to understand stiff systems and
19
00:02:07,189 --> 00:02:13,930
multi time scale modeling. We will do that
of course by considering again the same example,
20
00:02:13,930 --> 00:02:18,370
which we discussed in the previous class.
If we find time, by the end of the lecture
21
00:02:18,370 --> 00:02:25,800
we will also move on to numerical integration
methods. Just a brief overview.
22
00:02:25,800 --> 00:02:31,080
Now, if you recall what we did in the previous
class, we consider this particular circuit.
23
00:02:31,080 --> 00:02:36,470
It was consisting of a voltage source, which
is connected to a circuit consisting of two
24
00:02:36,470 --> 00:02:44,770
inductors, a capacitor. The values are of
importance. Here, you had a resistance of
25
00:02:44,770 --> 00:02:49,820
0.1 ohm 10 milli Henry inductor, 1 Henry,
this is a fat inductor and a 100 microfarad
26
00:02:49,820 --> 00:02:54,140
capacitor. And, we tried to analyze the transients
associated with this.
27
00:02:54,140 --> 00:02:59,500
Now, if you look at it from mathematical point
of view, it boils down to trying to solve
28
00:02:59,500 --> 00:03:05,370
this particular circuit, this particular system.
It is a… we write down the state space equations
29
00:03:05,370 --> 00:03:12,200
for this system. The states being the current
through the inductor 10 milli Henry, the current
30
00:03:12,200 --> 00:03:18,810
through the 1 Henry inductor and the voltage
across the capacitor. This of course, is the
31
00:03:18,810 --> 00:03:24,810
system we derived last time and we try to
find out what was the response if the initial
32
00:03:24,810 --> 00:03:30,450
conditions are these. I mean, this is the
initial conditions before the circuit is energized
33
00:03:30,450 --> 00:03:31,650
by an input.
34
00:03:31,650 --> 00:03:42,220
Now what we saw was, the response of course,
is given by this formula e raised to the power
35
00:03:42,220 --> 00:03:50,090
of A t into the initial conditions plus this
convolution integral. Eventually, the response
36
00:03:50,090 --> 00:04:02,400
of I l, I 2 and V c turns out to be this,
which was evaluated; effectively required
37
00:04:02,400 --> 00:04:13,190
us to evaluate this, where P is the Eigen
vector matrix and it required us to get lambda
38
00:04:13,190 --> 00:04:16,160
1, lambda 2 and lambda 3 and the Eigen vector
matrix.
39
00:04:16,160 --> 00:04:23,310
Now, this was got using Sci lab, you can also
use mat lab for getting the Eigen vector matrix
40
00:04:23,310 --> 00:04:34,620
A. Now, if you recall what A is, A is this.
Now the most notable feature here, when we
41
00:04:34,620 --> 00:04:45,830
did this study was lambda 1 was approximately
equal to this, lambda 2 was this, there was
42
00:04:45,830 --> 00:04:53,240
a conjugate, complex conjugate pair and you
had a real Eigen value which also had a negative
43
00:04:53,240 --> 00:04:59,860
part. So, it was a stable, this is a stable
system. Now, these were the Eigen values and
44
00:04:59,860 --> 00:05:05,500
the most important feature which struck us
then was that the magnitude of this Eigen
45
00:05:05,500 --> 00:05:11,610
value. These this pair of Eigen values and
this one is vastly different. This is a very
46
00:05:11,610 --> 00:05:15,900
small Eigen value. This is a very large magnitude
Eigen value.
47
00:05:15,900 --> 00:05:21,380
So, one thing which we had mentioned sometime,
a large magnitude Eigen value is associated
48
00:05:21,380 --> 00:05:28,250
with very fast rates of change. So, that is
one important thing. So, we have got, some
49
00:05:28,250 --> 00:05:33,090
part of the system is going to some part of
the response is going to move very fast and
50
00:05:33,090 --> 00:05:38,181
some of it very slowly. Another thing which
we saw when we looked again like the, at the
51
00:05:38,181 --> 00:05:43,400
Eigen vectors, the Eigen vectors were approximated
by this.
52
00:05:43,400 --> 00:05:48,590
This is of course an approximation. We did
some rounding off. These were very small values.
53
00:05:48,590 --> 00:05:54,500
So, we have done this rounded off, rounded
it off to 0. This was practically point, this
54
00:05:54,500 --> 00:05:59,750
is practically 0.1. This is minus j 0.1 and
so on.
55
00:05:59,750 --> 00:06:06,090
So, the approximate Eigen vectors were these.
And, the most important thing we saw was in
56
00:06:06,090 --> 00:06:13,690
the second state in the Eigen vector corresponding
to the second state and the first and second
57
00:06:13,690 --> 00:06:19,950
Eigen value, these are corresponding to the
first and the second Eigen values. These are
58
00:06:19,950 --> 00:06:25,090
in fact the complex pair. The second state
has no practically zero obser vability. These
59
00:06:25,090 --> 00:06:31,560
were of course rounded off, but we saw that
these were almost zero. So, you will not observe
60
00:06:31,560 --> 00:06:43,810
terms corresponding to e raise to lambda 1
and lambda 2 t in the response. In fact, lambda
61
00:06:43,810 --> 00:06:50,010
1 and lambda 2 t were complex numbers. So,
the first and second Eigen values in fact
62
00:06:50,010 --> 00:06:53,610
pointed towards oscillated response.
63
00:06:53,610 --> 00:06:59,460
And the final response, of course was evaluated
to be this. So, what do you notice here? Of
64
00:06:59,460 --> 00:07:05,760
course, here you have got this particular
oscillatory response. This comes out because
65
00:07:05,760 --> 00:07:13,060
of the complex pair of Eigen values, conjugate
pair of Eigen values. You also have this particular
66
00:07:13,060 --> 00:07:24,520
mode. This is practically not observable in
the variable i L 2. So, this is where we stopped
67
00:07:24,520 --> 00:07:28,710
last time.
