In this article we will discuss about:- 1. Introduction to Economic Operation of Power Plants 2. Methods of Loading Turbo-Alternators 3. Distribution of Load between Generating Units 4. Economic Load Neglecting Transmission Losses 5. Optimum Economic Operation 6. Economic Load Distribution between Different Plants in an Integrated System and Other Details.
Introduction to Economic Operation of Power Plants:
A modern power system is invariably supplied by a number of power plants. The purpose of economic operation of power system is to reduce the operating cost of generation to the minimum. The total generator operating cost includes fuel, labour and maintenance costs. For simplicity fuel cost is the only one considered to be variable.
The fuel cost is meaningful in case of thermal and nuclear power plants. Hydro plants have negligible operating cost, but are required to operate under constraint of availability of water for hydro generation in a given period of time. It is, however, unrealistic to neglect transmission losses particularly when long distance transmission of power is involved.
The control engineer or the load dispatcher has to consider a number of factors while interchanging energy from one power station to another station—how much energy to interchange, the cost of supplying energy to interconnections, and the cost of energy received from the interconnection, etc.
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It is necessary to arrange the operation of each generating unit on the system load curve every hour so as to reduce production costs during the year to a minimum. The factors directly affecting the economy of operation are many, such as variation in fuel cost, labour cost, and weather conditions, normal and emergency equipment ratings, reserve requirements, voltage limitations, characteristics of prime movers, transmission losses etc.
However, the major factor controlling the most desirable load allocation between different generating units is the total operating cost. The cost of fuel forms the major portion of the variable cost of the power plant, and the aim of scheduling the operation of generating units is to obtain minimum fuel cost.
By economic load scheduling, we mean to determine the generations of different power plants such that the total operating cost is minimum, and at the same time the total demand and losses at any instant is met by the total generation.
Methods of Loading Turbo-Alternators:
There are several methods of loading the turbo-alternators. Of course, the incremental loading, the most scientific and rational method of operation of units, will always result in the maximum operating efficiencies. In this method of loading, the total system load is divided among the generating units that all the units operate at equal incremental costs. The order in which the units are brought into action depends on the heat rates. Further, loading is adjusted by incremental rates.
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The other methods of loading are:
1. Base Load to Capacity:
In this method, the different units are successively loaded to capacity in the order of their efficiencies. Accordingly, all but the most efficient unit are operated at minimum loads until the most efficient one is loaded to capacity, and it is assumed that arrangement would result in highest overall efficiency. But such an assumption will be correct only if the division of load is the same as that obtained by the incremental rate method.
2. Base Load to Most Efficient Load:
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In this method, the various units are successively loaded to their most efficient loads in the ascending order of their heat rates. After that, all the units are loaded to capacity in the same order.
3. Loading Proportional to Capacity:
In this method of loading, the units are loaded in proportion to their capacities.
4. Loading Proportional to Most Efficient Load:
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In this method, the units are loaded in proportion to their most efficient loads. After the units are loaded up to their most efficient loads, the additional load on each unit is made proportional to the difference between the rated capacity and the most efficient load.
Distribution of Load between Generating Units in a Power Plant:
To compute economic load division between units, it is necessary to express the operating costs in terms of output. For the steam turbine, steam generator or steam station as a whole, the diesel engine, or the gas turbine, the input is expressed in of kJ per hour and the output or load in MW. For hydro plant, the input will be in terms of m3/s of water flow and output (or load) in MW.
The efficiency at any load can be measured by taking the input from the input/output curve corresponding to the load.
The efficiency is then given as:
Heat rate is defined as the ratio of the input to the output Input i.e., Heat rate = Input/output in kilojoules per kWh.
The incremental rate of a machine at any given output is the rate of change of the input with respect to the output i.e., the ratio of small change in input to a small change in output is known as incremental rate of a machine. Numerically it is equal to the slope of the input/output curve at a point corresponding to that output. Mathematically, it is the first derivative of the input with respect to the output.
