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SvM Lec Power Flow Intro — Transcript

by Prof. Zack Subin · 4,738 words · 860 segments · language en · Watch on YouTube

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0:02Hello everyone.
0:04This lecture is about power flow
0:06analysis.
0:07And in this first video, I'll give you a
0:10qualitative introduction to what it is
0:13we're trying to accomplish here.
0:15And then the next uh lecture will go
0:17into a bit more detail about the
0:20techniques uh that are used.
0:23So, in the big picture, what we're
0:26trying to do is we're trying to
0:28characterize
0:30an AC network,
0:33a synchronous uh AC network. This
0:35illustration
0:37uh is a simplified version of the
0:40Western Interconnect.
0:42And we're trying to identify the
0:45physical operating state of the network.
0:49And state is a term of art here. Uh the
0:52state variables
0:55are the voltages,
0:58where each voltage is a complex quantity
1:02or phasor with a magnitude and an angle.
1:06Um and the angle we might use
1:09uh lowercase delta or it could be theta
1:11as the variable
1:13that refers to the timing of the voltage
1:16waveform, right? Relative to the timing
1:20of the voltage
1:22somewhere else.
1:24Um and we're the the state of the the
1:28operating state of the system
1:31is defined by the set of voltages,
1:36magnitude and angle, everywhere at every
1:39node or bus of the network.
1:43And
1:44our job is to calculate
1:48those voltages, those state variables,
1:51based on the given input information,
1:54which is the power injection and
1:57withdrawal in terms of real power P and
2:00reactive power Q at each node
2:04or at each bus.
2:06And the power injection
2:09is much more along the lines of the way
2:11an economist thinks about the grid,
2:15right? Because that's how we capture our
2:17transactions.
2:19It's power being bought and sold into
2:21the grid.
2:23But physically
2:25we can't make power flow down a
2:29transmission line.
2:30The reason that power flows from
2:34node one to node two one end to the
2:37other of a transmission line is because
2:40there are different voltages maintained
2:42at either end.
2:44That's kind of the physical way uh to
2:47think about it. So, we are
2:50describing the physical state of the
2:53system as the set of voltages
2:56that are consistent with
2:59the power injections and withdrawals all
3:03across the grid.
3:05Um and as we'll see that's actually
3:08not a necessarily a unique one-to-one
3:11relationship.
3:13It this is its own topic because it's
3:15not a simple calculation
3:18um because it actually requires some
3:21trial and error or an iterative uh
3:24calculation
3:25and there's cases where it doesn't
3:27solve.
3:29Um
3:30so
3:32that's what I I want you to appreciate
3:35um the sort of subtlety and difficulty
3:37and really the profound takeaway here is
3:41that a physical way of of you know,
3:44thinking about power flow
3:46is very distinct from an economic way uh
3:49of thinking about power flow.
3:51This illustration is a vast
3:54oversimplification of the Western
3:56Interconnect. Um the real sort of
3:59professional grade representation has
4:02well over 10,000 nodes.
4:05Uh this is really distilled down to the
4:07major transmission paths and the major
4:10generation and load centers.
4:12Where the generators here are
4:13represented in circles.
4:15Uh the numbers are just
4:17cross-referenced. Uh they don't mean
4:19anything to us.
4:21Um and the arrows represent uh major
4:25load centers. So, you can appreciate the
4:27level of aggregation here, which is that
4:29in California we have, you know, the
4:31whole Bay Area is represented as a
4:33single load center.
4:35And then we have LA and so forth.
4:38Uh the big fat line here is our friend
4:41of Pacific DC Intertie.
4:44And uh you know, this is sort of a
4:48very simple representation, but just to
4:51give you sort of the mental picture
4:53of what
4:56um
4:57transmission operators um and uh
5:01authorities such as the California ISO
5:04would be looking at
5:06um and then they have computer
5:08algorithms running power flow analysis
5:12uh very frequently with updated values,
5:15where each analysis run is a snapshot of
5:19an in instantaneous operating state with
5:22some set of uh
5:25megawatt and megavar uh generation
5:29values across the system and some
5:31specific set of loads.
