# ECE 310.ppt by lovemacromastia

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```									         ECE 476
POWER SYSTEM ANALYSIS

Lecture 17
Optimal Power Flow, LMPs

Professor Tom Overbye
Department of Electrical and
Computer Engineering
Announcements

   Homework 7 is due now.
   Homework 8 is 7.1, 7.17, 7.20, 7.24, 7.27
   Should be done before second exam; not turned in
   Design Project is assigned today (see website for
details). Due date is Nov 20.
   Exam 2 is Thursday Nov 13 in class.
   Grainger Power Engineering Award Applications
Due Nov 1. See below for details:
http://energy.ece.uiuc.edu/Grainger.html
1
Back of Envelope Values

   Often times incremental costs can be approximated
by a constant value:
–   \$/MWhr = fuelcost * heatrate + variable O&M
–   Typical heatrate for a coal plant is 10, modern
combustion turbine is 10, combined cycle plant is 7 to 8,
older combustion turbine 15.
–   Fuel costs (\$/MBtu) are quite variable, with current
values around 2 for coal, 7 for natural gas, 0.5 for
nuclear, probably 10 for fuel oil.
–   Hydro costs tend to be quite low, but are fuel (water)
constrained

2
Aside: Levelized Cost of Generation
Technology                              \$/MWh (2007 Dollars) (IOU)
Wind – Class 5                                        67
Solar – Photovoltaic                                  686
Solar – Concentrating                                 434
Solar – Parabolic Trough                              281
Ocean Wave (Pilot)                                    838
Small Scale Hydro                                     118
Geothermal                                            63
Keep in mind these numbers involve LOTs of assumptions
that can drastically affect the value, and that many
technology costs are site dependent.
Source: California Energy Commission:
http://energyalmanac.ca.gov/electricity/levelized_costs.html
3
Area Supply Curve

The area supply curve shows the cost to produce the
next MW of electricity, assuming area is economically
dispatched   10.00

7.50

Supply
curve for     5.00

thirty bus
system        2.50

0.00
0   100                200             300   400
Total Area Generation (MW)

4
Economic Dispatch - Summary

   Economic dispatch determines the best way to
minimize the current generator operating costs
   The lambda-iteration method is a good approach for
solving the economic dispatch problem
–   generator limits are easily handled
–   penalty factors are used to consider the impact of losses
   Economic dispatch is not concerned with
determining which units to turn on/off (this is the
unit commitment problem)
   Economic dispatch ignores the transmission system
limitations
5
Optimal Power Flow

   The goal of an optimal power flow (OPF) is to
determine the “best” way to instantaneously operate
a power system.
   Usually “best” = minimizing operating cost.
   OPF considers the impact of the transmission system
   OPF is used as basis for real-time pricing in major
US electricity markets such as MISO and PJM.
   ECE 476 introduces the OPF problem and provides
some demonstrations.

6
Electricity Markets

   Over last ten years electricity markets have moved
from bilateral contracts between utilities to also
include spot markets (day ahead and real-time).
   Electricity (MWh) is now being treated as a
commodity (like corn, coffee, natural gas) with the
size of the market transmission system dependent.
   Tools of commodity trading are being widely

7
Electricity Futures Example

Source: Wall Street Journal Online, 10/30/08
8
“Ideal” Power Market

   Ideal power market is analogous to a lake.
Generators supply energy to lake and loads remove
energy.
   Ideal power market has no transmission constraints
   Single marginal cost associated with enforcing
constraint that supply = demand
–   buy from the least cost unit that is not at a limit
–   this price is the marginal cost
   This solution is identical to the economic dispatch
problem solution

9
Two Bus ED Example

Total Hourly Cost : 8459 \$/hr
Area Lambda : 13.02

Bus A                                                Bus B

300.0 MW                      300.0 MW
199.6 MW                          400.4 MW
AGC ON                             AGC ON

10
Market Marginal (Incremental) Cost
Below are some graphs associated with this two bus
system. The graph on left shows the marginal cost for each
of the generators. The graph on the right shows the
system supply curve, assuming the system is optimally
dispatched.
16.00                                           16.00

