NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Methodology for Constructing 

Reduced-Order Power Block 

Performance Models for CSP 

Applications

Preprint

Michael J. Wagner

National Renewable Energy Laboratory

To be presented at SolarPACES 2010

Perpignan, France

September 21-24, 2010

Conference Paper

NREL/CP-5500-49370

October 2010
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SolarPACES 2010
 

METHODOLOGY FOR CONSTRUCTING REDUCED-ORDER POWER

BLOCK PERFORMANCE MODELS FOR CSP APPLICATIONS

Michael J. Wagner

Mechanical Engineer, National Renewable Energy Laboratory, Golden, Colorado (USA)
 

Email: michael.wagner@nrel.gov, Phone: +1.303.384.7430
 

1 Introduction

Superheated steam Rankine cycle technology is widely used in utility-scale power generation and is well

understood for fossil-fuel, nuclear, and biomass applications. These fuel-consuming power plants typically

operate in a narrow range of conditions near their design point, but concentrating solar power (CSP)

plant operating conditions can vary much more widely depending on solar resource availability, storage

capacity, or ambient conditions. Consequently, CSP technology performance models must maintain a

high level of accuracy in off-design simulation to avoid significant error in annual electricity yield totals.

Yield analysis studies evaluating power plant economic feasibility have not traditionally required detailed

performance models of off-design heat absorption rates or varying heat source inlet temperatures. This

is the case since traditional plants spend most of their life operating at a fixed thermal load and resource

temperature; only variation in ambient temperature is of notable concern. Models that do accommodate

off-design thermal loads and temperatures tend to fall into two categories: they either are robust models

with detailed fundamental calculations like [1], [2], and [3], or they are simplified empirical models that

only capture top-level performance like [4].

The inherent variability of the solar resource presents a unique challenge for CSP systems. Incident

solar irradiation can fluctuate widely over a short time scale, but plant performance must be assessed

for long time periods. As a result, annual simulations with hourly (or sub-hourly) timesteps are the

norm in CSP analysis. A highly detailed power cycle model provides accuracy but tends to suffer from

prohibitively long run-times; alternatively, simplified empirical models can run quickly but don’t always

provide enough information, accuracy, or flexibility for the modeler.

The ideal model for feasibility-level analysis incorporates both the detail and accuracy of a first-principle

model with the low computational load of a regression model. The work presented in this paper proposes a

methodology for organizing and extracting information from the performance output of a detailed model,

then using it to develop a flexible reduced-order regression model in a systematic and structured way. A

similar but less generalized approach for characterizing power cycle performance has been presented in

[5], and a reduced-order modeling methodology for CFD analysis of heat transfer from electronic devices

is discussed in [6]. This paper builds on these publications and the non-dimensional approach originally

described in [7].1

2 Proposed modeling approach

Instead of relying on part-load polynomial curves or lookup tables to predict plant performance in offdesign conditions, this methodology uses a structured design-of-experiments approach [8] to characterize

plant behavior using detailed process modeling software, then extracts output dependence from that

information within the framework of statistical effects and interactions.

1The Alliance for Sustainable Energy, LLC (Alliance), is the manager and operator of the National Renewable Energy

Laboratory (NREL). Employees of the Alliance, under Contract No. DE-AC36-08GO28308 with the U.S. Dept. of Energy,

have authored this work. The United States Government retains and the publisher, by accepting the article for publication,

acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish

or reproduce the published form of this work, or allow others to do so, for United States Government purposes.

1
To illustrate this idea, consider the input and output of interest in a Rankine power cycle model. For

CSP plants, the modeler is primarily interested in the total power output, thermal absorption from the

heat transfer fluid (HTF), HTF return temperature, and cooling system load. These in turn depend

on inputs like HTF inlet temperature, turbine back-pressure, HTF mass flow rate, and others. If the

HTF flow rate varies exclusively, a simple part-load curve might adequately capture the performance

impact. However, accurate results are unlikely if the inlet HTF temperature and mass flow rate both vary

simultaneously and the two independent effect functions are overlaid. The proposed approach not only

captures the impact of varying a single input, but also captures competing interaction effects introduced

by concurrent off-design variation.

