Assignment title: Information
1
HW#7
Power System Economics and Deregulation
(Please send email to Wail Rashed for any questions ([email protected]))
Due date Thursday March 30, 2017 (9:00 am / Chicago time)
I will send you copy of the (Books pdf and Lecture note) to make it easier for your expert to solve below
Questions.2
Problem 1: (10 points)
The following problems are based on the following system.
For each problem, determine whether the given dispatch is
optimal or not.
The fuel cost functions for the three units are
2
F1(P 1) 749.55 7P 1 0.002P 1
2
F2(P2) 1285 8P2 0.001P2
2
F3(P3) 1531 6P3 0.002P3
The minimum and maximum capacities of the three units are
Unit No. Pi,min (MW) Pi,max (MW)
1 320 800
2 300 1200
3 275 1100
(1) If the load is 1300, the dispatch is P 1 387.5 , P2 275,
and P3 637.5 .
A. Yes, the dispatch is optimal.
B. No, the dispatch is not optimal.
(2) If the load is 2500, the dispatch is P 1 687.5 , P2 875 ,
and P3 937.5 .
A. Yes, the dispatch is optimal.
B. No, the dispatch is not optimal.
(3) If the load is 3000, the dispatch is P 1 812.5 , P2 1125 ,
and P3 1062.5 .
A. Yes, the dispatch is optimal.
B. No, the dispatch is not optimal.
(4) If the load is 1200, the dispatch is P 1 362.5 , P2 300 ,
and P3 612.5 .
A. Yes, the dispatch is optimal.
B. No, the dispatch is not optimal.
Problem 2: (10 points)
(1) In your own word, describe
What is restructuring, and
Why have some power systems been restructured?
(2) What are the benefits of interconnected systems?
(3) Name Six Independent System Operators (ISOs) operated
in the US (Full Name).3
Problem 3: (10 points)
Four areas are interconnected. The following table shows the
generation and cost at each area when each area independently
supplies its own load and when the four areas are centrally
dispatched by a pool control center.
Assume a split-savings pool agreement such that each area
exporting receives its increased costs of production plus its
proportionate share of 50% of the pool savings.
Calculate (1) the price of pool energy and (2) the net
operation cost for Area 1.
Area Load
Local
Generatio
n Supply
Load Cost
Pool
Generatio
n Supply
Load Cost
No MW MW $/h MW $/h
1 550 550 2,640 430 1,969
2 450 450 2,056 466 2,132
3 400 400 1,800 326 1,385
4 350 350 1,637 528 2,507
Total 1750 1750 8,134 1750 7,994
Problem 4: (20 points)
In the following two problems, in case that multiple solutions
exist,
Choose the solution that maximizes the traded volume, or
If many price levels give equal traded volume, choose the
price of the mid-point.
(1) Assume the following bidding information.
Demand Price Demand Volume
104 5000
105.5 6000
106 12000
Supply Price Supply Volume
105 12000
106 6000
107 6000
Determine the market clearing price (MCP), market
clearing quantity (MCQ), Social Welfare, Consumer
Surplus, and Producer Surplus.
(2) Assume the following bidding information.
Demand Price Demand Volume
104 2000
105 2000
106 500
Supply Price Supply Volume
104.5 2000
105 5000
Write down the LP formulation for clearing the bids.4
Problem 5: (10 points)
The following DC SF based LP formulation is used to
calculate LMPs for a 3-bus system shown in Figure 1.
r l
min max
r l
min max
l
min
r
max
T
T T
L L L η , η
P P P γ , γ
f SF * (A * P B * L) μ
SF * (A * P B * L) f μ
e A * P B * L
c * P b * L
( ) 0
s.t.
min
1 2
G1: 400MW @
10 $/MWh
G3: 200 MW @
20 $/MWh
L3: 450MW
3
Flow limit:
100MW
Figure 1
Determine the LMP at Bus 2.
The LP implementation in MATLAB is as follows.
