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rate curve with respect to the output, dVdx, which is the incremental heat rate curve. The fourth and fifth lines show the second derivative of the Input versus Output curves and the conditions that are required to make them positive.

In Section 11.2, it was shown that if the equation of input versus output I(x), incremental heat rate dI/dx, and the second derivative of the equation of input versus output d2 I/dx2, are continuous and d21/dx2>0,the minimum combined cost for a given combined output is obtained when the units operating in parallel are operating at outputs which correspond to the same incremental heat rate value.

In order to apply this principle, the incremental cost must always increase as the output increases. Therefore, the sum of the incremental fiel cost curve (the incremental heat rate multiplied by the fiel cost) and the incremental O&M cost must always increase as the output increases. If the incremental O&M cost is a fixed number (as is typical), then the incremental heat rate must always increase as the output increases.

The slope discontinuities at valve points and the non-convex nature along valve loops of a sequential valve machine will often conflict with the above principle imposed by the incremental dispatch model.

The economic dispatch computer model requires that an incremental cost curve be represented by a series of linear segments. Therefore, in order to have multiple line segments for the incremental heat rate, the input/output equation must be at least 3rd order. If the unit has a sequential arc turbine, usually one straight line is drawn between successive valve points. If the unit has a fill arc turbine, the curve is divided into straight segments to get the best fit. The Equation of the Line Segment Between ifiand ( i +1)~ valve points is as follows:

Owing to the non-convex nature of the input-output, an accurate performance checkout model will often conflict with the requirements imposed by the incremental dispatch model.

I(x) continuous dI/dx continuous d2I/dx2>0

Due to constraints imposed by the incremental dispatch algorithm, a 3rd order polynomial is fit to the data at each valve point.

d21/dx2=1.34E-4x - 2.66 and d21/dx2>0 when x>rnin load

Input vs Output

Figure 11.2 Smoothed Input vs. Output Curve

11.4.2 Develop Incremental Line Segments

Determine the equation for each line segment.

11.4.3 Incremental Heat Rate Curves

The curve below is a typical incremental heat rate curve for a unit in its original (performance guarantee) condition.

Incremental Heat

Rate

Figure 11.4 Incremental Heat Rate Curve

11.5Incremental Heat Rate Adjustment Factors

Because the heat rate of units changes over time, the incremental heat rate (and incremental cost) curves must be updated periodically. One method of doing this would be to run a 'full heat rate test on each unit annually or perhaps every 6 months. Although this is a very expensive and timeconsuming practice, there are several advantages to doing this.

1.Accurate, detailed information is obtained on the unit

2.The current position and shape of the actual perfbrmance of the unit, over the load range, is obtained.

A second procedure would be to have a real-time, continuous indication of not only the current heat rate of the unit, but an indication of what the heat rate curve is over the load range.

What is frequently done is to establish the original performance of the unit over the load range. Once this performance curve is made, it is left unchanged. Then an incremental heat rate adjustment factor is periodically calculated that can be multiplied by the reference curve to generate a curve that is representative of the current operation of the unit. The incremental heat rate adjustment factor is the ratio of actual heat rate to reference heat rate. Typically, this periodic calculation is done using full load data (actual heat rate and the reference heat rate at full load), and the resulting adjustment factor is assumed constant over the load range. However, adjustment factors could be calculated at several different loads, and instead of having a single factor, a curve of the incremental heat rate adjustment factor versus load could be developed.

where On= The fiactional effect on heat rate due to nLhoperational deviation fi-om expected level, then, substituting (1) into (2) yields

For the convenience of the dispatching data management, the reference heat rate curves (for each unit) are stored in the economic dispatching model as fixed bases. The incremental heat rate adjustment factors (one for each unit) are updated periodically, typically quarterly. They are then used to update the "input versus output" curve, the heat rate curve, the incremental heat rate curve and, ultimately the incremental cost curves, and entered into the dispatch algorithm.

Usually the incremental heat rate adjustment factor is calculated at full load. For example, if a 200 MW unit measures its actual heat rate daily, then the daily heat rates and reference heat rates would be calculated each day. For each day when the load was continuously high (between 185- 200 MW for example), the actual heat rate would be divided by the reference heat rate to calculate an incremental heat rate adjustment factor for that day. Those incremental heat rate adjustment factors are averaged to determine the incremental heat rate adjustment factor to be used in the dispatch algorithm for the next period.