Now, one of the important things which we
68
00:07:28,710 --> 00:07:34,790
can see here is, there is of course, this
is a very… this particular transient is
69
00:07:34,790 --> 00:07:39,880
associated with high rate of change. You look
at the real part as well as the imaginary
70
00:07:39,880 --> 00:07:46,449
part; rather the frequency as well as the
rate of decay is quite high. So, if you look
71
00:07:46,449 --> 00:08:23,860
at, just we will redraw this circuit.
So, what we see is i L 1 is this and i L 2
72
00:08:23,860 --> 00:08:32,120
is this. Now in i L 2, we are hardly seeing
any oscillatory component. That is because
73
00:08:32,120 --> 00:08:38,339
of course, I mean intuitively, this being
a very large inductor, it does not happy with
74
00:08:38,339 --> 00:08:46,260
large rates of change. So, you will find that.
You know, this large inductor prevents a large
75
00:08:46,260 --> 00:08:52,019
fast rates of change of the current. So, that
is why in i L 2 1 is not surprised to see
76
00:08:52,019 --> 00:08:58,629
that the faster transient is not visible.
So, in fact if you look at the response of
77
00:08:58,629 --> 00:09:23,370
i L 2, you will find that it starts from zero.
Of course, if we and it grows to 10 amperes.
78
00:09:23,370 --> 00:09:36,190
So, it starts with 0 and grows to 10 amperes.
You can see this; as t tends to infinity the
79
00:09:36,190 --> 00:09:41,690
value of i L 2 becomes 10.
80
00:09:41,690 --> 00:09:48,019
This is roughly four times the time constant
associated with this Eigen value. So, Eigen
81
00:09:48,019 --> 00:09:54,889
value is approximately 0.1, the magnitude
of the Eigen value. So, the time constant
82
00:09:54,889 --> 00:10:01,899
associated with this is roughly 10 seconds.
So, around 40 seconds you will see that this
83
00:10:01,899 --> 00:10:07,699
builds up to near about to the final value.
So, this will be around 40 seconds. You hardly
84
00:10:07,699 --> 00:10:26,059
see any oscillatory response in this. Now,
what about i L 1? If you look at i L 1, the
85
00:10:26,059 --> 00:10:34,220
response is primarily consisting of this component.
And, a small component which is oscillatory,
86
00:10:34,220 --> 00:10:42,550
a decaying oscillatory form. So, what you
have got is a decaying oscillation and then
87
00:10:42,550 --> 00:10:52,329
again, this settles down to 10 amperes in
40, around 40 seconds.
88
00:10:52,329 --> 00:11:01,470
So, this is the response of i L 2. And, of
course V c; V c is interesting. We look at
89
00:11:01,470 --> 00:11:12,149
how V c looks like. It consists of this as
well as this. So, what it has here is exponentially
90
00:11:12,149 --> 00:11:18,579
decaying component as well as a decaying oscillatory
component of high frequency. So, what you
91
00:11:18,579 --> 00:11:37,850
are likely to find is something of this kind;
roughly around 40 seconds, it will become
92
00:11:37,850 --> 00:11:50,930
a very small value. And in the beginning,
there is an oscillatory component. Now when
93
00:11:50,930 --> 00:11:59,800
you are studying the circuit, a thing which
will get suggested automatically is, can we
94
00:11:59,800 --> 00:12:06,111
simplify the analysis of this circuit. For
example, if you look at the state space equation
95
00:12:06,111 --> 00:12:16,610
of the system, it appears that the response
for i L 2, the slowly varying variable will
96
00:12:16,610 --> 00:12:24,140
be preserved even if I make certain approximations
in this. Just look at what I am trying to
97
00:12:24,140 --> 00:12:39,600
say. What I am saying is, as far as the response
of i L 2 goes is the 1 Henry inductor goes,
98
00:12:39,600 --> 00:12:45,069
I can obtain it simply by trying to studying
this circuit. In fact, what is the response
99
00:12:45,069 --> 00:12:54,540
of this circuit? It is exactly what we got
for i L 2. The time constant is, of course
100
00:12:54,540 --> 00:13:00,060
10 seconds; the lambda for this circuit is
0.1, minus 0.1.
101
00:13:00,060 --> 00:13:07,470
So, what we… you know a very simple, a simplification
of the circuit does not seem to affect the
102
00:13:07,470 --> 00:13:13,970
response of i L 2. So, it appears that we
can actually make a simplification.
103
00:13:13,970 --> 00:13:36,610
In fact looking at it a bit formally, if I
set d i L 1 in this, I artificially set it
104
00:13:36,610 --> 00:14:00,329
to be 0. I mean, set this value here to be
0. The rest of the things will remain exactly
105
00:14:00,329 --> 00:14:23,939
the same. So, what I have done is just set
these two things to 0. In that case, what
106
00:14:23,939 --> 00:14:37,279
we have here is only one differential equation
corresponding to i L 2 and i L 1 and V c can
107
00:14:37,279 --> 00:14:44,800
in fact be written in terms of i L 2. So,
what we have done is, we have got now d i
108
00:14:44,800 --> 00:15:01,709
L 2 by d t is equal to V c. This is a differential
equation. And, i L 1 is nothing but i L 2.
109
00:15:01,709 --> 00:15:08,579
From this last equation, since this has been
set to 0 you have 10000 i L 1 is equal to
110
00:15:08,579 --> 00:15:15,509
minus 10 plus, sorry, 10000 i L 1 minus 10
thousand i L 2 is equal to 0; which actually
111
00:15:15,509 --> 00:15:19,639
gives you i L 1 is equal to i L 2.
112
00:15:19,639 --> 00:15:37,199
And from this, we will get V c is equal to
what? What we have here is minus 10 times
113
00:15:37,199 --> 00:16:02,519
i L 1 minus 100 V c plus 100 is equal to 0.
So, what we have from that is V c is equal
114
00:16:02,519 --> 00:16:28,680
to 100; which is, nothing but V c is equal
1 minus 0.1 i L 2 because i L 1 is equal to
115
00:16:28,680 --> 00:16:45,779
i L 2. So, this is what V c is, so it is algebraically,
so i L 1 and V c are algebraically related
116
00:16:45,779 --> 00:16:58,839
to i L 2. So, what we have is basically a
state space equation d i L 2 by d t is equal
117
00:16:58,839 --> 00:17:04,549
to V c, which is nothing but 1 minus 0.1 i
L 2.