5:34And I want to distinguish that
5:36uh from optimal power flow, which we're
5:40not going to learn at all about how
5:42that's done, uh but you hear that
5:44terminology, and I want you to uh
5:47recognize the distinction here.
5:49The power flow analysis is answering a
5:52physics question. It's answering the
5:55physics question, what state
5:58of what or what set of state variables
6:02uh of voltage magnitudes and angles will
6:05produce the desired Ps and Qs
6:10all across the system.
6:12Optimal power flow
6:15answers the question
6:17based on some criteria or preferences
6:20that we define
6:21which dispatch, which combination of Ps
6:25and Qs at different locations do we like
6:28the best?
6:30And so to answer that question, it has
6:33to run many many power flow analyses
6:37with different scenarios that can then
6:40be compared according to the criteria
6:42such as least cost um and security
6:47constraints that uh will say a little
6:49bit more about later and so forth. But
6:51so optimal power flow is about finding a
6:54solution that we like the best that
6:57would give guidance uh to operations.
7:00Power flow analysis is what's running
7:02under the hood.
7:04And I want you to appreciate
7:07uh the complexity of power flow
7:10analysis, to appreciate the difference
7:14uh between thinking about power flow
7:17in a physical way rather than thinking
7:20about it in an economics way. And what
7:23we'll see is that
7:26you know, getting power to flow along a
7:29transmission link is not at all the same
7:33as sending a truckload of cargo from
7:35point A to point B
7:37because everything in the system
7:41is interactive
7:43to some degree.
7:45Uh but so power flow analysis would
7:48answer the type of question if if we
7:50make some change of power injection or
7:53power demand at some location I just
7:56arbitrarily picked some place
7:59what will be the effect on the way that
8:02power flows all across that network?
8:05Because any change at any one bus or
8:09node of this system
8:12any change in a voltage anywhere
8:15will result in a different set of state
8:20variables, a different maybe slightly
8:22different uh but different set of power
8:27flows, different set of currents through
8:30all these transmission lines. And so we
8:32might discover by running power flow
8:35analysis
8:36that if some market participants over
8:39here would like to
8:41change um their transaction power bought
8:44and sold somewhere
8:46that will have implications for the
8:49loading of a transmission line somewhere
8:51else in the system.
8:53And you don't know whether it does or
8:56not or whether that's consequential or
8:59not until you actually
9:02in effect simulate the whole system.
9:04You're in
9:05actually running the whole numbers to
9:08see again what set of voltages um
9:13will be consistent with that new uh
9:17desired state. And what I should say is
9:21that um we're starting with information
9:24about the transmission network which for
9:27purposes of this analysis we're holding
9:29fixed. There's no switches being thrown
9:32or anything.
9:33Um and once we know the voltages at
9:36every bus or every node and here all the
9:38buses are the horizontal lines in this
9:41uh toy illustration.
9:43Once we know all those voltages we
9:47already know all the uh impedances of
9:51all the transmission lines. So from that
9:53it's then straightforward to calculate
9:55the current
9:57uh in each line and the and the power
9:59flow in each line.
10:01But the hard part is getting from power
10:04injections to an entire profile of state
10:09variables, of voltage magnitudes and
10:11angles. And that's what the uh power
10:14flow
10:15analysis is doing.
10:18So,
10:20um this slide just summarizes exactly
10:22what I told you. We have um a set of
10:25impedances that could be expressed at in
10:27the inverse as admittances
10:30that uh represent
10:33all the properties of all the
10:35transmission lines and how they're
10:36interconnected.
10:38Um and in practice that's represented uh
10:41in a matrix format in linear algebra.
10:45And then we have a set of generation and
10:48loads that have some specified real and
10:51reactive power.
10:53Um the power flow analysis then cranks
10:57through the math to determine all the
10:59bus voltages and all the transmission
11:03link uh currents throughout the network.
