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Heat Rate Modelling

Contents

  1. Definitions
  2. Heat Rate Properties
  3. Heat Input Function and Convexity
  4. Separable Programming and Convexity
  5. Schemes Without Load Point
    1. Constant Average and Marginal Heat Rates
    2. Constant Marginal Heat Rate
    3. Polynomial Heat Input Functions
  6. Schemes With Load Point
    1. Marginal Heat Rates with Intercept
    2. Marginal Heat Rates with Average at Min Stable Level
    3. Average Heat Rates
  7. Heat Rate Formula Conversions
  8. Heat Rate Output
  9. References

1. Definitions

The following definitions are used throughout this article:

  1. The heat input function is given by y = f (x) where y is the heat input (amount of fuel consumed) to produce at megawatt level x for one hour i.e. to produce x megawatt hours of output.
  2. The marginal heat-rate function is the derivative of the heat input function m (x) = δy / δx
  3. The average heat rate function is a (x) = f(x) / x
  4. The term average marginal heat rate refers to the average value of the marginal heat rate function between two points x1 and x2. The average marginal heat rate is equal to the marginal heat rate at the mid-point between x1 and x2 in all cases except if the heat input function is third-order.
  5. The heat input at a national zero megawatt output is referred to as the no load cost.

Note that the fuel units shown in the examples here are sometimes GJ (giga-joules - Metric) and sometimes MMBTU (million British Thermal Units - Imperial U.S.), but all examples apply to either unit system. Note that "giga" means billion units, and the "M" in "MMBTU" refers to the Roman numeral "M" meaning one thousand, thus "MM" is a way of saying thousand thousands or a million since there is no Roman numeral for million other than the rather awkward convention of writing M with an over-bar.

2. Heat Rate Properties

In the simulator:

Where the heat input function is defined as a polynomial function y = a + bx + cx2 + dx3, the a term (intercept of the heat input function with the y-axis) is Heat Rate Base, b (linear term) is Heat Rate Incr, c (quadratic term) is Heat Rate Incr2, and the optional d (tertiary term) is Heat Rate Incr3.

Broadly speaking the term "heat rate function" might refer to the marginal or average heat rate function. Both are derived from the heat input function. There is wide variation on the way heat data are provided for use in simulation models. For convenience then the simulator provides six different methods of input by specifying one of the schemes in Table 1.

Table 1: List of available heat input function schemes.

Number Description Parameters
1. A linear heat input function with constant average and marginal heat rates Heat Rate or Heat Rate Incr
2. A linear heat input function with constant marginal heat rate Heat Rate Base and Heat Rate Incr
3. A polynomial heat input function Heat Rate Base, Heat Rate Incr,
Heat Rate Incr2, and optionally
Heat Rate Incr3
4. Pairs of load points and marginal heat rates Load Point / Heat Rate Incr pairs
and optional Heat Rate Base
5. Pairs of load points and marginal heat rates with no-load cost defined as an average heat rate Load Point / Heat Rate Incr pairs plus
the average Heat Rate
6. Pairs of load points and average heat rates Heat Rate / Load Point in multiple bands

Note that hydro generator efficiency is defined similarly with properties Efficiency Base and Efficiency Incr however the efficiency is the inverse of the marginal heat rate used for thermal generators.

3. Heat Input Function Polynomial

The schemes in Table 1 are simply a method for defining the heat input function. Having received these input the simulator:

  1. Computes a third-order polynomial fit to the heat input function i.e. no matter which method you use for input the result is always converted to scheme 3 (with exception noted below).
  2. Divides the space between Min Stable Level and Max Capacity into even length 'tranches' with a tranche reserved for generation below Min Stable Level.
  3. Calculates a step approximation to the marginal heat rate function based on the average marginal heat rate in each tranche.

The Production settings Fuel Use Function Precision and Max Heat Rate Tranches control the accuracy and number of tranches used in this step-wise function. The setting Generator Max Heat Rate Tranches provides the same control by Generator object. As mentioned above, one tranche is used for generation below Min Stable Level.