118
00:17:04,549 --> 00:17:12,799
So, this is my one and only differential equation,
if I make this approximation. So, if I make
119
00:17:12,799 --> 00:17:21,339
this approximation you have reduced your system
size to this. And, if you want to get V c
120
00:17:21,339 --> 00:17:30,190
and i L 1, you have to use this algebraic
equation and this algebraic equation.
121
00:17:30,190 --> 00:17:37,630
So, what we have? If you look at this particular
system, if you look at the response of this
122
00:17:37,630 --> 00:17:53,090
particular system we have this. It is the
same or practically the same responses before
123
00:17:53,090 --> 00:18:04,340
what about i L 1. i L 1 also follows the same
response because i L 1 is equal to i L 2 and
124
00:18:04,340 --> 00:18:21,640
Vc is 1 minus 0.1i L 2. So, if you look at
the response of V c, it simply this is one.
125
00:18:21,640 --> 00:18:33,860
So, you will have this 10. This is not to
scale, like this. So, this is Vc. Now if you
126
00:18:33,860 --> 00:18:40,190
look at, we had drawn these responses before.
If you consider the whole system in its full
127
00:18:40,190 --> 00:18:50,840
glory, you have got this. If you make the
approximation which I made, you will get this.
128
00:18:50,840 --> 00:19:00,809
So, the basic issue which I would like to
point out here is that if you have got a system,
129
00:19:00,809 --> 00:19:08,510
please note carefully, if you have got a system
in which there are fast and slow transients
130
00:19:08,510 --> 00:19:18,299
and your interest is in the slow transients,
then you can get the slow transient behavior.
131
00:19:18,299 --> 00:19:23,090
Roughly correctly; you know, you will get
it almost correct. If you assume the rate
132
00:19:23,090 --> 00:19:30,269
of change of the variables, which you think
are associated with the fast transients is
133
00:19:30,269 --> 00:19:34,480
set to zero.
So, what I have done is that, I have artificially
134
00:19:34,480 --> 00:19:46,789
set the derivatives in this equation to 0
and made i L 1 and i V c algebraically related
135
00:19:46,789 --> 00:19:55,429
to i L 2. So, you can make a general statement
of this kind. So, if you have got some fast
136
00:19:55,429 --> 00:20:30,960
variables, then as far as the behavior, the
slow behavior of the system can be obtained
137
00:20:30,960 --> 00:20:45,169
simply by replacing this algebraic equation.
Rather this differential equation by an algebraic
138
00:20:45,169 --> 00:20:50,750
equation; that is, I am making x 1 and x 2,
x 2 algebraically related to x 1.
139
00:20:50,750 --> 00:20:57,880
So, in such a case, I will be able to capture
the slow transients without making too many
140
00:20:57,880 --> 00:21:03,130
errors. So, the basic point is, first thing
you need is this is a clear separation of
141
00:21:03,130 --> 00:21:08,039
fast and slow transients. There is a mixture
of fast and slow transients in the response
142
00:21:08,039 --> 00:21:15,309
as is seen in our system here. So, this is
a very very important, you know modeling principle
143
00:21:15,309 --> 00:21:21,880
or modeling simplification which we are making.
Now, this we, in fact this kind of modeling
144
00:21:21,880 --> 00:21:26,029
simplification, we seem to making all the
time without knowing it. For example, you
145
00:21:26,029 --> 00:21:44,860
just take a simple case of a bulb connected
to the plug
146
00:21:44,860 --> 00:21:55,539
and you are, this is a bulb. And, the basic
point is I want to study how this bulb lights
147
00:21:55,539 --> 00:22:01,600
up.
Now when you switch this on, the bulb practically
148
00:22:01,600 --> 00:22:07,830
lights up instantaneously. It is only a resistive
kind of load, but of course a purist may say,
149
00:22:07,830 --> 00:22:14,779
well, there are parasitic inductances and
capacitance is associated with this, the wire
150
00:22:14,779 --> 00:22:22,301
which connects the bulb to the plug point.
So, what you somebody may say is, well, if
151
00:22:22,301 --> 00:22:28,019
you really want to study how this bulb is
going to light up, you ought to model all
152
00:22:28,019 --> 00:22:34,360
the inductances, the distributed inductances
and capacitance as in a transmission line.
153
00:22:34,360 --> 00:22:40,580
Now the point is usually, you do not really
need to do that because the transients associated
154
00:22:40,580 --> 00:22:49,769
with this die down very soon. So, that is
why we can practically treat this wire as
155
00:22:49,769 --> 00:22:57,529
if it is directly a simple resistance less
connection to this point resistance and 0
156
00:22:57,529 --> 00:23:02,340
dynamics associated with this. Suppose, of
course if you are interested, in fact if one
157
00:23:02,340 --> 00:23:08,460
really are looking at a time scales of a few
nano seconds or a micro seconds, you may actually
158
00:23:08,460 --> 00:23:15,409
see some transients evolving even in this
simple circuit. But, usually we may not be
159
00:23:15,409 --> 00:23:17,690
interested in those fast transients. Ok.
160
00:23:17,690 --> 00:23:23,980
So, in such a case, remember you can make
an approximation. You can actually neglect
161
00:23:23,980 --> 00:23:31,049
the dynamics associated with that fast transients
by simply taking the state equations and setting
162
00:23:31,049 --> 00:23:37,830
the right hand side… writing that d by d
t is equal to 0 in those states, which we
163
00:23:37,830 --> 00:23:42,539
think are associated with the fast transients.
Of course, now this seems to be a kind of
164
00:23:42,539 --> 00:23:54,509
a chicken and egg story because to know which
elements or which states in a circuit are
165
00:23:54,509 --> 00:23:59,570
associated with slow or fast transients itself,
requires you to do an analysis of the system.
166
00:23:59,570 --> 00:24:04,720
So, you have to do full blown analysis and
then find out which states are associated
167
00:24:04,720 --> 00:24:09,130
with the fast transients and then you can
make the modeling simplification.