11:06We are not looking at changes or
11:09transitions from one state into another.
11:11We're not looking at disturbances. This
11:14is strictly a steady state analysis.
11:17Um now one point I do want to make um
11:22and I won't get into too much detail
11:24here because you won't be doing
11:26this uh you won't be performing this uh
11:29calculation
11:31in detail, but I want you to appreciate
11:33that this is a non-linear
11:35um technique. And the a consequence of
11:39that is that there is not necessarily
11:43one solution.
11:45Uh it may be that there are multiple
11:48solutions that are consistent. Or it may
11:51be that there's no solution.
11:53And the set of power injections and
11:56demands that you specified would not
11:59actually work in physical practice.
12:05So, one way to describe non-linearity
12:08here
12:09is uh to just with this very simple uh
12:13toy example
12:15um to see that power and complex power
12:18would be letter S
12:20um for a branch here
12:25um
12:25in the middle. So, we're calling this uh
12:27current three
12:29um I three through this device, whatever
12:33it is,
12:34uh and voltage V3 across this device.
12:39This current branch will consume or
12:43produce some amount of complex power S
12:46which would be given by the product of
12:49current times voltage, right? Or using
12:51complex uh numbers, it's the complex
12:53conjugate of current uh times voltage, I
12:56star V.
12:58Um
12:59but because current and voltage are
13:02related to each other, it's really a
13:04quadratic function. So, we could say um
13:08it's really a function of voltage
13:10squared over the impedance
13:13uh or I squared times the impedance.
13:18Um
13:20It is It is not just a linear function
13:22because current and voltage are
13:25dependent on each other.
13:28It is
13:29the impedance of this branch that's sort
13:32of the the fixed quantity, but
13:34we're specifying the power, which is the
13:38product of current times voltage.
13:41So, it turns out in this case,
13:44there is uh if if we're given voltages
13:48V1 um
13:50and
13:52uh and V2 from two different sources and
13:56uh two currents are combining here,
13:59there are actually multiple ways to
14:01satisfy
14:03um
14:03some given amount of power that we want
14:06in this third branch.
14:09And it is that that is the non-linearity
14:13of the problem, so that this will force
14:16us to use a kind of trial and error
14:20uh method an iterative solution method
14:23to actually come up with um
14:25with the profile of voltages.
14:28Now, I'm going to expand
14:31this equation for complex power
14:35in uh a way that may look a little
14:37shocking.
14:39Um but these are the explicit power
14:44injections
14:45real power P and reactive power Q for
14:49each bus in the network or each node in
14:53the network uh where each bus has a
14:55subscript I.
14:57And here's
15:02what they are equal to. They are equal
15:05to the sum of power flows between node I
15:09and all the other nodes that node I
15:13whichever one were
15:15examining at the moment uh all the other
15:17nodes that it's connected to.
15:19So, how it's connected to the rest of
15:22the nodes in the network is expressed in
15:25terms of the G's and the B's which are
15:29the real and imaginary parts of the
15:31admittance.
15:33Admittance being the inverse of
15:34impedance. So, there's a conductance and
15:38a susceptance B um that being the
15:42imaginary part.
15:44And so, there may be some nodes some
15:48node pairs that aren't connected to each
15:50other. So, there's no conductance and
15:52susceptance between them. So, those
15:54terms don't show up, but summing over
15:58all the K nodes all the other nodes in
16:01the system
16:03um from one to uh to N and there should
16:08be an N on the top of the summation uh
16:10symbol here.
16:12Um
16:13we're going to add together all the
16:17power flows from node I to
16:22node K for all the Ks, which will be
16:25given by the product of the voltage
16:27magnitude. So, the um
16:29the absolute value signs uh represent
16:32the magnitude times the conductance and
16:35the cosine
16:37of the voltage phase angle difference
16:39between those nodes.
16:42And then another term that's related to
16:45the susceptance
16:47and the sign of that voltage phase angle
16:49difference between those nodes. And then
16:52I add that up for all the K nodes that
16:55my node I is connected to,
16:58and that will be equal to the power
17:00injection at node I.