Notes for input scheme 4:

Notes for input scheme 6:

4. Separable Programming and Convexity

The step-wise approximation to the marginal heat rate function is based on the Fan Approximation method-see DRAYTON 1997 [1, 2]. This method uses separable programming to cast the polynomial heat input function as a linear approximation. Separable programming relies on certain characteristics both of the marginal heat rate tranches and the optimization's objective function and other constraints. Firstly the marginal heat rate 'steps' must be monotonically non-decreasing in cost. When this is not the case we refer to the function as being 'non-convex'. Secondly their cost should not be zero, nor should any other constraint or cost in the optimization problem imply that the marginal value of steps is monotonically non-decreasing. In simple terms, the optimization must prefer the tranche one marginal heat rate first, tranche two second, and so on. Any other order and the approximation will yield an inaccurate modelled heat input function.

To solve this issue the simulator allows you to switch to a variation of the Fan Approximation that includes additional constraints that ensure that the tranches are used in order. Because this version requires integer variables and additional constraints it is not enabled by default, however warnings from the simulator will indicate when this integer Fan Approximation might be needed:

You can switch to the integer Fan Approximation globally by setting Production Heat Rate Error Method to "Allow Non-convex". The setting Generator Formulate Non-convex provides the same function for individual objects.

With this option enabled there is no restriction on the 'convexity' of the marginal heat rate function, which allows you to model marginal heat rate functions of practically any shape. Examples occur often when modelling CCGT plant as aggregate generator objects.

By default Heat Rate Error Method is set to only warn about the non-convexity and "adjust" the marginal heat rate function to make it convex. The adjustment procedure is as follows. Assume the step-wise marginal heat rate function has steps (tranches) numbered from one being the tranche starting at notional zero generation to N being the tranche that runs up to Max Capacity:

  1. Set n=N-1
  2. Decrement n. If n=0 then stop.
  3. If the marginal heat rate in tranche n is less-than-or-equal-to the marginal heat rate in tranche n+1 then go to step 2.
  4. Set the marginal heat rate in tranche n and n+1 equal to the lesser of the average of values in tranche n and n+1 and the maximum of all values above n+1.
  5. Go to step 2.

5. Schemes Without Load Point

5.1. Constant Average and Marginal Heat Rates

Figure 1 shows an example of a simple linear heat input function in which the average and marginal heat rate curves are equal i.e. the heat input function has the form y = bx This type of function can be entered by setting the Heat Rate property of the generator as in the following example:

Property Value Metric Units Value Imperial Units
Units 1 - 1 -
Max Capacity 250 MW 250 MW
Min Stable Level 70 MW 70 MW
Heat Rate 9.487 GJ/MWh 9487 Btu/kWh

Note that Heat Rate Incr could also be used here.

Figure 1: Linear Heat Input Function with Constant Marginal and Average Heat Rate

5.2. Constant Marginal Heat Rate

Figure 2 shows an example of a linear heat input function in the form y = a + bx i.e. with constant marginal heat rate but variable average heat rate. This type of function can be entered by setting the Heat Rate Base and Heat Rate Incr properties of the generator as in the following example:

Property Value Metric Units Value Imperial Units
Units 1 - 1 -
Max Capacity 250 MW 250 MW
Min Stable Level 70 MW 70 MW
Heat Rate Base 78 GJ 78 mmBtu
Heat Rate Incr 9.175 GJ/ MWh 9175 Btu/ kWh
Figure 2: Linear Heat Input Function with Constant Marginal Heat Rate

5.3. Polynomial Heat Input Functions

Figure 3 shows an example of quadratic heat input function. The coefficients are entered using the properties, Heat Rate Base, heat Rate Incr, and Heat Rate Incr2 and in this example:

Property Value Metric Units Value Imperial Units
Units 1 - 1 -
Max Capacity 250 MW 250 MW
Min Stable Level 70 MW 70 MW
Heat Rate Base 78 GJ 78 mmBtu
Heat Rate Incr 7.97 GJ/MWh 7970 Btu/kWh
Heat Rate Incr2 0.00482 GJ/MWh^2 0.00482 Btu/kWh^2

To simulate this curve, PLEXOS will automatically develop a piecewise linear approximation to the quadratic function, and you can specify how many linear tranches are used to make the approximation. This is described below.