168
00:24:09,130 --> 00:24:15,460
But, actually engineers do not do that. In
most situations, it is fairly obvious to an
169
00:24:15,460 --> 00:24:21,259
engineer, what are the things which decay
fast and which other thing which are the transients,
170
00:24:21,259 --> 00:24:26,380
which really move very fast or which are the
transients which move very slowly and which
171
00:24:26,380 --> 00:24:31,519
are the states associated with them, even
at in a rough way. So, even through engineering,
172
00:24:31,519 --> 00:24:37,940
one should be able to find this out. For example,
in this particular circuit it is quite clear
173
00:24:37,940 --> 00:24:47,840
that the the large inductance, the magnetizing
inductance in that particular circuit. 1 Henry
174
00:24:47,840 --> 00:24:54,240
is much larger compared to the, you know the
leakage; that is the 0.1 Henry inductance
175
00:24:54,240 --> 00:24:59,160
and 1100 micro capacitance lightly, you know
it looks a large value.
176
00:24:59,160 --> 00:25:03,500
So, I could have guessed that I could have
probably made these assumptions. It was an
177
00:25:03,500 --> 00:25:08,919
engineer, I would not have waited. I would
have made this assumption and in so far as
178
00:25:08,919 --> 00:25:14,929
a slow transients are concerned, I would have
made very little error, even if I had just
179
00:25:14,929 --> 00:25:20,840
taken this circuit. So, if I wanted to represent,
only represents my slow transient correctly,
180
00:25:20,840 --> 00:25:29,990
it was adequate to have a circuit of this
kind. Now, what we really have done is there
181
00:25:29,990 --> 00:25:35,340
was an inductor here, but it was a very small
value. So, we assumed that it goes into steady
182
00:25:35,340 --> 00:25:41,710
state right away. It kind of is algebraically
related to the current here. And, we also
183
00:25:41,710 --> 00:25:48,490
have a capacitor here, which you have basically
said it is open circuited.
184
00:25:48,490 --> 00:25:57,630
So, we have basically got this particular
circuit which is marked in red by making the
185
00:25:57,630 --> 00:26:04,830
d by d t s associated with these states equal
to 0. But remember, if you do not want, if
186
00:26:04,830 --> 00:26:09,620
you want to make this approximation without
actually doing the analysis of the circuit,
187
00:26:09,620 --> 00:26:14,919
then you have to rely on engineering intuition.
Now, you can also make a converse kind of
188
00:26:14,919 --> 00:26:22,409
analysis. Suppose, you have got a system which
has got a mixture of slow and fast transients
189
00:26:22,409 --> 00:26:28,649
and you are interested in, what happens just
after the transient has occurred. You do not
190
00:26:28,649 --> 00:26:34,590
want to; we are not really interested in the
fast rates, the slow variations or even the
191
00:26:34,590 --> 00:26:41,190
final steady state. But, you are interested
in… what happens? Just a few, may be just
192
00:26:41,190 --> 00:26:48,000
for a short while after a transient has occurred.
For example, in this circuit, remember that
193
00:26:48,000 --> 00:26:59,730
your
V c varies like this. This is around 1 and
194
00:26:59,730 --> 00:27:06,800
this is around 40 seconds. Now, if my aim
is to study this fast transient here, so,
195
00:27:06,800 --> 00:27:11,610
of course there could be situation you are
interested in the fast transients. So, if
196
00:27:11,610 --> 00:27:16,571
you are interested in the fast transients,
can you guess what has to be done? Well, one
197
00:27:16,571 --> 00:27:21,270
thing is, you need to know the states which
are associated with the fast transient. Again,
198
00:27:21,270 --> 00:27:25,909
this is an intuitive thing. How you know which
states are associated with the fast transients?
199
00:27:25,909 --> 00:27:29,710
You need to really look at something more
than what we have studied till now. I will
200
00:27:29,710 --> 00:27:32,460
just mention that in a few more minutes from
now.
201
00:27:32,460 --> 00:27:37,960
But, right, now let us try to do it intuitively.
So, if I want to study the fast transients,
202
00:27:37,960 --> 00:28:04,519
probably if I assume, we will just first write
it down. These are the… if we assume that
203
00:28:04,519 --> 00:28:12,039
this particular state remains where it is
during this extremely short time period, you
204
00:28:12,039 --> 00:28:20,480
know that this is a very fast transient. So,
it will decay very fast if it is stable. So,
205
00:28:20,480 --> 00:28:27,140
rather I should say that I want to see the
transient in a very small window. In that
206
00:28:27,140 --> 00:28:36,370
window, i L 2 does not change at all. So,
what I will do is, I will rewrite my circuit
207
00:28:36,370 --> 00:28:50,340
in this fashion. I will just rewrite it on
another page; d i L 1 by d t d i L 2 by d
208
00:28:50,340 --> 00:29:20,590
t and d V c by d t is equal to… what I will
do is, this is the original state space equation.
209
00:29:20,590 --> 00:29:27,240
What I will do is, I will set this equal to
0. This is also as zero.
210
00:29:27,240 --> 00:29:36,710
So, what it means is that, i L 2 d i L 2 is
equal to 0. The rest of the thing is, of course
211
00:29:36,710 --> 00:29:52,039
remain the same. So, what we have is i L 2.
See, if I set d i L 2 is equal to 0, it implies
212
00:29:52,039 --> 00:30:00,249
that i L 2 is equal to as a function of time
is nothing but i L 2 0. So, what I am going
213
00:30:00,249 --> 00:30:02,409
to do is make this approximation.
214
00:30:02,409 --> 00:30:09,520
So, when I am interested in, what happens
for a short while? After a particular disturbances
215
00:30:09,520 --> 00:30:14,540
or the initial period after a transient has
occurred or some disturbance occurred, then
216
00:30:14,540 --> 00:30:21,169
I can make this assumption. If I know that
this variable is associated with slow transients
217
00:30:21,169 --> 00:30:29,529
only. So, if I know that, this is what I can
do. So, in such a case, you will have d i
218
00:30:29,529 --> 00:30:39,139
L 1 by d t d V c by d t. you know, what i
L 2 is because I have assumed it to be 0.
