Download Production Cost Calculation Cont…

PRODUCTION COSTING
W.D. Prasad
Lecturer
Dept. of Electrical Engineering
University of Moratuwa
Introduction
• Electricity generation system planning requires minimization of the total cost
of supplying the demand during a specified period of time.
• Short term, Medium term or Long term.
• Medium term and Long term planning
• initial investment cost + production costs
• Short term planning
• production cost only
Load and Generator Models
• Production costing includes probabilistic treatment of the system load and the
generation unit availability in almost all planning models.
• Chronological load curve
Load duration curve
• Chronological load curve is modified with a plot of load versus the duration
that the system load exceeds that load level.
• This curve can be converted to a probability curve, F(x) by dividing the
horizontal axis (x- axis) by the total duration of the chronological load profile, T
and rearranging the axes (Load Duration Curve, LDC).
Load and Generator Models Cont…
• Generators are normally represented by a two-state model where either a generating
unit is available at its full capacity or not available.
pi - Availability
qi -FOR
0
Ci ;
Ci = Capacity
• Probability associated with the state where the unit is not available is called Forced
Outage Rate (FOR).
Production Cost Calculation
• Generators are first ranked according to their average incremental costs so that the
units with the lower costs are placed at the top of the list (Merit Order).
• These units are now gradually loaded onto the LDC in merit order.
• After loading each generator the Effective Load Duration Curve (ELDC), F
obtained.
i
can be
F i x   qi F i 1x   pi F i 1 x  Ci 
• The area under each of these ELDCs multiplied by the normalizing value, T, directly
indicates the energy not served in the system.
• Unserved energy (UEi) after loading the generator i is given by,
UEi  T
Lmax
i
F
 x dx
0
Where T is the total duration
Production Cost Calculation Cont…
• Once the unserved energies are known the difference in unserved energies before and
after loading a generator can be used to obtain the energy served by that generator.
• Energy produced by generator i, Ei is given by,
Ei  UEi 1 UEi
• Corresponding production cost, Costi is given by
Costi  Ei ICi
where ICi is the incremental cost of generator i
• Total production cost, TC is given by
ng
TC   Costi
i 1
Where ng is the number of generator units
Multiple Availability States of Generators
• In most practical circumstances some of the generation units are likely to be deliberately
operated at output levels below their full capacities during operation.
• Consider a generating unit model with two availability states.
pi2
p i1
qi
Ci1
0
Ci2
• New ELDC will be



F i x   qi F i 1 x   pi1F i 1 x  Ci1  pi2 F i 1 x  Ci2

• In the case of a generator with multi-level availability states
F  x   qi F
i
i 1
n
x    p
k 1
k
i
F
i 1
x  C 
k
i
where n is the no. of
availability levels
System Unserved Energy and Loss of Load Probability
• After loading all the generating units onto the load curve there will be a final ELDC left
behind.
F n x 
LOLP
x
Max Load
• Loss of Load Probability (LOLP) is the probability that the system generation is not able
to supply the system load either fully or partially. This can be directly obtained from the
final ELDC.
LOLP  F n x  0
• The total energy left to be served after loading all the generating units is called the
Expected System Unserved Energy.
Expected System Unserved Energy  UEn  T
Lmax
n
F
 xdx
0
• Average cost of losses due to the power supply failures is called the Value of Lost Load
(VLL) which is given in Rs/ kWh not supplied.
Expected Total Cost of Not Supplying Load  VLL  UEn
• System planners need to add new generating units into the system until the following
condition is satisfied.
Expected Total cos t of installati on
and operation of the unit
 VLL  UEn
Example
1) [a] Determine LOLP, EUE and total production cost if the system load given in Table 1 is
supplied with generators in Table 2.
Table 1: Load Variation
Time
(hrs)
00-03
03-06
06-09
09-12
12-15
15-18
18-21
21-24
Load
(MW)
300
300
400
600
600
600
400
300
Table 2: Generator Data
Generator
Incremental Cost
(Rs/MWh)
Capacity (MW)
Forced Outage Rate
Generator 1
800
300
0.05
Generator 2
1000
250
0.05
Generator 3
1200
200
0.1
Answer
Cost (Rs/MWh)
800
1000
1200
FOR
0.05
0.05
0.1
Capacity
300
250
200
Load (x)
Duration (hrs)
F0(x)
F1(x)
F2(x)
F3(x)
0
24
1
0.64375
0.418125
0.076125
50
24
1
0.64375
0.061875
0.0405
100
24
1
0.40625
0.05
0.0224375
150
24
1
0.40625
0.038125
0.00521875
200
24
1
0.40625
0.038125
0.00465625
250
24
1
0.40625
0.038125
0.00465625
300
15
0.625
0.03125
0.019375
0.00278125
350
15
0.625
0.03125
0.001563
0.001
400
9
0.375
0.01875
0.000938
0.00009375
450
9
0.375
0.01875
0.000938
0.00009375
500
9
0.375
0.01875
0.000938
0.00009375
550
9
0.375
0.01875
0.000938
0.00009375
600
0
0
0
0
0
10500
3660
802.875
189.3
Energy Served (MWh)
6840
2857.125
613.575
Production Cost (Rs)
5472000
2857.125
736290
Unserved Energy (MWh)
LOLP
7.6 %
EUE (MWh)
189.3
Total Production Cost (Rs)
9065415
b] Comment on possible changes to the answers in (a) if generator 2 and 3 are replaced
with generator 4 given in Table 3 having an incremental cost of 1100 Rs/MWh
Table 3: Generator 4 Data
Capacity
(MW)
0
200
250
450
Probability
0.005
0.045
0.095
0.855
Answer
When generators 2 and 3 are combined the resultant generator will have an
availability distribution with four levels of operation as given below.
Generation
Level (MW)
Probability
0
q2  q3  0.005
200
q2  p3  0.045
250
450
p2  q3  0.095
p2  p3  0.855
This distribution is exactly the same as the generator 4 distribution given. Thus
even though the generators 2 and 3 are replaced with the generator 4, the final
probability distribution will not change. This means that the LOLP and the
expected system unserved energy also will not be modified. However, the
production cot will change due to the modified incremental cost.
Total energy served by generator 2 & 3
Total production cost of generator 2 & 3
= 3470.7 MWh
= Rs 3593415
Energy served by generator 4
Production cost of generator 4
= 3470.7 MWh
= Rs 3817770