Methods of determining depreciation of Power Plant

There is a reduction in the value of the equipment and other property of the plant every year due to depreciation. Therefore, a suitable amount (known as depreciation charge) must be set aside annually so that by the time the life span of the plant is over, the collected amount equals the cost of replacement of the plant. The following are the commonly used methods for determining the annual depreciation charge:

(i) Straight line method 

(ii) Diminishing value method 

(iii) Sinking fund method

Straight Line Method:

In this method, a constant depreciation charge is made every year on the basis of total depreciation and the useful life of the property.Obviously, annual depreciation charge will be equal to the total depreciation divided by the useful life of the property.

Annual depreciation charge = total depreciation ÷ useful life 

= (P—S) ÷ n

Where,

P = Initial cost of equipment

n = Useful life of equipment in years

S = Scrap or salvage value after the useful life of the plant.

The straight line method is extremely simple and is easy to apply as the annual depreciation charge can be readily calculated from the total depreciation and useful life of the equipment. The figure below shows the graphical representation of the method. It is clear that initial value P of the equipment reduces uniformly, through depreciation, to the scrap value S in the useful life of the equipment.

The depreciation curve (PA) follows a straight line path, indicating constant annual depreciation charge. However, this method suffers from two defects. Firstly, the assumption of constant depreciation charge every year is not correct. Secondly, it does not account for the interest which may be drawn during accumulation.

Diminishing Value Method: 

In this method, depreciation charge is made every year at a fixed rate on the diminished value of the equipment. In other words, depreciation charge is first applied to the initial cost of equipment and then to its diminished value.

Let say,

P = Capital cost of equipment

n = Useful life of equipment in years

S = Scrap value after useful life

Suppose the annual unit depreciation is x. It is desired to find the value of x in terms of P, n and S.

Value of equipment after one year = P − Px = P (1 − x) 

Value of equipment after 2 years = Diminished value − Annual depreciation 

= [P − Px] − [(P − Px)x] 

= P − Px − Px + Px^2 

= P(x 2 − 2x + 1) 

= P(1 − x)^2 

∴ Value of equipment after n years = P(1 − x)^n

But the value of equipment after n years (i.e., useful life) is equal to the scrap value S. 

∴ S = P(1 − x)^n 

Or (1 − x)^n = S/P 

Or 1 − x = (S/P)^1/n 

Or x = 1 − (S/P)^1/n ...(i) 

From exp. (i), the annual depreciation can be easily found. Thus depreciation to be made for the first year is given by: 

Depreciation for the first year = xP

= P[1 − (S/P)^1/n ]

Similarly, annual depreciation charge for the subsequent years can be calculated. This method is more rational than the straight line method.The figure below shows the graphical representation of diminishing value method. The initial value P of the equipment reduces,through depreciation, to the scrap value S over the useful life of the equipment. The depreciation curve follows the path PA. It is clear from the curve that depreciation charges are heavy in the early years but decrease to a low value in the later years.This method has two drawbacks. Firstly, low depreciation charges are made in the late years when the maintenance and repair charges are quite heavy. Secondly, the depreciation charge is independent of the rate of interest which it may draw during accumulation. Such interest money, if earned, are to be treated as income.

Sinking Fund Method:

In this method, a fixed depreciation charge is made every year and interest compounded on it annually. The constant depreciation charge is such that total of annual installments plus the interest accumulations equal to the cost of replacement of equipment after its useful life. 

Let 

P = Initial value of equipment 

n = Useful life of equipment in years 

S = Scrap value after useful life 

r = Annual rate of interest expressed as a decimal 

Cost of replacement = P − S 

Let us suppose that an amount of q is set aside as depreciation charge every year and interest compounded on it so that an amount of P− S is available after n years. An amount q at annual interest rate of r will become q(1 + r)^n at the end of n years. 

Now, the amount q deposited at the end of first year will earn compound interest for n − 1 years and shall become q(1 + r)^n − 1 i.e., 

Amount q deposited at the end of first year becomes = q (1 + r)^n − 1

Amount q deposited at the end of 2nd year becomes = q (1 + r)^n − 2 

Amount q deposited at the end of 3rd year becomes = q (1 + r)^n − 3 

Similarly amount q deposited at the end of (n − 1) year becomes = q (1 + r)^n − (n − 1) 

                                                                                                               = q (1 + r) 

∴ Total fund after n years = q (1 + r)^n − 1 + q (1 + r)^ n − 2 + .... + q (1 + r) 

                                             = q [(1 + r)^n − 1 + (1 + r)^n − 2 + .... + (1 + r)] 

This is a G.P. series and its sum is given by : 

Total fund = [q(1 + r)^n −1]÷ r 

This total fund must be equal to the cost of replacement of equipment i.e., P − S. 

∴ P − S = [q(1 + r)^n −1]÷ r 

Or Sinking fund, q = (P—S)[r ÷ (1 + r)^n −1]...(i) 

The value of q gives the uniform annual depreciation charge. The parenthetical term in eq. (i) is frequently referred to as the “sinking fund factor”. 

∴ Sinking fund factor = r ÷ (1 + r)^n −1 

Though this method does not find very frequent application in practical depreciation accounting, it is the fundamental method in making economy studies.

Introduction to power station economics.