219
00:30:39,139 --> 00:31:13,750
So, you will have simply the state space equations
as… this is of course 0 because I have assumed
220
00:31:13,750 --> 00:31:16,820
that i L 2 does not change in the short period
which we have.
221
00:31:16,820 --> 00:31:22,259
So, this is the kind of a, converse kind of
approximation which we can make. Now, this
222
00:31:22,259 --> 00:31:27,929
particular circuit or this particular set
of equations, in fact this describes if this
223
00:31:27,929 --> 00:31:35,190
is set to 0, then in fact it describes this.
224
00:31:35,190 --> 00:31:49,009
So, in the fast period, when we can say that
this, the current in this does not move at
225
00:31:49,009 --> 00:31:54,499
all. So, we have for all practical purposes,
disconnected this. The current through this
226
00:31:54,499 --> 00:32:01,059
stays at 0. It does not change at this in
that case; you have got this fast dynamical
227
00:32:01,059 --> 00:32:07,029
equivalent of the circuit. And, the Eigen
values of this will have to be found out.
228
00:32:07,029 --> 00:32:13,899
In fact if you do a study of this system,
the Eigen values you will get are or I will
229
00:32:13,899 --> 00:32:18,129
just write down the directly the response
of this circuit. It will turn out to be i
230
00:32:18,129 --> 00:32:35,789
L 1 is equal to 0.5 e raise to minus 5 t sine
999.9; that is almost 1000 and V c of t. This
231
00:32:35,789 --> 00:32:41,790
is the final time response for this circuit.
I am just writing it down directly, you can
232
00:32:41,790 --> 00:33:04,629
take out the Eigen values and Eigen vectors
using Sci lab, Cos plus 0.005 sine…
233
00:33:04,629 --> 00:33:20,029
So, this is our response. So, what you see
is that i L 1… V c is this. If you look
234
00:33:20,029 --> 00:33:27,710
at… actually find out what V c is, if you
evaluate this it is this. Whereas, what we
235
00:33:27,710 --> 00:33:38,250
had got using the complete system
was something like this. So, actually only
236
00:33:38,250 --> 00:33:42,499
in the initial portion this gives the correct
result.
237
00:33:42,499 --> 00:33:50,070
So, if you are only interested in the initial
behavior; the very in a very short time period,
238
00:33:50,070 --> 00:33:56,429
if you are interested in the behavior, you
can assume that the slow state or the state
239
00:33:56,429 --> 00:34:03,370
associated with the slow variable it just
frozen at its pre disturbance value. So, that
240
00:34:03,370 --> 00:34:06,399
is an important modeling point, which you
should appreciate.
241
00:34:06,399 --> 00:34:12,360
Now, well, one of the simple reasons why we
would need to do this modeling approximation
242
00:34:12,360 --> 00:34:16,830
is that, if you have a system consisting of
fast and slow transients you can actually
243
00:34:16,830 --> 00:34:21,940
reduce the size of the system you want to
study. In fact, the differential as we have
244
00:34:21,940 --> 00:34:27,280
in this; of course simple example, we did
not get much of a improvement as far as the
245
00:34:27,280 --> 00:34:33,120
size of the system is concerned. in the…
we manage to get rid of two variables, two
246
00:34:33,120 --> 00:34:38,370
state variables and we are left with one state
variable d i L 2 by d t, when we wanted to
247
00:34:38,370 --> 00:34:43,460
understand the slow transients. And, if you
wanted to understand the fast transients,
248
00:34:43,460 --> 00:34:50,450
you could assume that state associated with
slow transients i L 2 was frozen at its pre
249
00:34:50,450 --> 00:34:54,100
disturbance value.
So, in this particular example, there seems
250
00:34:54,100 --> 00:35:01,410
to be no very great advantage computation,
otherwise of making this modeling simplification,
251
00:35:01,410 --> 00:35:07,400
but when you come to large systems, you know
consisting of very many diverse components,
252
00:35:07,400 --> 00:35:12,490
it does not make sense to make these modeling
simplification. So, one important thing you
253
00:35:12,490 --> 00:35:18,990
should note, which is true of any engineering
modeling is, you do not have to model the
254
00:35:18,990 --> 00:35:23,860
system in its full glory. You do not have
to start from Maxwell’s equation and model
255
00:35:23,860 --> 00:35:30,420
everything in terms of p d s and so on. You
can make some approximation, get some simplified
256
00:35:30,420 --> 00:35:37,630
models. You can even neglect the transients
d by d t is associated with some elements,
257
00:35:37,630 --> 00:35:43,300
which we know from engineering judgment, you
know they are associated only with the slow
258
00:35:43,300 --> 00:35:48,450
transients. So, you can actually make this
kind of time scale distinction and the modeling
259
00:35:48,450 --> 00:35:49,450
can be the approximate.
260
00:35:49,450 --> 00:35:54,970
Even in a power system, since we have many
diverse components, the huge variation in
261
00:35:54,970 --> 00:36:00,050
time scale. So, you can shift your attention
to this particular slide. So, I have drawn
262
00:36:00,050 --> 00:36:06,730
a kind of a transient and control spectrum.
So, if you just look at the kind of transients,
263
00:36:06,730 --> 00:36:10,140
you are likely to study, in fact you will
be studying some of these transients in this
264
00:36:10,140 --> 00:36:17,480
particular course, as the fastest transients,
the fastest transients in a power system are
265
00:36:17,480 --> 00:36:22,760
those associated with travelling wave phenomena
on the transmission lines, in fact in such
266
00:36:22,760 --> 00:36:27,230
a case, you will have to model transmission
lines in a great amount of detail. You know,
267
00:36:27,230 --> 00:36:30,940
you may have your model m s p d e’s partial
differential equations.
268
00:36:30,940 --> 00:36:37,980
But, of course when you are studying lightning
and switching transients, you can assume some
269
00:36:37,980 --> 00:36:43,010
things like the speed of the synchronous generators,
etcetera, are not going to vary much in 0.01
270
00:36:43,010 --> 00:36:49,030
seconds or 0.0001 seconds. So, you can make
modeling simplifications. So, if you are only
271
00:36:49,030 --> 00:36:54,020
interested in looking at lightning and switching
transients, you can assume that the slow transients
272
00:36:54,020 --> 00:37:01,880
are frozen at the pre disturbance values.