3 Off-design response framework

The ultimate goal of this method is to convert a complex and detailed power block model into a descriptive

simplified model. To accomplish this task, a thorough information set is required that describes the

cycle’s off-design performance as a function of operating conditions, and a detailed process simulation

software tool like IPSEproTM, GateCycleTM, or ThermoFlowTMprovides this information. One purpose

of this methodology is to minimize the modeling effort that is required to develop a thorough system

performance map, and this is achieved through strategic selection of the simulation parameters. The

response framework provides a design for the analytical experiment that meets these criteria.

The first step in constructing the response framework is establishing which quantities impact cycle performance (model inputs) and which are required to fully describe the observed performance (model

outputs). Cycle performance is fundamentally expressed in terms of power output (W ̇ ) and heat absorption (Q ̇); the relationship between the independent variables and these two values allows the calculation

of secondary information including cycle efficiency, heat rejection load, and HTF temperature drop. For

any given cycle design, performance is a function of three independent variables2: HTF inlet temperature (Tin), condenser steam pressure (Pc), and HTF mass flow rate (m ̇ ). Deviation among these three

variables from their design-point values adversely impacts the heat-to-power conversion efficiency of the

cycle.

3.1 Parameter normalization

The input and output variables are readily expressed in normalized terms, and the regression methodology

presented below requires this generalized state. The power output, heat input, and HTF mass flow rate

are simply divided by their design-point values such that they are equal to 1.0 when they match the design

rate. The two exceptions to this approach are the condenser pressure and the inlet HTF temperature.

Condenser pressure does not generally scale up or down as the size of the power block varies, so it is left

in dimensional form.

Inlet HTF temperature is normalized with respect to the superheater temperature rise. For cycles with

superheat, most of the heat transfer from the HTF to the steam working fluid occurs in boiling and

superheat. Steam evaporates at a constant temperature for a given boiler pressure, and this temperature represents the effective minimum temperature at which the HTF could enter the power block and

still provide heat flow into the steam. Therefore, the boiler steam temperature (not the HTF outlet

temperature) is considered to be the zero-value for normalized HTF inlet temperature.

Tin − Tboil Tˆ

in = (1) Tin,des − Tboil

2This statement assumes that the turbine inlet pressure is either fixed or varies systematically with HTF inlet temperature.

2
3.2 Experimental design

The three independent variables are varied parametrically within the response framework to obtain the

required performance information. Each independent variable has a direct impact on performance, but

simultaneous variation of the independent variables can also induce secondary impacts that adversely

affect accuracy. To measure these “interaction” effects, the analysis includes parametric runs where the

inputs are evaluated concurrently at design and off-design conditions.

Table 1 provides the experimental design framework for this methodology. Each run in the table represents a parametric set of simulations where the variable indicated with “�” is varied stepwise from the

lower to upper bounds. The other two variables are held fixed at either their lower bound (-), upper

bound (+), or design-point value (0). Runs 2, 5, and 8 measure the primary effects of one of the three

independent variables while the other two are held at their design-point value. The remainder of runs

1-9 aid in measuring the impact of two-variable interaction (denoted A⟩B), and runs 10-13 measure the

impact of the three-variable interaction (which is often negligible and has been neglected in this paper).

The upper and lower bounds are selected based on practical operation limits of the power cycle. For

example, if the minimum turn-down state for the modeled cycle is 30% of design (i.e. a 70% reduction

in operation), the lower bound on m ̇ should be 0.3 × m ̇ des.

Run Tin Pc m ̇ Measures

1 � 0 - f(m ̇ ⟩Tin)

2 � 0 0 f(Tin)

3 � 0 + f(m ̇ ⟩Tin)

4 - � 0 f(Tin⟩Pc)

5 0 � 0 f(Pc)

6 + � 0 f(Tin⟩Pc)

7 0 - � f(Pc⟩m ̇ )

8 0 0 � f(m ̇ )

9 0 + � f(Pc⟩m ̇ )

10

11

12

13

-

-

+

+

-

+

-

+

�

�

�

�

f(Tin⟩Pc⟩m ̇ )

f(Tin⟩Pc⟩m ̇ )

f(Tin⟩Pc⟩m ̇ )

f(Tin⟩Pc⟩m ̇ )

Table 1: The experimental design for characterizing power block performance. The experiment is adapted from a

three-level full-factorial design.