17:03And an analogous equation for the
17:07reactive power injection and reactive
17:09power flows, except that sign uh and
17:13cosine are reversed and there's a
17:16a sign change.
17:18So, you will not have to worry about,
17:22you know, crunching through examples
17:25with this.
17:26The takeaway that I really want you to
17:29get uh from
17:32appreciating this uh these this pair of
17:35equations
17:37um is that if I give you the state
17:42variables, if I give you
17:45all the V sub I or V sub I V sub K,
17:50and I give you all the deltas,
17:53with that input information being armed
17:56with the knowledge of the transmission
17:58network, which is characterized by the
18:01Gs and the Bs for each branch connecting
18:05uh nodes,
18:08from that information, you could in a
18:11straightforward way uniquely calculate
18:14what are the Ps and Qs.
18:17Right? Because the Ps and the Qs are
18:19functions
18:20of
18:21the V's and the deltas.
18:24Where the G's and the B's are constants.
18:28But that's not what we're doing. We're
18:30going the other way around.
18:33We're given we're starting from the
18:36power injections and we're in a way
18:38trying to unpack or reverse engineer
18:41these equations
18:43to extract what is the physical with the
18:47set of physical state variables V and
18:50delta
18:51that would produce these P's and Q's.
18:54And so this is where uh it these
18:57functions are non-linear and we can't
19:00just
19:02turn them inside out. We can't just turn
19:05the signs and cosines uh inside out and
19:07solve. There's no closed-form solution.
19:11What we're going to have to do is a
19:12process of
19:15iterative computation that's essentially
19:17trial and error of plugging in different
19:20sets of V's and deltas to see if they
19:22give us the answer that we're looking
19:24for.
19:25Um
19:26now a qualitative takeaway
19:30again without really diving into the the
19:33details of how do you perform this
19:35calculation but a qualitative takeaway
19:38that is really worth remembering
19:41is that the real power
19:45injection and real power transfer
19:47between nodes
19:49will be more sensitive
19:52to the voltage phase angles delta. In
19:54other words, it'll be more sensitive to
19:56the timing of the voltage.
19:59And that is consistent with what we
20:01talked about in the context of
20:03generators.
20:05Uh consistent with this idea that if a
20:08generator is pushing harder
20:10and injecting more power into the system
20:14its rotor is going in the more forward
20:17position relative to other generators in
20:20the system.
20:21So that corresponds to a slight
20:24advancement in the timing of the voltage
20:27phase angle.
20:29So, that is one takeaway here is that
20:32the real power is going to be sensitive
20:35to the voltage phase angle delta or
20:38theta,
20:40however we call it.
20:42Whereas the reactive power Q is going to
20:45be more closely related to the voltage
20:48magnitude. And again, that is something
20:51that's consistent with our physical
20:53understanding of generator controls
20:55where the reactive power has to do with
20:59the excitation current and the generator
21:02bus voltage magnitude.
21:04Um and so
21:07it is
21:08to a first-order simplification,
21:11if we want to affect the generator bus
21:16voltage magnitude, we would change
21:20uh or adjust the real power. Excuse me.
21:24Stop. Rewind. If we want to change the
21:29generator bus voltage magnitude, we
21:31would adjust the reactive power output,
21:34and we would in practice do that with
21:35the excitation current in a synchronous
21:37machine.
21:39Um with an inverter, you do it
21:41digitally.
21:42Uh but it would the reactive power is
21:45what's related to the voltage magnitude
21:47primarily,
21:49and the real power is going to be
21:52related to the timing of the voltage.
21:55And with this delta, we're talking about
21:58angles, angle differences that are very
22:02subtle. Uh and in fact, on this
22:06just illustrative example with uh six
22:09buses,
22:10you can see just, you know, the types of
22:12angles we're talking about. It's just
22:14single-digit degrees. I mean, for a
22:16larger system, it could be tens of
22:18degrees.