As you move towards the right in the spectrum,
273
00:37:01,880 --> 00:37:07,420
after lightning and switching transients you
have slower network transients as well as
274
00:37:07,420 --> 00:37:13,660
some torsional transients associated with
the shaft of turbines and generators.
275
00:37:13,660 --> 00:37:21,001
Even at a slower time scale, say from half
a second to around 10 seconds, you have what
276
00:37:21,001 --> 00:37:26,670
are known as relative angle dynamics. The
dynamics associated with machines staying
277
00:37:26,670 --> 00:37:32,090
in synchronism. So, if you give a… for example,
a push to a synchronous machine which is connected
278
00:37:32,090 --> 00:37:40,010
to another synchronous machine. You are likely
to excite oscillations of around 1 hertz or
279
00:37:40,010 --> 00:37:45,490
so. And also, if you give a very large disturbance,
you may have loss of stability. All these
280
00:37:45,490 --> 00:37:51,540
phenomena take place in a period of around
half a second to 10 seconds. So, all these
281
00:37:51,540 --> 00:37:58,200
things are visible in that if you take a snap
shot in this period. In fact if you are studying
282
00:37:58,200 --> 00:38:04,810
slow relative angle dynamics, you need not
model a transmission line by; you know the
283
00:38:04,810 --> 00:38:08,720
partial differential equations. In fact, you
do not even, you can even neglect the d by
284
00:38:08,720 --> 00:38:16,150
d t s associated with the transmission line
inductances and capacitances.
285
00:38:16,150 --> 00:38:24,320
At even slower scale at the prime mover dynamics…
slow frequency and load changes. These are
286
00:38:24,320 --> 00:38:29,900
slower, much slower. In fact, although it
is true that I can cause a load change by
287
00:38:29,900 --> 00:38:36,700
simply going and switching off a bulb, on
an aggregate sense, you know if you look at
288
00:38:36,700 --> 00:38:43,830
a substation, extra high voltage substation
or the power system as a whole, the over all
289
00:38:43,830 --> 00:38:50,420
power, you know power being consumed at a
given time does not vary very dramatically.
290
00:38:50,420 --> 00:38:58,030
I mean, you can almost predict how the load
changes. In fact, you know at unearthly hours
291
00:38:58,030 --> 00:39:03,840
like around 1 or 2 am, this load is very low.
And, after 10’ o clock in the morning, it
292
00:39:03,840 --> 00:39:10,140
starts rising. There is a kind of peak at
around 10:30 or 11. And, may be after 8’
293
00:39:10,140 --> 00:39:14,190
o clock in the night there is another big
peak that, is the highest peak in the system.
294
00:39:14,190 --> 00:39:20,660
This is a typical kind of loading scenario.
So, these of course things, these load changes
295
00:39:20,660 --> 00:39:26,080
can be considered as very slow changes. There,
you cannot even call them as disturbance unless
296
00:39:26,080 --> 00:39:32,740
you have a sudden load trip of say, you know
100 mega watt in a in a, say 20 giga watts
297
00:39:32,740 --> 00:39:38,440
system. So, this kind of thing can be considered
as a disturbance. But, generally load changes
298
00:39:38,440 --> 00:39:41,680
slowly and it is not really changing at a
very fast rate.
299
00:39:41,680 --> 00:39:46,630
Similarly, you can even, you know these are
the inherent, you know elements in the system,
300
00:39:46,630 --> 00:39:53,620
the power elements in the system. There are
elements which we put into the power system.
301
00:39:53,620 --> 00:39:58,770
These are usually control system designed
by us. For example, protection systems are
302
00:39:58,770 --> 00:40:04,030
a kind of special controls which are put in
a power system, which kind of isolate faultier
303
00:40:04,030 --> 00:40:13,800
equipment. So, the equipment protection can
be in the range of around 1 cycle to around
304
00:40:13,800 --> 00:40:21,600
1 second. So, this is typically the equipment,
range of equipment protection the transient,
305
00:40:21,600 --> 00:40:24,760
rather the time at which equipment protection
acts.
306
00:40:24,760 --> 00:40:31,140
Power electronic controls on the other hand
also, rather they also are very fast. I mean,
307
00:40:31,140 --> 00:40:35,910
you are trying to control. For example, a
rectifier in a h V d c station. That also
308
00:40:35,910 --> 00:40:41,610
is a very fast kind of moving system and the
controls associated with them are also fast.
309
00:40:41,610 --> 00:40:47,780
You are trying to control the firing of the
electronic wall. So, basically you find that
310
00:40:47,780 --> 00:40:53,830
the control systems, there also are fairly
fast. You also have other control systems
311
00:40:53,830 --> 00:41:01,310
like, for example, the excitation in a synchronous
machines is also controlled by what is known
312
00:41:01,310 --> 00:41:07,790
as automatic voltage regulator. That also,
you know, you consider as acting in the range
313
00:41:07,790 --> 00:41:13,420
of 0.1 to around 1 or 2 seconds.
So, that is a typical. You can say this response.
314
00:41:13,420 --> 00:41:19,080
I am I am deliberately not giving too much
precision into what I am trying to say. When
315
00:41:19,080 --> 00:41:24,390
we come to the topic later in this course,
when we model a V r’s automatic voltage
316
00:41:24,390 --> 00:41:28,840
regulator excitation systems, you will come
to know about the physical elements concerned
317
00:41:28,840 --> 00:41:36,030
with these systems. You also have system protection
schemes like; you know, I mentioned that if
318
00:41:36,030 --> 00:41:40,680
a system looses synchronism, you have got
synchronism machines loosing synchronism with
319
00:41:40,680 --> 00:41:46,340
other synchronous machines. You may wish to
do islanding; that is controlled system separation
320
00:41:46,340 --> 00:41:51,360
of the system or you can do under frequency
load shading and so and are different circumstances.
321
00:41:51,360 --> 00:41:57,440
For example, when under frequency load shading
is done, when there is a sudden, very gross
322
00:41:57,440 --> 00:42:02,570
kind of load generation imbalance.