In the input-output characteristic if input is expressed in kJ/hour and output in MW, the slope of the curve gives incremental fuel rate in kJ/MWh and if the input is expressed in Rs per hour, the slope of the curve gives incremental fuel cost in Rs/MWh. Mostly incremental fuel cost is taken as the incremental cost (IC) and forms the basis of load division.
If the equation of the input-output curve is given in the form of an algebraic equation, then, the incremental rate can be easily determined by differentiating the equation with respect to the output.
The incremental rate can also be determined graphically, from the input/output curve. Corresponding to a given output, a point can be located on the curve. Then the slope of the tangent to the curve through this point gives the incremental rate at the given output.
Alternative method of determination of incremental rate is by taking input values corresponding to the series of output values from the input/output curve. The interval between two consecutive values of the outputs should be small and constant. The incremental rate is then calculated as the ratio of the input difference to the output difference and it is assumed to be a function of the mid-point.
The shapes of the heat rate curve and the incremental rate curve are shown in Fig. 9.2. Here, the heat rate curve and the incremental rate curve are plotted on the common coordinates. It is clear that the two curves cross each other at a point where heat rate is minimum.
Economic Load Neglecting Transmission Losses:
Two Generation Units:
Consider two turbo alternators operating in parallel and sharing the load P1 and P2 and costing C1 and C2.
Total output, P = P1 + P2 …(9.2)
Total cost, C = C1 + C2 …(9.3)
For most economical loading of alternator, total cost C should be minimum and it will be minimum when the first differential coefficient of Eq. (9.3) with respect to P1 is zero.
Thus, the condition for minimum total cost of operation of two alternators is that incremental costs of two alternators should be equal.
The same conclusion can be arrived at through a qualitative analysis. Let the two units operate at different incremental costs. Let us transfer some load from the unit operating at higher incremental cost to the unit operating at lower incremental cost. Reduction of load on the unit with higher incremental cost will mean a greater reduction in cost than the increase in cost owing to the increase in load on the unit with lower incremental cost. In the limiting case, when the distribution of load among two units is such that both operate at the same incremental cost, maximum economy will be achieved.
Larger Number of Units:
The above qualitative discussion about the two unit system can be extended to a system with a multiple generating units. For achieving economical loading the loads should be so allocated among the different generating units that every generating unit operates at the same incremental cost.
This can be illustrated as below:
For a plant with m generating units.
Since, the input or cost of any generating unit depends only on the output of that unit, the total input/cost would be minimum only when,
The economic distribution of load among m generating units operating in parallel in a plant can alternatively be found by using Lagrangian method. The total power received by the plant bus and transferred to the load is given by Eq. (9.7) whereas the total input/cost to the m generating units is given by Eq. (9.8).
Since the input to any generating unit depends only on the output of that unit, the total input/cost would be minimum only when dP is zero-
Multiplying Eq. (9.9) by λ, subtracting the resulting equation from Eq. (9.12) and rearranging the terms, we have:
The above equation will be satisfied only when each term equals zero. Since the input to a generating unit depends on the output of that unit only, the partial derivative becomes full derivative-
The constant λ is referred to as Lagrangian multiplier and here it is the incremental rate of the unit. For optimum economy every generating unit should be operated at the same incremental rate.
The condition for economic loading among several generating units within a plant is represented by a graphical interpretation in Fig. 9.3.
The load allocation among several generating units can be found by preparing operating schedules for different values of λ.
The operating schedule that gives the time sum of the outputs of the generating units equal to the total load on the system will be the operating schedule for distribution of loads among the generating units for the given load.
With the increase in load, the individual load will increase. One or several units will attain their rated capacities earlier than other units. After this happens, the remaining units should be operated at equal incremental rate costs. If the minimum and maximum loads of some or all the units are specified, some units will not be able to operate at the same incremental cost and still meet the constraint of the minimum and maximum outputs.