4 Formulating a model

The output of the runs in Table 1 is used to generate a model that can express both heat absorption and

power output as functions of the three independent variables and their interactions. The intermediate

result of each run is a table that relates heat absorption or power output to a particular arrangement

of Tin, Pc, and m ̇ . For example, Run #1 will produce a pair of functions that vary with HTF inlet

temperature at a fixed lower-bound value for HTF mass flow rate. Substituting the general output

variable Y  ̇  ̇ for Q and W: ( ) YRun1 = f Tin, m ̇ (−) (2)

Figure 1 corresponds to runs 4-6 in Table 1 and shows how power (left) and heat (right) output vary as

a function of parameterizing Pc at three distinct levels of Tin. The central lines in Figure 1 represent the

situation where only Pc is varied and both Tin and m ̇ are left at their design-point values (run #5).

This methodology asserts that the upper and lower curves observed in Figure 1 can be approximated

by multiplying the main effects of YME(Tin) and YME(Pc), where the two-variable interaction effect

YINT (Tin⟩Pc) scales the impact of the second main effect. Since the purpose of the interaction effect is

3
0.00 0.05 0.10 0.15 0.20 0.25

Condenser pressure [bar] (Pc )

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Normalized Power Output (

ÇW)

Tin at (+)

Tin at (0)

Tin at (-)

0.00 0.05 0.10 0.15 0.20 0.25

Condenser pressure [bar] (Pc )

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Normalized Heat Input (

ÇQ)

Tin at (+)

Tin at (0)

Tin at (-)

Figure 1: Data showing how the power and heat output varies with condenser pressure [bar] at three distinct levels of

HTF inlet temperature.

to adjust the magnitude of main effect’s impact on the output, the interaction effect is applied only to

the difference between the calculated main effect and the basis normalized value (i.e. YME − 1). Since

all effect functions in the set Y are normalized about 1, the main effect must first be shifted from 1 to

0 before being multiplied by the interaction effect. The product is then readjusted to 1.0 as shown in

Eq.[3]. In general terms, the calculated output variable Y * is determined for any set of inputs Xi with

the relationship:

∏

nX

Y * = [(YME(Xi) − 1) · YINT (Xj ⟩Xi) + 1] (3)

i=1

In Eq.[3], Xj represents the input variable that interacts with Xi. For example, in Table 1, m ̇ interacts

with Tin, and this effect is measured in runs #1-3. In this example, Xj ≡ m ̇ and Xi ≡ Tin.

Interaction effects measure the relative difference between the calculated value that is predicted by

multiplying only main effects and the actual value observed from the detailed modeling simulation.

Mathematically speaking, the interaction is the ratio of values (Observed - Main Effect)/(Predicted ­

Main Effect), ˆ as shown in Eq.[4] (the Y notation indicates a data point obtained from the detailed

model). The interaction equation can be formulated for either the lower or upper bound runs (±),

assuming either the lower or upper bound of Xˆj is used.

( ) Yˆ (±)

( ˆ Xi) YINT

(±) Xˆj

(±)

⟩Xˆi = X

(

i) − YME( ˆ ) (4) (±) YME(Xˆi) YME(Xˆj ) − 1

The interaction calculations provide a function that indicates how variation in one independent variable

influences the effect of another varying independent variable. The relative impact of an interaction effect

applies consistently to the main effect that it’s adjusting. In other words, if two independent variables Xi

and Xj simultaneously deviate from their design, the magnitude of the interaction effect that is applied

to the main effect is only a function of the value of the first independent variable Xi. Consequently,

interaction effects that are calculated for the lower and upper limit values for a particular effect (i.e.

runs #1 and #3 for run #2) both represent the same interaction effect and can be averaged to estimate

a better interaction curve as shown in Eq.[5].