Figure 3: Quadratic Heat Input Function

An extension to this case is to add a third-order coefficient as in the following data and illustrated in Figure 4:

Property Value Metric Units Value Imperial Units
Units 1 - 1 -
Max Capacity 250 MW 250 MW
Min Stable Level 70 MW 70 MW
Heat Rate Base 78 GJ 78 mmBtu
Heat Rate Incr 7.97 GJ/MWh 7970 Btu/kWh
Heat Rate Incr2 0.00482 GJ/MWh^2 0.00482 Btu/kWh^2
Heat Rate Incr3 -0..009 GJ/MWh^3 -0.000009 Btu/kWh^3

To simulate this curve, PLEXOS will develop a piecewise linear approximation, but here the marginal heat rate is not monotonically non-decreasing thus the function is not separable. PLEXOS solves this problem using MIP as described earlier.

Figure 4: Third-order Polynomial Heat Function

6. Schemes With Load Point

PLEXOS allows for the specification of heat rate functions using pairs of heat rate (either marginal or average) along with megawatt load points. The following sections illustrate three methods of input based on the quadratic heat input function in Figure 3 i.e. these all produce the same outcome but use different combinations of input variables. These methods allow the input of heat rate functions that do not take simple functional forms like those in the preceding cases.

Note that the Load Point may take any value and can overlap or skip parts of the 'normal' operating range of the unit if required. Further, as with the polynomial functions, PLEXOS does not require that the marginal heat rates are monotonically non-decreasing, thus any set of values can be entered.

The user can also specify separate heat rate functions for different timeslices. In such a case, it should be ensured that the number of bands remains the same across all timeslices and the heat rate and load point information for all bands must be input.

The following properties are common to all the examples:

Property Value Units
Units 1 -
Max Capacity 250 MW
Min Stable Level 70 MW

6.1. Marginal Heat Rates With Intercept

The Heat Rate Base is specified along with multiple pairs of Load Point and Heat Rate Incr. Heat Rate Base is the intercept of the heat input function with the y-axis. Heat Rate Incr is the marginal heat rate at the mid-point of the band not at the Load Point itself. In this method, the data are describing the steps of a marginal cost function that PLEXOS should use to model the generator's heat input function and PLEXOS will use the function verbatim.

An example for the function show in Figure 3 is:

Property Value Units Band
Heat Rate Base 78 GJ/hr 1
Load Point 70 MW 1 Heat Rate Incr 8.3074 GJ/MWh 8307.3 Btu/kWh 1
Load Point 90 MW 2 Heat Rate Incr 8.7412 GJ/MWh 8741.2 Btu/kWh 2
Load Point 110 MW 3 Heat Rate Incr 8.934 GJ/MWh 8934.0 Btu/kWh 3
Load Point 130 MW 4 Heat Rate Incr 9.1268 GJ/MWh 9126.8 Btu/kWh 4
Load Point 150 MW 5 Heat Rate Incr 9.3196 GJ/MWh 9319.6 Btu/kWh 5
Load Point 170 MW 6 Heat Rate Incr 9.5124 GJ/MWh 9512.4 Btu/kWh 6
Load Point 190 MW 7 Heat Rate Incr 9.7052 GJ/MWh 9705.2 Btu/kWh 7
Load Point 210 MW 8 Heat Rate Incr 9.898 GJ/MWh 9898.0 Btu/kWh 8
Load Point 230 MW 9 Heat Rate Incr 10.0908 GJ/MWh 10090.8 Btu/kWh 9
Load Point 250 MW 10 Heat Rate Incr 10.2836 GJ/MWh 10283.6 Btu/kWh 10

6.2. Marginal Heat Rates with Average at Min Stable Level

The Heat Rate is specified at the Min Stable Level along with multiple pairs of Load Point and Heat Rate Incr. Heat Rate is the average heat rate at Min Stable Level and is provided only on the band 1 Load Point. The first Load Point must always be at Min Stable Level and the last must be at Max Capacity. The Heat Rate Incr is the marginal heat rate at the mid-point of the band.