So, you may, these are all called system protection
323
00:42:02,570 --> 00:42:09,510
schemes, prime mover controllers and governors,
they act between the range. The actuators
324
00:42:09,510 --> 00:42:15,760
associated these things required around 1
to 100 seconds. And, of course manual control
325
00:42:15,760 --> 00:42:21,860
is much slower of the order of minutes. You
do have continuous monitoring of the system
326
00:42:21,860 --> 00:42:26,980
and manual control actions. But, usually manual
control actions are associated with changing
327
00:42:26,980 --> 00:42:34,581
the set points of various automatic regulators
in the system as well as in some, you know
328
00:42:34,581 --> 00:42:42,020
unusual cases, you may even want to, you know
actuate your protection system manually.
329
00:42:42,020 --> 00:42:48,270
So, manual controls, of course are much much
slower. And, you know, they can take a longer
330
00:42:48,270 --> 00:42:55,420
longer time. Now, of course one of the important
points which we have come across is that a
331
00:42:55,420 --> 00:43:00,280
very important thing infact is the chicken
and egg story. You know, when do you really
332
00:43:00,280 --> 00:43:08,120
say that a particular state is associated
with a particular Eigen value or a mode. See,
333
00:43:08,120 --> 00:43:15,090
the basic point which we really try to tell
you today was that if you have got a slow
334
00:43:15,090 --> 00:43:22,740
transients, find out the, you know state variable
which are associated with the slow transients
335
00:43:22,740 --> 00:43:27,930
and the fast transients. Then in the fast
transients, you set the d by d t equal to
336
00:43:27,930 --> 00:43:34,030
0. I mean, set the right hand side rather
the left hand side of your state equation
337
00:43:34,030 --> 00:43:37,600
corresponding to the derivatives of those
fast variables equal to zero.
338
00:43:37,600 --> 00:43:42,030
So, I have converted your algebraic equations,
rather the differential equations into algebraic
339
00:43:42,030 --> 00:43:49,750
equations. Eliminate them if you can and just
work with the slow subsystem with the x 2
340
00:43:49,750 --> 00:43:54,820
the faster variable, simply algebraically
related to your slower variables. So, this
341
00:43:54,820 --> 00:44:01,470
is basically what we have learnt. But, the
whole point is how do you, which state is
342
00:44:01,470 --> 00:44:06,250
associated with, you know the faster and slower
transients. I told you, you have to do it
343
00:44:06,250 --> 00:44:12,900
intuitively. But, that is not a very, you
know comforting answer for students who are
344
00:44:12,900 --> 00:44:15,640
learning an Engineering subject for the first
time.
345
00:44:15,640 --> 00:44:22,420
So, let us just first quickly look at the
Eigen vectors associated. Perhaps, they will
346
00:44:22,420 --> 00:44:27,630
give us a clue. So, if you look at the Eigen
vectors associated with the slow and fast
347
00:44:27,630 --> 00:44:35,150
transients, remember lambda 1 and lambda 2.
Lambda 1 is minus 5 plus or minus plus or
348
00:44:35,150 --> 00:44:47,050
was it minus, it was plus j, lambda 2 was
minus 5 minus j and lambda 3 was point one,
349
00:44:47,050 --> 00:44:58,360
minus 0.1. What you see here is, in the second
state variable you do not observe lambda 1
350
00:44:58,360 --> 00:45:03,990
and lambda 2. There is very little observability.
So, perhaps it is practically zero.
351
00:45:03,990 --> 00:45:11,390
So, if such a situation occurs you can say,
may be you can say that i L 2 is not associated
352
00:45:11,390 --> 00:45:17,860
with the fast transients. But, there is a
pit fall. In just looking at these Eigen vectors
353
00:45:17,860 --> 00:45:22,670
to come to this conclusion, let look at the
converse thing; which are the variables are
354
00:45:22,670 --> 00:45:27,560
associated with the slow transients. So, you
will directly say, oh i L 2 was associated
355
00:45:27,560 --> 00:45:33,560
with the slow transients. But, look at the
components corresponding to i L 2, corresponding
356
00:45:33,560 --> 00:45:44,270
to lambda 3, the Eigen vector components corresponding
to the three states and the Eigen value minus
357
00:45:44,270 --> 00:45:51,860
0.1, this is a slow transients. You will see
that, in fact i L 1 also is associated with
358
00:45:51,860 --> 00:45:57,580
it. You will, you will observe this slow transient
is not only in i L 1, but also observe it
359
00:45:57,580 --> 00:45:59,280
in V c.
360
00:45:59,280 --> 00:46:05,340
So, just looking at these Eigen vector, components
of P will not tell you that i L 2 is associated
361
00:46:05,340 --> 00:46:11,830
with the slow transient. So, what you need
to do here really is, look at both P and P
362
00:46:11,830 --> 00:46:20,760
inverse. If you look at both P and P inverse,
so we will do this computation using Sci lab.
363
00:46:20,760 --> 00:46:36,630
So, I will just write down the the A matrix
first. The A matrix was minus 10 0 minus 100;
364
00:46:36,630 --> 00:46:53,430
0 0 1… So, this is A. so, the Eigen values
of A are obtained from this command. This
365
00:46:53,430 --> 00:46:59,130
is what we got. So, this is the Eigen value
0.1. We rounded off it to 0.1, minus 0.1.
366
00:46:59,130 --> 00:47:07,750
This is minus, approximately minus 5 plus
or minus j 1005 j or i is i or j are square
367
00:47:07,750 --> 00:47:12,570
root of minus one.
No, if you want to get the right Eigen values,
368
00:47:12,570 --> 00:47:21,180
right Eigen vectors, you have to use this
command. And, P gives you the right Eigen
369
00:47:21,180 --> 00:47:30,670
vectors. In fact, you see that this is roughly
j into 0.1. This is a small value. This is
370
00:47:30,670 --> 00:47:37,160
an extremely small value. So, we have approximated
it to be 0. In fact, if you look at what I
371
00:47:37,160 --> 00:47:42,540
have wrote here on the sheet, these are just
approximations of what you have seen there.