22:20Um but it's a very subtle shift in the
22:23timing of that voltage sine wave. A
22:26little ahead means a lot more power
22:28injected, a little behind, a lot less
22:30power injected.
22:32So, um I kind of walked through this
22:35example in the book in uh chapter 12.
22:39Um
22:41these are just a again, and this is an
22:43example from Power World, which is a
22:45fabulous
22:47um
22:48teaching uh tool, and there's a free
22:51download version
22:53um that you could actually play with if
22:55you want to spend the time. You don't
22:57need to for this class.
22:59Um but um
23:02in this example, there are loads here
23:05with real and reactive specifications,
23:08generators with real and reactive
23:10specifications,
23:12and then each bus, or each of these
23:14heavy horizontal lines,
23:17um is characterized by a voltage
23:20magnitude, and that's specified here in
23:22per unit.
23:24Um so, it's independent of the actual
23:29um you know,
23:31nominal operating voltage of the system.
23:33It's in terms of whatever the nominal
23:36is. So, this one is 4% high, 1.04 per
23:40unit.
23:41Um so, if this whole system were
23:44operated nominally at 100 kilovolts,
23:47the voltage magnitude here would be 104
23:50kilovolt kilovolts per um
23:53uh root mean squared, and 2.77 degrees
23:58uh would be the angle, the positive
24:00angle, meaning it's ahead of time.
24:04Um and that's because this particular
24:07bus has more generation than load. So,
24:12intuitively, where there is a more
24:15positive
24:16voltage phase angle,
24:18that will tend to be associated with a
24:21more positive real power injection.
24:26Um and then we have real and reactive
24:29line flows. So, here the real ones are
24:31blue, the reactive ones are red.
24:34Um,
24:37the bus down here is more of a load bus,
24:39so that has a more negative voltage
24:42angle, a little behind.
24:45Um, and then there's this bus number one
24:48is a slack bus, and that has a
24:51significance.
24:52It's where we choose to uh
24:56define the time as being t equal to
24:58zero, where we press the button of the
25:00stopwatch, because voltage phase angle
25:04shifts need a reference. They need
25:07exactly one reference is where we draw
25:09the zero in our uh or how we choose a
25:13coordinate system when we're comparing
25:15multiple voltage phasors. So, we're just
25:17saying we're calling the voltage uh
25:20phase angle at this bus
25:22equal to zero,
25:24and that will also be the bus
25:27um that we um allow an adjustment
25:31um
25:32of power injection to make up for line
25:34losses. And
25:37why is that necessary? Because
25:41when we first specify power injections
25:44and demands,
25:45we don't know yet how those flows will
25:50um distribute themselves across the
25:52network.
25:54And until we know that, we can't know
25:56what the line losses are going to be.
25:59So, this is a simplified version that's
26:02not showing line losses, because what
26:04goes into a transmission line on one end
26:07equals what comes out at the other end.
26:10That's actually
26:12quite a simplification. So, if we go to
26:15the next level of detail, uh for each of
26:18these transmission lines here, which had
26:21been assigned some particular
26:24um
26:25real and imaginary impedance or
26:28admittance value,
26:30we can now calculate and the
26:33the power flow analysis will produce the
26:36values of the real and reactive line
26:39losses.
26:40Um where it's important to actually
26:44specify that the real line losses are
26:49a physical process of dissipating heat.
26:54That's the I squared R heating.
26:56And that is real money
26:59and real physical energy transfer.
27:02The reactive line losses are more of an
27:06accounting device.
27:09They can still be represented as I
27:12squared X.
27:14But as we've learned when we talked
27:17about reactive power,
27:19it's not really a quantity that's
27:21consumed or produced anywhere. So the
27:24term loss doesn't mean that physically
27:27something is getting lost. It means that
27:30this transmission line is acting as a
27:34predominantly inductive device.
27:37And so it is acting to quote consume
27:42reactive power
27:44in the context in which we've defined
27:46that where inductive things are
27:49consumers and capacitive things are
27:51producers of reactive power.