An example for the function show in Figure 3 is:

Property Value Units Band
Load Point 70 MW 1 Heat Rate 9.4216 GJ/MWh 8307.4 Btu/kWh 1
Load Point 90 MW 2 Heat Rate Incr 8.7412 GJ/MWh 8741.2 Btu/kWh 2
Load Point 110 MW 3 Heat Rate Incr 8.934 GJ/MWh 8934.0 Btu/kWh 3
Load Point 130 MW 4 Heat Rate Incr 9.1268 GJ/MWh 9126.8 Btu/kWh 4
Load Point 150 MW 5 Heat Rate Incr 9.3196 GJ/MWh 9319.6 Btu/kWh 5
Load Point 170 MW 6 Heat Rate Incr 9.5124 GJ/MWh 9512.4 Btu/kWh 6
Load Point 190 MW 7 Heat Rate Incr 9.7052 GJ/MWh 9705.2 Btu/kWh 7
Load Point 210 MW 8 Heat Rate Incr 9.898 GJ/MWh 9898.0 Btu/kWh 8
Load Point 230 MW 9 Heat Rate Incr 10.0908 GJ/MWh 10090.8 Btu/kWh 9
Load Point 250 MW 10 Heat Rate Incr 10.2836 GJ/MWh 10283.6 Btu/kWh 10

6.3. Average Heat Rates

Multiple Load Point and Heat Rate pairs are specified. Heat Rate is the average heat rate at each Load Point. The Load Point values may take any value and can overlap or skip parts of the 'normal' operating range of the unit if required. Thus, in this method, the data are describing points on the heat input function and PLEXOS will generate a piecewise linear model of the marginal heat rate function from these data.

Note: When using Heat Rate and Load points - the inputs will be approximated with a polynomial first and then tranches are created



An example for the function show in Figure 3 is:

Property Value Units Band
Load Point 70 MW 1 Heat Rate 9.4216 GJ/MWh 9421.6 Btu/kWh 1
Load Point 90 MW 2 Heat Rate 9.2705 GJ/MWh 9270.5 Btu/kWh 2
Load Point 110 MW 3 Heat Rate 9.2093 GJ/MWh 9.2093 Btu/kWh 3
Load Point 130 MW 4 Heat Rate 9.1966 GJ/MWh 9196.6 Btu/kWh 4
Load Point 150 MW 5 Heat Rate 9.213 GJ/MWh 9231.0 Btu/kWh 5
Load Point 170 MW 6 Heat Rate 9.2482 GJ/MWh 9248.2 Btu/kWh 6
Load Point 190 MW 7 Heat Rate 9.2963 GJ/MWh 9296.3 Btu/kWh 7
Load Point 210 MW 8 Heat Rate 9.3536 GJ/MWh 9353,6 Btu/kWh 8
Load Point 230 MW 9 Heat Rate 9.41775 GJ/MWh 9417.75 Btu/kWh 9
Load Point 250 MW 10 Heat Rate 9.487 GJ/MWh 9487.0 Btu/kWh 10

7. Heat Rate Formula Conversions

To manually convert heat rate functions between formats, start by computing the heat input (in GJ or MMBTU) at each load point. The average heat rate function is simply the heat input at the load point divided by the generation at the point. The incremental function starts by computing the heat input at min stable level (either as an average heat rate using the Heat Rate property, or as the No Load input as Heat Rate Base), and then computing the incremental heat input for the incremental generation. The incremental heat rate is then the change in heat input from point n to point n-1 divided by the change in generation from point n to point n-1.

Always remember unit conversion in the Imperial system. Because heat rates are expressed in BTU/kWh. Heat input in MMBTU is computed as the product of generation (MWh) times heat rate (BTU/kWh) times 0.001. The metric system units do not require conversion.

Our Heat Rate examples above show the units of GJ/MWh = 1000 BTU/kWh. This is a convenience for the examples only, the actual conversion factor is 1055.06

8. Heat Rate Output

The following outputs are based on the step-wise approximation to the marginal heat rate function:

The following outputs relate to the heat rate at the given Generator Generation level:

9. References

  1. G.R. DRAYTON. Coordinating Energy and Reserves in a Wholesale Electricity Market, Doctoral Thesis, University of Canterbury, New Zealand. 1997.
  2. G.R. DRAYTON. The Fan Approximation. Drayton Analytics Research Paper Series.