A power station is built due to deliver or provides power to different consumers to meet their requirements or demand. But, obviously, generation & distribution of power to its consumers is not free. So that, while designing and building a power station, power engineers give their efforts to  achieve overall economy to minimize per unit cost of generation may as low as possible due to ensure reliable service and make profit for the generation company. These generating stations, known as central station generators, are often located in remote areas, far from the point of consumption.

As we discuss about the economics of power generation which is largely a matter of costing and there are several factors which influence the production cost such as cost of land and equipment, depreciation of equipment, interest on capital investment etc. That's why we need a careful study to calculate the cost of production. But before entering in details of power station economics we need to know following terms which I will use in my next posts.

• Interest:  In simple meaning interest is a payment made by a borrower to the lender for the money borrowed and is expressed as a rate percent per year.

A power generating station is constructed by investing a huge capital. This money is generally borrowed from banks or other financial institutions and the supply company has to pay the annual interest on this amount. Even if company has spent out of its reserve funds, the interest must be still allowed for, since this amount could have earned interest if deposited in a bank. Therefore, while calculating the cost of production of electrical energy, the interest payable on the capital investment must be included.

• Depreciation: Depreciation is a method of allocating some cost of a tangible long term asset over its useful life and thus, refers to the decrease in value of the asset (except land) during its lifetime. 

Depreciation is only calculated when the asset fulfills the following requirements:

1. The Depreciable assets have a limited useful life,
2. The useful life of the asset is more than one year,
3. The assets should not be the sales inventory i.e. it should be used in production of goods and services to run a business

If the power station equipment were to last forever, then interest on the capital investment would have been the only charge to be made. However, in actual practice, every power station has a useful life ranging from fifty to sixty years. From the time the power station is installed, its equipment steadily deteriorates due to wear and tear so that there is a gradual reduction in the value of the plant. This reduction in the value of plant every year is known as annual depreciation. Due to depreciation, the plant has to be replaced by the new one after its useful life. Therefore, suitable amount must be set aside every year so that by the time the plant retires, the collected amount by way of depreciation equals the cost of replacement. It becomes obvious that while determining the cost of production, annual depreciation charges must be included.

Power Station : Terms & Factors. Part-2

(Continue from previous post........)

It is matter of fact that generated electrical power cannot be stored, that's why power stations need to produce as and when demanded to fulfill the requirements of the consumers. As we know demand of  consumers varies time to time what makes the power engineer to consider some factors, as we discussed before. Today we discuss next level of factors. 

• Load Factor: Ratio of average load to maximum demand for a given period is known as Load Factor.

Hence,

Load factor = Average load ÷ Maximum Demand.

for a given period T, 

Load factor= (Average load x T)÷(maximum demand x T)

                   = (Units generated in T hours)÷(maximum demand x T)

It will always less then 1 because average load is smaller then the maximum demand. This factor help to determine the cost per unit generation. If load factor of power station goes higher the per unit cost will be lesser because high load factor means low maximum demand and low maximum demand means lower capacity of plant which reduce the cost of plant.

• Diversity Factor: The ratio of the sum of individual maximum demand to the maximum demand on power station is known as Diversity Factor.

Hence,

Diversity factor = Sum of individual maximum demand ÷ Maximum demand on power station.

This factor will always be greater then 1 and higher the diversity factor lesser the cost of per unit generation.

• Capacity Factor: The ratio of actual energy generated or supply over a period of time to the possible energy that would have been produced if the plant had operated continuously at its maximum rating.

Hence,
Capacity factor = Actual energy produced ÷ Max. energy that could have been produced.

for a given period T, 

Capacity factor = (Average demand x T) ÷ (maximum demand x T)

                          = Average demand ÷ Plant capacity

Annual Capacity factor= Annual kWh output ÷ (plant capacity x 8760)

• Plant Use-factor or Utilization Factor: It is the ratio of energy (kWh) produced or generated to the product of plant capacity and the number of hours for which the plant was in operation.

Hence,
Plant Use-factor= Station o/p in kWh ÷ (plant capacity x hours in operation)

Power Station : Terms & Factors. Part-1

The main function of a power station is to generate power and deliver it to different types of consumers. It is not necessary to state that, this delivery system depends on consumers due to their demands and activities. This variation in supply system introduce some important terms and factors in power plant engineering.

• Connected Load: Different consumers install or use different machinery/ equipment at his own premises. All this equipment are connected to power station. these are known as connected load. We can say, it is the sum of continuous rating of all equipment connected to supply/delivery system.
• Maximum Demand: The load in a power station varies time to time as changing the demand of consumers. Thus the highest demand load on a power station for a given period of time known as maximum demand. But generally this load is less then the connected load because all consumers do not open or operate their equipment at the same time.
• Demand Factor: It is the ratio of maximum demand on a power station to its connected load. 
  Hence, Demand factor = Maximum demand / Connected load
• Average Load: The average of load (as per demand of consumers) occurring in the power station for a given period (day, month or year) is known as average load or average demand. In other word, the average of unit (kWh) generation for a given period (day, month or year) is known as average load. 
  Hence,
  Daily Average load = Units (kWh) generated in a day ÷ 24 hrs.
  Monthly Average load = Units (kWh) generated in a Month ÷ hrs. in a month.
  Annual Average load= Units (kWh) generated in a year ÷ hrs. in a year

(To be continued...)