% LMP calculation, DC PTDF based method (bus 1 is
the reference bus)
%x=[G1 G3 L3]'
% matrix of PTDFs, ref bus = 1
SF=[0 -2/3 -1/3;
0 -1/3 -2/3;
0 1/3 -1/3];
% Bus-unit incidence matrix
A=[ 1 0;
0 0;
0 1];
% Bus-load incidence matrix
B=[ 0;
0;
1];
% Flow limits vector
%[f12 f13 f23]
flow_limits=[1000 1000 100]';
% unit vector
e=[1 1 1]'; %the dimension is number of buses
Aeq=[e'*A -e'*B];
beq=[0];
A1=[SF*A -SF*B;
-SF*A SF*B];
b1=[flow_limits;
flow_limits];
f=[10 20 0]';
lb=[0 0 450]';
ub=[400 200 450]';
[x,fval,exitflag,output,lambda]=linprog(f,A1,b1,Aeq,
beq,lb,ub);
x
lambda.eqlin
lambda.ineqlin
The solutions from LP is
x =
300.0000
150.0000
450.0000
lambda.eqlin =
-10.0000
lambda.ineqlin =
0.0000
0.0000
30.0000
0.0000
0.0000
0.00005
Problem 6: (20 points)
The following DC SF based LP formulation is used to
calculate LMPs for a 3-bus system, shown in Figure 2, with
consideration of losses. Losses are always balanced at Bus 3.
r l
min max
r l
min max
l
max
r
max
T
T
T T
L L L η , η
P P P γ , γ
SF * A * P B * L D * f μ
SF * A * P B * L D * f μ
LF * A * P B * L
e * A * P B * L
c * P b * L
( )
( )
( )
( )
s.t.
min
Loss
Loss
offset Loss
Loss
1
2
G1: 400MW @
10 $/MWh
G2: 200 MW @
20 $/MWh
L3: 450MW
3
Flow limit:
255MW
Figure 2
If we choose Bus 2 as the reference bus, the LMPs at the three
buses are
Bus No LMP ($/MWh)
1 10
2 20
3 31.7308
Determine the energy components, loss components,
congestion components, and LMPs at all buses if we choose
Bus 1 as the reference bus.
If we choose Bus 1 as the reference bus, the LP
implementation in MATLAB is as follows.
% LMP calculation, DC PTDF based method with
consideration of losses
%x=[G1 G2 L3 Loss]'
% Loss distribution factors
D = [0 0 1]';
% Loss factors
LF=[0 -0.02 -0.1]';
offset=-25;
% Shifting factors
SF=[0 -2/3 -1/3;
0 -1/3 -2/3;
0 1/3 -1/3];
% Bus-unit incidence matrix
A=[ 1 0;
0 1;
0 0];
% Bus-load incidence matrix
B=[ 0;
0;
1];
% Flow limits vector
%[f12 f13 f23]
flow_limits=[1000 255 1000]';
% unit vector
e=[1 1 1]'; %the dimension is number of buses
Aeq=[ e'*A -e'*B -1;
LF'*A -LF'*B -1];
beq=[ 0;
-offset];
A1=[ SF*A -SF*B -SF*D;
-SF*A SF*B SF*D];
b1=[ flow_limits;
flow_limits];
f=[10 20 0 0]';
lb=[0 0 450 -inf]';
ub=[300 200 450 inf]';
[x,fval,exitflag,output,lambda]=linprog(f,A1,b1,Aeq,be
q,lb,ub);
x
lambda.eqlin
lambda.ineqlin
The solutions from LP is
x =
298.3654
168.2692
450.0000
16.6346
lambda.eqlin =
-10.0000
28.8462
lambda.ineqlin =
0.0000
28.2692
0.0000
0.0000
0.0000
0.00006
Problem 7: (20 points)
Write down the LP formulation for calculating LMP for
the following system with consideration of losses. Assume
that system losses are balanced at Bus 2 (25%) and Bus 3
(75%).
Network Data
Line
Index
From
Bus
To
Bus
Resistance
(p.u.)
Reactance
(p.u.)
Flow Limit
(MW)
1 1 2 0.002 0.01 1000
2 2 3 0.002 0.01 250
3 1 3 0.002 0.01 1000
Generation Bid Data
Unit No Bus No MW Price
($/MWh)
1 1 200 14
2 1 300 15
3 2 300 30
Load Bid Data
Load No Bus No MW Price
($/MWh)
1 2 150 40
2 3 450 40
3
0 2
3
1
3
0 1
3
1
3
0 1
3
2
SF
0.078
0
0.028
LF , Offset=-20
Problem 8: Extra (20 points)
The following DC SF based LP formulation can be used to
calculate LMPs.
r l
min max
r l
min max
r l
min max
T
T T
f f f μ ,μ
L L L η ,η
P P P γ , γ
f SF* (A * P B * L) μ
e A * P B * L
c * P b * L
( ) 0
s.t.
min
Determine the formula for LMP and prove the revenue
adequacy of the LMP-based market.