372
00:47:42,540 --> 00:47:47,500
So, just just rounded it off, you know, so
this is approximately 0.
373
00:47:47,500 --> 00:48:13,220
And, this is what we get is P. So, P is, I
will just rewrite it. P is… now, what is
374
00:48:13,220 --> 00:48:18,520
P inverse? P inverse is what we called as
Q.
375
00:48:18,520 --> 00:48:36,720
So, if you look at Q which is nothing but
P inverse, this is just roughly, I will just
376
00:48:36,720 --> 00:48:40,891
write it down roughly. So, it is roughly,
I will just write down while you can note
377
00:48:40,891 --> 00:48:56,450
it down also. It is minus j 5, plus j 5 approximately.
And, this is roughly 0.015 and this is j 5
378
00:48:56,450 --> 00:49:04,890
minus 5. It is not surprised that this Eigen
vector, this is the complex conjugate of this,
379
00:49:04,890 --> 00:49:20,270
sorry, minus 1.403, this is a complex conjugate,
1.043. I encourage you to download Sci lab
380
00:49:20,270 --> 00:49:28,540
and just try out these example, try out these
particular example yourself and this is almost
381
00:49:28,540 --> 00:49:29,540
zero.
382
00:49:29,540 --> 00:49:35,330
So, what we will do is, do an element by element
multiplication of this kind. We will multiply
383
00:49:35,330 --> 00:49:47,100
this by this, this by this, this by this and
we multiply this by this, this by this, this
384
00:49:47,100 --> 00:49:57,160
by this and so on. So, what we are doing effectively
is a kind of an element by element multiplication
385
00:49:57,160 --> 00:50:09,290
of P and P inverse transpose. So, what we
will do is use this command; P dot star inverse
386
00:50:09,290 --> 00:50:16,010
of P dot transpose. This is a transpose. So,
what we will get in such a case? So, what
387
00:50:16,010 --> 00:50:24,570
we have done is P dot star into P inverse
transpose. This dot star implies the element
388
00:50:24,570 --> 00:50:26,200
by element multiplication.
389
00:50:26,200 --> 00:50:31,680
\
So, this is given by 0.5, 0.5, roughly 0,
390
00:50:31,680 --> 00:50:51,170
roughly 0, roughly 0, 0.99. So, this is almost
1 and you have got 0.5, 0.5 and 0. So, if
391
00:50:51,170 --> 00:51:04,800
I compute P, I compute P inverse and I do
the computation P dot star; that is the element
392
00:51:04,800 --> 00:51:11,750
by element multiplication of P and P inverse
transpose. This is what I get. What we see
393
00:51:11,750 --> 00:51:18,020
here is, this is of course corresponding state
1, state 2 and state 3; this is Eigen value
394
00:51:18,020 --> 00:51:37,760
1, Eigen value 2 and Eigen value 3. This is
I L1, I L 2 and V c. What we see is that,
395
00:51:37,760 --> 00:51:48,630
this particular thing is called a participation
matrix participation matrix. And, it gives
396
00:51:48,630 --> 00:51:55,960
you the participation of a state in a mode,
for example, i L 1 is associated with the
397
00:51:55,960 --> 00:52:04,650
complex pair of modes. V c is also associated
with the complex pair of mode and i L 2 is
398
00:52:04,650 --> 00:52:12,980
associated fully, it is 1. It is a normalized
measure is completely associated with lambda
399
00:52:12,980 --> 00:52:18,730
three.
So, actually this particular matrix, P dot
400
00:52:18,730 --> 00:52:26,780
star P inverse transpose. This is the element
by element multiplication of these two matrices,
401
00:52:26,780 --> 00:52:32,760
actually gives you an interesting matrix called
the participation matrix. And, this participation
402
00:52:32,760 --> 00:52:41,510
matrix, in fact gives you correctly the association
of various states to various modes. So, if
403
00:52:41,510 --> 00:52:50,040
the participation of state number 3 in Eigen
value, Eigen value 3 is 0. It means that this
404
00:52:50,040 --> 00:52:53,840
state is not participating in that particular
Eigen value.
405
00:52:53,840 --> 00:53:02,190
So, if you relook at this participation matrix,
what you see is i L 1 participate in the complex
406
00:53:02,190 --> 00:53:10,330
mode, complex pair of mode, Eigen values and
i L 2 participates in this. In fact, remember
407
00:53:10,330 --> 00:53:17,670
that i L 3 is the slow mode and lambda 1 and
lambda 2 correspond to the fast transient.
408
00:53:17,670 --> 00:53:25,860
So, i L 1 and V c correspond to the fast transients
and i L 2 corresponds to the slow transient.
409
00:53:25,860 --> 00:53:32,320
So, we can actually, if you take out this
participation matrix, we are able to tell
410
00:53:32,320 --> 00:53:37,990
which states are associated with which modes.
Of course, there may be situations in which
411
00:53:37,990 --> 00:53:42,590
many states are associated with a certain
modes like, for example, i L 1 and V c are
412
00:53:42,590 --> 00:53:48,250
both associated with the fast transients.
So, you have to consider them together, whenever
413
00:53:48,250 --> 00:53:54,270
you do fast modeling, fast transient modeling.
Fast transient modeling means the slow transients
414
00:53:54,270 --> 00:53:57,380
are assumed to be frozen at the pre disturbance
state.
415
00:53:57,380 --> 00:54:04,450
So, this is a very important concept called
participation. We will, I will give you a
416
00:54:04,450 --> 00:54:11,900
reference for this in the next class again,
when we recap this particular part of the
417
00:54:11,900 --> 00:54:16,740
subject again. In the next class, I shall,
we shall move on to the next part of this
418
00:54:16,740 --> 00:54:23,481
course; that is, numerical integration techniques,
for you know numerically integration integration
419
00:54:23,481 --> 00:54:28,320
techniques for dynamical systems. Numerical
integration techniques will be required in
420
00:54:28,320 --> 00:54:34,480
systems which are too complicated to handle
by linearized analysis of the kind I have
421
00:54:34,480 --> 00:54:51,500
shown you today. So, with this, we will end
today’s lecture.