For example, if the incremental heat rate adjustment factor is 1.15, then the original reference curve and the curve representing the current incremental heat rate would look like Figure 11.5. The adjusted curve would be used for dispatch.

Incremental Heat Rate

Figure 11.5 Incremental Heat Rate Curves, "Reference" and "Adjusted" for Actual Condition

11.6Example

The following is an example of dispatching based on the "Equal Incremental Method", comparing two unit's incremental heat rate curves (see Figure 11.6).

The units would be dispatched so that their combined output would meet the demand while having the same incremental rate.

If the demand was 190 MW, the outputs of "Unit A" and "Unit B at the same incremental rate of 8.55 would be 60 MW and 130 MW, respectively.

Incremental Heat Rate

Figure 11.6 Incremental Heat Rate Curves, Unit A and Unit B

11.7Conclusions

Units are dispatched based on the "Equal Incremental Cost Method."

The Incremental Cost Curves are derived from the incremental heat rate curve, and the variable O&M cost.

The minimum cost for a given combined output is obtained when the incremental costs of each generating unit are equal.

Even though the "Input versus Output" curve may follow a convex shape, the dispatch model forces the valve loops to fit in at least squares form with a positive slope for each segment.

The incremental cost curve is the sum of the "incremental heat rate curve" plus the incremental "variable operation and maintenance" costs.

Incremental curves should be updated periodically to better reflect the unit's current condition.

1 1 References

Economy Loading of Power Plants and Electric Systems, by Max J. Steinberg and Theodore H. Smith, published by John Wiley & Sons, Inc.

Optimal Power Dispatch - A Comprehensive Survey, by H. H. Happ, published in the IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, no. 3, MayIJune

1977.

SECTION 12 TYPICAL AREAS FOR HEAT RATE IMPROVEMENTS

12.1Introduction

This section reviews some areas where heat rate improvements are possible at many plants. It should not be used in place of reviewing all heat rate deviations, investigating the problem to determine the root cause, and implementing the appropriate solution. However, these eleven areas are typical opportunities for improving efficiency, reducing emissions, reducing maintenance, and obtaining other additional benefits. Not all plants have problems in each of these areas, but in many plants, these problems are commonly encountered.

Some of these problems show up as "unaccountabley' heat rate deviations, which are not readily apparent, therefore, they often go unnoticed (such as cycle isolation). Other potential improvement areas are overlooked because the true "expected" performance level is not defined (such as condenser performance or boiler outlet 02).

Most of these improvements require little, and in some cases such as variable pressure operation, no expense (material or manpower). Usually, these items can be implemented and improvements realized in a short amount of time.

12.2Improved Condenser Cleanliness

In almost all plants, there are opportunities for increased thermal efficiency is by increasing the cleanliness of the condenser. Even on units that have a closed loop condenser circulating water system, with treated water, over time, deposits (organic, inorganic, or both) will form on the internal diameter of the condenser tubes. The deposit or "foulingyydoes not have to be very thick, it may not even be apparent to the eye, for it to "insulate7'the tubes. The additional resistance to heat transfer causes the condenser pressure (and steam temperature) to increase, thereby increasing the temperature differential, so that the heat of condensation can flow through the tube, to the circulating water. This increase in condenser pressure increases the heat rate and decreases unit load. If the unit is not restricted by steam flow, coal flow, air flow, etc., the firing rate may be increased to maintain the unit load, but at an additional heat rate penalty. Other times, especially in the summer, if the steam flow cannot be increased, the unit output will decrease.

Many plants clean their condensers only once a year, during annual outages. For almost all units, this is insufficient, and results in higher production costs (and emissions) than necessary. A cost versus benefit analysis should be done, comparing the cost of cleaning a condenser to the heat rate improvement that will result, to find the optimum cleaning cycle that minimizes the total cost (cost of cleaning and cost due to the heat rate penalty from dirty tubes).

This approach requires that the "true expected" condenser pressure be known for a unit. This is not a constant ("designyy)value. The "true expected" condenser pressure is frequently below the "designyypressure, and sometimes above. It varies with the heat load on the condenser (primarily