15.00                                           15.00

14.00                                           14.00

13.00                                           13.00

12.00                                           12.00
0   175        350          525   700           0   350            700        1050     1400
Generator Power (MW)                            Total Area Generation (MW)

Current generator operating point
11
Real Power Markets

   Different operating regions impose constraints --
total demand in region must equal total supply
   Transmission system imposes constraints on the
market
   Marginal costs become localized
   Requires solution by an optimal power flow

12
Optimal Power Flow (OPF)

   OPF functionally combines the power flow with
economic dispatch
   Minimize cost function, such as operating cost,
taking into account realistic equality and inequality
constraints
   Equality constraints
–   bus real and reactive power balance
–   generator voltage setpoints
–   area MW interchange

13
OPF, cont’d

   Inequality constraints
–   transmission line/transformer/interface flow limits
–   generator MW limits
–   generator reactive power capability curves
–   bus voltage magnitudes (not yet implemented in
Simulator OPF)
   Available Controls
–   generator MW outputs
–   transformer taps and phase angles

14
OPF Solution Methods

   Non-linear approach using Newton’s method
–   handles marginal losses well, but is relatively slow and
has problems determining binding constraints
   Linear Programming
–   fast and efficient in determining binding constraints, but
can have difficulty with marginal losses.
–   used in PowerWorld Simulator

15
LP OPF Solution Method

   Solution iterates between
–   solving a full ac power flow solution
 enforces real/reactive power balance at each bus
 enforces generator reactive limits
 system controls are assumed fixed
 takes into account non-linearities
–   solving a primal LP
 changes system controls to enforce linearized
constraints while minimizing cost

16
Two Bus with Unconstrained Line

With no
Total Hourly Cost : 8459 \$/hr
OPF matches                    Area Lambda : 13.01         line is not
dispatch
Bus A                   13.01 \$/MWh    Bus B         13.01 \$/MWh

300.0 MW                      300.0 MW
197.0 MW                          403.0 MW
AGC ON                             AGC ON

Marginal cost of supplying
power to each bus
(locational marginal costs)
17
Two Bus with Constrained Line

Total Hourly Cost : 9513 \$/hr
Area Lambda : 13.26

Bus A                   13.43 \$/MWh    Bus B         13.08 \$/MWh

380.0 MW                      300.0 MW
260.9 MW                          419.1 MW
AGC ON                             AGC ON

must be supplied locally, causing the marginal costs to
diverge.
18
Three Bus (B3) Example

   Consider a three bus case (bus 1 is system slack),
with all buses connected through 0.1 pu reactance
lines, each with a 100 MVA limit
   Let the generator marginal costs be
–   Bus 1: 10 \$ / MWhr; Range = 0 to 400 MW
–   Bus 2: 12 \$ / MWhr; Range = 0 to 400 MW
–   Bus 3: 20 \$ / MWhr; Range = 0 to 400 MW
   Assume a single 180 MW load at bus 2

19
B3 with Line Limits NOT Enforced

60 MW                   60 MW
Bus 2                                     Bus 1
10.00 \$/MWh

0.0 MW   10.00 \$/MWh
120 MW           180.0 MW
120%
0 MW
60 MW
120%   120 MW
Total Cost 60 MW                              Line from Bus 1
1800 \$/hr
Bus 3             10.00 \$/MWh     to Bus 3 is over-
0 MW                      have same
marginal cost
20
B3 with Line Limits Enforced

20 MW                   20 MW
Bus 2                                     Bus 1
10.00 \$/MWh

60.0 MW 12.00 \$/MWh
100 MW            120.0 MW
100%
0 MW
80 MW
100%   100 MW
Total Cost 80 MW
1920 \$/hr                                   LP OPF redispatches
Bus 3             14.00     \$/MWh
to remove violation.
180 MW
Bus marginal
0 MW                    costs are now
different.
21
Verify Bus 3 Marginal Cost

19 MW                   19 MW
Bus 2                                     Bus 1
10.00 \$/MWh

62.0 MW 12.00 \$/MWh
100 MW            119.0 MW
81%                        100%
0 MW
81 MW
81%    100%   100 MW    One additional MW
Total Cost 81 MW
1934 \$/hr                                   of load at bus 3
Bus 3             14.00     \$/MWh
raised total cost by
181 MW
14 \$/hr, as G2 went
0 MW                    up by 2 MW and G1
went down by 1MW
22
Why is bus 3 LMP = \$14 /MWh