The incremental costs are usually non-linear functions of the output. Most of the time conversion of these non-linear functions into linear functions is possible without a loss of much accuracy and the operating schedule can be had analytically. However, if this is not possible, use of a digital technique becomes essential.
Optimum Economic Operation of Power Systems Considering Transmission Losses:
To introduce the complexity of transmission losses, let us consider the simplest possible example. Let one unit (or plant) be connected to a bus that also supplies the one-system load and let another unit (or plant) feed the same bus over a transmission line. Now when the unit (or plant) connected to the line increases its generation (output) by an increment, the same increment is not delivered to the bus.
The change in power at the bus is the original increment less the incremental losses in the line. In this simple case the incremental losses incurred in the transmission line are function of the power of the one generator (or plant) and the line characteristics. It is evident that the incremental cost curve at the generator (or plant) could be modified to include the cost of the incremental losses in the line.
Thus the new curve would be the incremental cost curve for the unit (or plant) for supplying power over the transmission line to the bus and would include the incremental cost of transmission losses. The new curve would be used in conjunction with the curve from the other unit (or plant) to load the system.
For the sake of simplicity let us neglect stunt capacitance and shunt conductance of the line and assume impedance of the line Z (in magnitude) = 
ohms per phase-
Obviously, P1 and P2 will be the function of the bus voltages V1 and V2 and power angle ∂. If bus voltages are maintained to be constant, the transmission losses then become function of ∂ only.
However, if the constancy of bus voltages V1 and V2 have not been assumed constant, the situation will be very complicated.
The transmission losses, PL can be computed utilising loss formula coefficients or “B-coefficients”. These B coefficients for a given system are assumed to remain constant and their evaluation depends upon the open-circuit impedance matrix of the system network.
The B coefficients are dependent on the parameters of the system network, configuration, plant power and voltages etc. Numbers of assumptions are necessarily made for simplification of the otherwise cumbersome equations.
Economic Load Distribution between Different Plants in an Integrated System Considering Transmission Losses:
Since the plants are generally long distance apart in an integrated system, it becomes imperative to consider transmission losses in deciding the load allocation to different plants. This leads to the dispatch of power in an economic way so as to make the overall cost to be the minimum.
Let there be m plants in a system integrated by transmission lines and tie. Let PG1, PG2,… PGm be the generation output of the plants in MW, PD be total load in MW and PL the losses in transmission lines.
Since transmission losses PL are function of bus voltages and thus reactive power flow in the system. In our analysis, let us ignore the effect of reactive power flow or we assume the magnitudes of all voltages to be constant. Thus transmission losses PL are function of plant outputs (PG1, PG2, PG3,… PGn).
The above Eq. (9.29) represents the modified economics operation criterion for the plants with transmission losses considered.
The minimum fuel cost for a given total load is obtained by solving simultaneous equations represented by Eq. (9.29).
The n optimum loading equations along with the power balance Eq. (9.25) are sufficient to determine the (n + 1) unknown. (PG1, PG2, PG3 … PGn and λ).
λ is thus given by:
So, minimum fuel cost is obtained when the incremental fuel cost of each plant multiplied by its penalty factor is the same for all the plants.
Coordination Equations:
Coordination equations represent the coordination between various generating units as regards to their load allocation.
If C is the total cost, PG is the total generation for the system having n units, the coordination equations can be formulated as below:
Iterative Procedure of Solving Coordination Equations:
The coordination Eq. (9.44) cannot be solved directly, because the incremental transmission loss (ITL) is dependent on power outputs from all the units. So, for solving the coordination equation an iterative procedure involving a method of successive approximations is adopted.
The iterative procedure is as follows:
(i) Assume PG2 = 0.
(ii) Determine PG1 from Eq. (9.47) taking PG2 = 0.
(iii) Determine PG2 from Eq. (9.48) using the value of PG1 determined in above step (step (ii)).