27:53And in fact, you can see that there's
27:56other transmission lines here that are
27:58capacitive in character.
28:01And they have
28:03negative reactive losses. In other
28:06words, they're generating VARs.
28:08Um
28:10and that will in practice that would be
28:12because they're lightly loaded
28:15um in this example. And in fact, it is
28:17it is because these are somewhat
28:20capacitive lines that the rules of thumb
28:24don't exactly
28:26work out across this whole um toy
28:30network, but it's worth um still
28:33remembering the general
28:37general rules of thumb that a more
28:39positive voltage phase angle will be
28:41associated with buses that produce more
28:44real power.
28:45A greater magnet voltage magnitude will
28:48be associated with buses that produce
28:51more reactive power.
28:53And conversely, if I have a big reactive
28:56load at some place in the network, that
28:59will tend to draw the voltage magnitude
29:01down a lot.
29:03Um and if I have a big real load, that
29:05will tend to pull the voltage phase
29:08angle behind or more negative at that
29:11location. Those are the general
29:14qualitative rules of thumb. And if you
29:17know just that much and you walk into a
29:21room with transmission operators,
29:23they'll respect you for knowing that
29:25much.
29:27Um
29:29Now, these rules are most apt for a
29:33network where the transmission lines
29:36are predominantly
29:38inductive.
29:39Um which, as we talked about with
29:42transmission line models, is usually the
29:45case um and is especially the case if if
29:49the transmission lines are more heavily
29:51loaded, their inductive properties will
29:54will dominate.
29:56Um
29:57And if transmission lines have uh a lot
29:59of resistance, that's more often the
30:02case for distribution actually than uh
30:04transmission lines.
30:06But, if they're behaving in a capacitive
30:08way, those rules don't work so well.
30:11Um
30:13So,
30:14with that, I want to come back to the
30:18optimal power flow.
30:20Uh so, now that you have a kind of a
30:23sense of what power flow analysis does,
30:26um
30:28we can have a little bit clearer
30:30distinction or to understand how So,
30:34power flow and individual power flow
30:36analysis run would be contributing
30:39to an optimal power flow analysis that
30:43runs many, many scenarios
30:46of of power flow.
30:48So, in OPF, we're trying to
30:51um
30:52optimize some quantity we've defined.
30:55So, maybe it's total generation cost and
30:58we want to find the minimum.
31:00Um but we also now want to take into
31:03account constraints as in what are we
31:07actually allowed to do in the real world
31:09or not.
31:11Um
31:12so, there are
31:15first of all, equality constraints,
31:17which are
31:19uh the real and reactive power balance.
31:22Um what goes in must come out including
31:25the losses.
31:27Um there are generator voltage set
31:29points, uh which is actually how in
31:32practice we're going to enforce
31:35uh the megavar injections that are
31:37necessary to keep the system uh
31:40balanced.
31:41And then there also as we uh learned in
31:44the previous uh set of lectures, um
31:47interchanges between areas, between
31:50neighboring areas,
31:52um that have to be accounted for.
31:56There are inequality constraints that
31:58basically say we have to keep certain
32:00quantities below some threshold, below
32:03some limit.
32:04So, we would have transmission line
32:06loading limits.
32:08Um they could be thermal limits or they
32:11could be stability limits of some sort.
32:15Um but
32:16uh
32:17there would be some applicable
32:19constraints that if we come up with some
32:22scenario of megawatts injected here and
32:24consumed there,
32:26and we run the power flow analysis and
32:28the power flow says, "Oh, you're going
32:30to have such and such a current through
32:32here or number of megawatts flowing
32:34through here."
32:36If that crosses a line, if that exceeds
32:40a threshold that in our engineering
32:43analysis we've decided that's some safe
32:47limit,
32:49then we have to go back and say,
32:52"Sorry, you're not allowed to make that
32:55transaction. We can't authorize that
32:59transaction if injecting so many
33:01megawatts here and selling them down
33:04there because that would violate a
33:08transmission
33:11loading limit.