   All lines have equal impedance. Power flow in a
simple network distributes inversely to impedance
of path.
–   For bus 1 to supply 1 MW to bus 3, 2/3 MW would take
direct path from 1 to 3, while 1/3 MW would “loop
around” from 1 to 2 to 3.
–   Likewise, for bus 2 to supply 1 MW to bus 3, 2/3MW
would go from 2 to 3, while 1/3 MW would go from 2 to
1to 3.

23
Why is bus 3 LMP \$ 14 / MWh, cont’d

   With the line from 1 to 3 limited, no additional
power flows are allowed on it.
   To supply 1 more MW to bus 3 we need
–   Pg1 + Pg2 = 1 MW
–   2/3 Pg1 + 1/3 Pg2 = 0; (no more flow on 1-3)
   Solving requires we up Pg2 by 2 MW and drop Pg1
by 1 MW -- a net increase of \$14.

24
Both lines into Bus 3 Congested

0 MW                       0 MW
Bus 2                                        Bus 1
10.00 \$/MWh

100.0 MW12.00 \$/MWh
100 MW           100.0 MW
100%                         100%
0 MW
100 MW                            For bus 3 loads
100%    100% 100 MW
Total Cost MW
100
above 200 MW,
2280 \$/hr
Bus 3                         the
204 MW     supplied locally.
Then what if the
4 MW
bus 3 generator
opens?
25
Profit Maximization: 30 Bus Example

52.45 MW                            69.58 MW
slack

A

1                                           2                                                                                                                          84%
MVA
18
A                                                                                                              15
68%

A
MVA

A            A        A
1.000                            A
A

62%
MVA
MVA
MVA                                                                                                                                                                                           19
MVA        MVA          MVA
14           A
28                                                       A                                                                                                                     67%

MVA
MVA
35.00 MW

A
A

MVA
3                      A
4

A
16 MW
A

MVA
A
Gen 13 LMP
MVA
56%
82%
MVA
MVA

8
7                                                  5           MVA
12                          13                        A

MVA
33.46 \$/MWh
6                                                                                                                       A
A                                                                                                                                                                               A
A                                                                                           MVA

MVA                                                                                                                                                                              MVA
MVA
9                                                                                                                                                                    16                               17
11
11 MW
A

MVA
19 MW
A
11 MW
MVA                                                                                                         A

MVA

10 MW

A
20
66%
MVA
21 MW
26                                                                     10
A                                                         24.00 MW                                     A

MVA                                                                                                MVA                                                                                                                         23
A

73%
MVA

25
22                                                21                          24
A                                                                                                                           2 MW
52%                                                                                                                                               A                                  16.00 MW
MVA

MVA

40.00 MW                                                                                                            A

52%
A                                 MVA

87%
MVA

29                                                       30
27
A
A

MVA
A                                       MVA

MVA

26
Typical Electricity Markets

   Electricity markets trade a number of different
commodities, with MWh being the most important
   A typical market has two settlement periods: day
offers for the next day; OPF is used to determine who
gets dispatched based upon forecasted conditions.
Results are financially binding
–   Real-time: Modifies the day ahead market based upon
real-time conditions.

27
Payment

   Generators are not paid their offer, rather they are
paid the LMP at their bus, the loads pay the LMP.
   At the residential/commercial level the LMP costs
are usually not passed on directly to the end
consumer. Rather, they these consumers typically
pay a fixed rate.
   LMPs may differ across a system due to
transmission system “congestion.”

28
MISO LMP Contours – 10/30/08

29
Why not pay as bid?

   Two options for paying market participants
–   Pay as bid
–   Pay last accepted offer
of both?
   Talk about supply and demand curves, scarcity,
withholding, market power

30
Market Experiments

31
Limiting Carbon Dioxide Emissions

• There is growing concern about the need to limit
carbon dioxide emissions.
•   The two main approaches are 1) a carbon tax, or 2)