(iv) Determine PG1 from Eq. (9.47) using the value of PG2 determined in above step (step (iii)).
(v) Repeat steps (iii) and (iv) one after the other till the values of PG1 and PG2 determined in any previous step match values of PG1 and PG2 determined later.
Optimal Unit Commitment:
It is not economical to operate all the units available all the time. The unit commitment deals with the problem of determination of a unit of generating power plant which should operate for a particular load to minimize the operating cost.
The unit commitment problem is applicable to thermal power plants only because for other types of generating plants like hydro, the operating cost and start-up times are negligible; therefore, their on-off status is not significant.
A simple but an inaccurate method of solving unit commitment problem is assignment of priority to the generating units such that the most efficient unit is loaded first to be followed by the most efficient unit from the remaining units as the load increases.
Another method of solving this problem is computation of most economical operating cost for all the possible combinations of the units and then selects that combination which has the least operating cost among these. This method is very cumbersome and highly time consuming.
Considerable computational savings can be achieved by adopting some optimization techniques such as branch and bounce or a dynamic programming method for arriving at minimum operating cost combination as certain combinations need not be tried at all in these methods.
Optimal Unit Commitment Using Dynamic Programming Method:
In practice, the unit commitment problem is solved for the complete load cycle. The total number of units available, their individual operating cost characteristics and the load cycle on the generating stations are assumed to be known in advance. Further, it is assumed that the load on each unit or combination of units changes in suitably but uniform step of size ΔPD MW (e.g., 1 MW).
Let a cost function FN(x) be defined as below:
FN(x) = The minimum cost in Rs/hour of generating x MW by N units
FN(y) = Cost of generating y MW by the Nth unit (known in advance)
FN-1 (x -y) = the minimum cost of generating (x – y) MW by the remaining (N – 1) units
[for N = 2, F2-1(x – y) = F1(x – y) = f1(x – y) known in advance]
The application of dynamic programming results in the following recursive relations:
Here, the value of y is varied from minimum load level to the maximum load level in the discrete steps of ΔPD providing a set of values of the expression {fN(y) + fN-1 (x – y)}. The minimum of these values is optimum value of fN(x).
By use of recursive relation given by Eq. (9.49), we can easily find the combination of units which needs minimum operating costs for load ranging in convenient steps from the minimum permissible load of the smallest unit to the sum of the capacities of all the available units. In this process the total minimum operating cost and the load shared by each unit of the optimal combination are automatically found for each load level.
Starting arbitrarily with two units, the most economical combination is found for all the discrete plant load levels using Eq. (9.49) where for N = 2, FN-1, (x – y) = F1(x – y) =f1(x – y) which is known. At each plant load level the most economic solution may to run either unit or both units with a certain load sharing between the two. It may also be possible that all the combinations considered are infeasible for certain load levels.
The most economical cost curve in discrete form for the two units thus obtained can be viewed as the cost curve of a single equivalent unit. The third unit is now added and the process is repeated for determination of the cost curve of the combination of the three units considering the first two units as single equivalent unit.
It may be noted that in this process the operating combinations of third and first, also third and second are not required to be worked out resulting in considerable saving in computational effort. The process is repeated, till all available units are exhausted. The advantage of this approach is that having obtained the optimal way of loading k units, it is quite easy to find the optimal manner of loading (k + 1) units.
From the results obtained by the above procedure a unit commitment table, which indicates the units in operation requiring minimum operating cost for the different loads, is prepared. This unit commitment table is prepared once and for all for a given set of units. If the load cycles on a plant load changes, it will need to change the starting and stopping times of the units with the basic unit commitment table remaining unchanged.
Using the unit commitment table and increasing load in steps, the most economical plant operating cost is determined for the complete range of plant capacity by equal incremental cost criterion. The result is the overall plant cost characteristic in the form of set of data points. A quadratic equation (or higher order equation, if required) can then be fitted to this data for later use in economic load sharing among generating plants.