33:13Um so, there are limits on all kinds of
33:17equipments, right? Limits on
33:18transformers that we talked about. Those
33:21would be mostly thermal limits.
33:23Uh there could be uh flow limits at the
33:25interface at the borders between uh
33:28jurisdictions.
33:29And especially for long transmission
33:31lines, there could be different physical
33:33factors
33:35uh
33:36that would say, you know, you cannot
33:38transfer more than a certain amount of
33:42power, more than some number of
33:44megawatts on this link.
33:46Sometimes those transmission line limits
33:49uh could be articulated in terms of an
33:51angle difference. That would be a
33:52stability limit.
33:55Where, you know, if the power flow
33:57solution comes out with, you know, your
34:00delta one voltage angle is one number
34:02here and so many degrees different um on
34:06the other side of this transmission
34:08line,
34:09that may be exceeding some stability
34:12limit. And so then you have to go back
34:13and say,
34:15"That set of power injections is not
34:18allowed."
34:19And let's make adjustments to uh the
34:23dispatch or uh the the way that the
34:27market was supposed to clear because
34:30this arrangement violates our
34:32constraints, so we have to pull the
34:34generation from somewhere else.
34:37Um
34:38the generators obviously have physical
34:41limits of what they can produce. They
34:43will have limits of how much reactive
34:45power they can
34:47um
34:48inject. And then there are
34:51um limits on the bus voltage magnitudes
34:54that are acceptable. So perhaps I want
34:56to keep all of the bus voltages with
34:59within plus or minus 5% of nominal.
35:03So now you get the picture
35:06of how we want to have some dispatch
35:10of megawatts and megavars from
35:14thousands of generators across the
35:16system
35:17to meet the load across the system.
35:21But to check whether that's feasible and
35:24okay with the transmission network,
35:28we have to run power flow analysis.
35:31We have to check that it meets all of
35:35the constraints. It doesn't violate
35:38anything.
35:40And then if we say, "All right, that
35:43works well."
35:45We might run another scenario and say,
35:48"Okay, well let me tweak the generation
35:50values some because maybe there are some
35:53contracts that are not up for
35:55negotiation, but there's some generators
35:58that as a system operator I can dispatch
36:01and let's see if I can come to another
36:04dispatch arrangement that would be uh
36:08less costly but still not violate any of
36:11these constraints."
36:13And so then I would iterate and run many
36:16many power flow calculations until I
36:19come up with the one that's aha, optimal
36:22according to my criteria.
36:25Um
36:26in orange here is a side comment uh
36:29which is that n minus one security
36:31analysis
36:33is an an important piece of these
36:36constraints. And that will be a separate
36:40um piece of a lecture,
36:43but um just in a nutshell, n minus one
36:46means one piece breaks.
36:50And the whole thing shouldn't the whole
36:53grid shouldn't fall apart. No customers
36:55should be out of power.
36:57Anyone generator, anyone transmission
37:00link
37:01could get struck by lightning or
37:04suddenly be taken out of commission for
37:06reasons we don't know.
37:08And
37:10the overall dispatch or power flow
37:12scenario that we choose should be robust
37:15with respect to any one of those
37:17contingencies happening.
37:20Um n minus two would be that well, two
37:23things could go wrong and we'd still be
37:25fine. So, all of these inequalities
37:28would still have to be satisfied
37:31even if we lose any one and typically we
37:36want to
37:38say well, what's the worst that could
37:39happen? What if the biggest generator
37:42right in the middle goes out or the
37:45biggest transmission link goes out?
37:47Um
37:48and in fact, choosing the contingencies
37:53for a contingency analysis is
37:57is a whole settled business and mistakes
37:59have been made.
38:01Where something gets overlooked that
38:03seems like oh, this would be a small
38:05failure, but it's actually
38:06consequential.
38:08And again, that's because the power flow
38:09problem is so
38:11counterintuitive.
38:14Um so, I will pause here and the next
38:18lecture will go a little bit more into
38:19the math.

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