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Economics of Power Generation

The art of determining the per unit (i.e., one kWh) cost of production of electrical energy is known as economics of power generation.

The economics of power generation has assumed a great importance in this fast developing power plant engineering. A consumer will use electric power only if it is supplied at reasonable rate. Therefore, power engineers have to find convenient methods to produce electric power as cheap as possible so that consumers are tempted to use electrical methods. Before passing on to the subject further, it is desirable that the readers get themselves acquainted with the following terms much used in the economics of power generation :

(i) Interest:

The cost of use of money is known as interest.

A power 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. The rate of interest depends upon market position and other factors, and may vary from 4% to 8% per annum.

(ii) Depreciation:

The decrease in the value of the power plant equipment and building due to constant use is known as depreciation.

If the power station equipment were to last for ever, 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.

Cost of Electrical Energy

The total cost of electrical energy generated can be divided into three parts, namely;

(i) Fixed cost;            (ii) Semi-fixed cost ;               (iii) Running or operating cost.

(i) Fixed cost:

It is the cost which is independent of maximum demand and units generated.

The fixed cost is due to the annual cost of central organization, interest on capital cost of land and salaries of high officials. The annual expenditure on the central organization and salaries of high officials is fixed since it has to be met whether the plant has high or low maximum demand or it generates less or more units. Further, the capital investment on the land is fixed and hence the amount of interest is also fixed.

(ii) Semi-fixed cost:

It is the cost which depends upon maximum demand but is independent of units generated.

The semi-fixed cost is directly proportional to the maximum demand on power station and is on account of annual interest and depreciation on capital investment of building and equipment, taxes, salaries of management and clerical staff. The maximum demand on the power station determines its size and cost of installation. The greater the maximum demand on a power station, the greater is its size and cost of installation. Further, the taxes and clerical staff depend upon the size of the plant and hence upon maximum demand.

(iii) Running cost:

It is the cost which depends only upon the number of units generated.

The running cost is on account of annual cost of fuel, lubricating oil, maintenance, repairs and salaries of operating staff. Since these charges depend upon the energy output, the running cost is directly proportional to the number of units generated by the station. In other words, if the power station generates more units, it will have higher running cost and vice-versa.

Expressions for Cost of Electrical Energy

The overall annual cost of electrical energy generated by a power station can be expressed in two forms viz three part form and two part form.

(i) Three part form: In this method, the overall annual cost of electrical energy generated is divided into three parts viz fixed cost, semi-fixed cost and running cost i.e.

Total annual cost of energy = Fixed cost + Semi-fixed cost + Running cost

= Constant + Proportional to max. demand + Proportional to kWh generated.

= Rs (a + b kW + c kWh)

Where,

a = annual fixed cost independent of maximum demand and energy output.

b = constant which when multiplied by maximum kW demand on the station gives the annual semi-fixed cost.

c = a constant which when multiplied by kWh output per annum gives the annual running cost.

(ii) Two part form:

It is sometimes convenient to give the annual cost of energy in two part form. In this case, the annual cost of energy is divided into two parts viz., a fixed sum per kW of maximum demand plus a running charge per unit of energy. The expression for the annual cost of energy then becomes :

Total annual cost of energy = Rs. (A kW + B kWh)

Where,

A = a constant which when multiplied by maximum kW demand on the station gives the annual cost of the first part.

B = a constant which when multiplied by the annual kWh generated gives the annual running cost.

It is interesting to see here that two-part form is a simplification of three-part form. A little reflection shows that constant “a” of the three part form has been merged in fixed sum per kW maximum demand (i.e. constant A) in the two-part form.

Methods of Determining Depreciation

There is 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.

(i) 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. Thus, if the initial cost of equipment is Rs 1,00,000 and its scrap value is Rs 10,000 after a useful life of 20 years, then,

In general, the annual depreciation charge on the straight line method may be expressed as:

Annual depreciation charge =

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. Fig. 4.1 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.

(ii) 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. As an example, suppose the initial cost of equipment is Rs 10,000 and its scrap value after the useful life is zero. If the annual rate of depreciation is 10%, then depreciation charge for the first year will be 0·1 × 10,000 = Rs 1,000. The value of the equipment is diminished by Rs 1,000 and becomes Rs 9,000. For the second year, the depreciation charge will be made on the diminished value (i.e. Rs 9,000) and becomes 0·1 × 9,000 = Rs 900. The value of the equipment now becomes 9000 − 900 = Rs 8100. For the third year, the depreciation charge will be 0·1 × 8100 = Rs 810 and so on.

Mathematical treatment

Let,

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 + Px2

= P(x2 − 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. Fig. 4.2 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 moneys, if earned, are to be treated as income.

(iii) 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 instalments 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:

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

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

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

The load factor plays a vital role in determining the cost of energy. Some important advantages of high load factor are listed below:

(i) Reduces cost per unit generated: A high load factor reduces the overall cost per unit generated. The higher the load factor, the lower is the generation cost. It is because higher load factor means that for a given maximum demand, the number of units generated is more. This reduces the cost of generation.

(ii) Reduces variable load problems: A high load factor reduces the variable load problems on the power station. A higher load factor means comparatively less variations in the load demands at various times. This avoids the frequent use of regulating devices installed to meet the variable load on the station.