PRODUCTION COSTING

W.D. Prasad

LecturerDept. 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 ; C

i = 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 i can be obtained.

ii

ii

ii CxFpxFqxF 11

• 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,

max

0

Li

i dxxFTUE 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,

iii UEUEE 1

• Corresponding production cost, Costi is given by

iii ICECost where ICi is the incremental cost of generator i

• Total production cost, TC is given by

ng

iiCostTC

1Where 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. 2ip

1ip

iq

1iC 2

iC0• New ELDC will be

2121111i

iii

ii

ii

i CxFpCxFpxFqxF

• In the case of a generator with multi-level availability states

n

k

ki

iki

ii

i CxFpxFqxF1

11 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.

xF n

xLoadMax

LOLP

• 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.

0 xFLOLP n

• The total energy left to be served after loading all the generating units is called the Expected System Unserved Energy.

max

0

Ln

n dxxFTUEEnergyUnservedSystemExpected

• 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.

nUEVLLLoadSupplyingNotofCostTotalExpected

• System planners need to add new generating units into the system until the following condition is satisfied.

unittheofoperationand

oninstallatioftTotalExpected cosnUEVLL

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.

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 1: Load Variation

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

Table 2: Generator Data

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

Unserved Energy (MWh) 10500 3660 802.875 189.3

Energy Served (MWh) 6840 2857.125 613.575

Production Cost (Rs) 5472000 2857.125 736290

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.

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 = 3470.7 MWhTotal production cost of generator 2 & 3 = Rs 3593415

Energy served by generator 4 = 3470.7 MWhProduction cost of generator 4 = Rs 3817770

Generation Level (MW)

0 200 250 450

Probability 005.032 qq 045.032 pq 095.032 qp 855.032 pp