( ) ( ) (+) (+) (−) (−) Y X INT j ⟩Xi + Y X INT j ⟩Xi

YINT (Xj ⟩Xi) = (5) 2

4
Note that the resulting expression for YINT (Xj ⟩Xi) contains arguments for Xj and Xi; however, the

function is only truly dependent on Xi, retaining Xj in the nomenclature merely as a reference to the

two interacting variables. Figure 2 shows the heat absorption and power output as a function of HTF

temperature for three mass flow rates and the associated interaction effect curves. This shows that even

for cases where the output appears to be linearly related to the independent variables, interaction effects

can have a significant relative impact.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF Inlet Temperature (Tin)

0.0

0.5

1.0

1.5

2.0

Power output (

ÇW)

mÇ at (-)

mÇ at (0)

mÇ at (+)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF Inlet Temperature (Tin)

0.0

0.5

1.0

1.5

2.0

Heat absorption (

ÇQ)

mÇ at (-)

mÇ at (0)

mÇ at (+)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF inlet temperature (Tin)

0.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

Interaction value

(mÇ |Tin) for power output

(mÇ |Tin) for heat abs.

Figure 2: Data showing how the power and heat output varies with HTF temperature at three distinct levels of HTF

mass flow rate (top), and the associated interaction effects (bottom).

4.1 Other calculated output

Since the power block model regresses only power output and heat input as functions of HTF temperature,

condenser pressure, and mass flow rate, other performance values need to be calculated. The calculation

of power cycle efficiency is straightforward. Thermodynamic power cycle efficiency is determined by

simply dividing the work output by the required heat input.

W ̇ 7cycle = (6) Q ̇

The outlet HTF temperature from the power cycle is calculated by considering the HTF mass flow rate,

the heat input, the inlet temperature, and the specific heat of the HTF. The specific heat is assumed to

vary linearly over the range of the inlet and outlet temperatures. Thus, an average specific heat value is

5
used and is recalculated throughout the simulation.

Tout Tin − Q ̇

m ̇ cp,ave

(7)

Since the power block does not directly include heat rejection equipment, it must supply information to

an external cooling model on how much heat should be rejected. This value is equal to the difference

between work output and heat absorption, or alternatively:

q ̇rej = (1 − 7cycle) Q ̇ (8)

5 Model validation

To test how well the regression model fits data from the detailed model, 1000 random combinations of

the three input variables Tin, Pc, and m ̇ spanning the full range of operation of the power cycle were

generated and modeled using both a detailed modeling software tool and the corresponding regression

model. The difference between the value predicted by the regression model and the actual value from

the modeling software represents the methodology error. Table 2 quantifies the error for the two outputs

W ̇ and Q ̇ .

W ̇ Q ̇

Upper max. error (%) 0.69 0.15

Lower max. error (%) -2.01 -0.16

Average error(%) -0.24 -7.4×10−5

Error standard deviation (%) 0.49 0.04

Table 2: Quantification of the error from 1000 runs of the regression model as compared to the detailed modeling

software. Errors are shown for both heat absorption and power output.

The analysis shows good regression model agreement, with standard deviation in error between the two

models of less than half of one percent for both power output and heat absorption. A histogram plot

of the error shown in Figure 3 indicates that the majority of the simulation points produce errors very

near to zero. 0.200.150.100.05 0.00 0.05 0.10 0.15 0.20

% Heat Absorption Error in Regression Model

0

50

100

150

200

250

Frequency (of N=1000) 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0

% Power Output Error in Regression Model

0

50

100

150

200

250

Frequency (of N=1000)

=

Figure 3: Histogram plots of the % error in the regression model for Heat Absorption and Power Output.

Figure 4 presents plots of the error for each of the three independent variable combinations, where the

size of the marker is proportional to the magnitude of the modeling error. The inputs for each simulation

were selected randomly within the range handled by the model, and each simulation is plotted as a single

6
point with the unweighted points provided in the leftmost plot as a location reference. The size of the

largest markers are equal to the maximum error values in Table 2. The error for each simulation is

plotted three times once for each unique two-variable combination of the input variables Pc, Tin, and

m ̇ . The plots in Figure 4 and the values in Table 2 indicate that the heat absorption regression fares

somewhat better than the power output, though both exhibit reasonable modeling errors in light of the

information in Table 2.

-

Figure 4: Plots mapping the relative error in the regression model for each of the three unique independent variable

combinations. The leftmost plot shows the unweighted points, and the right two plots in each group show error in

heat absorption and power output predictions.

7
6	 Conclusions

This paper presents a regression methodology that is useful in characterizing detailed systems like the

Rankine power cycle. Since regression models are less computationally demanding than detailed iterative

models, they are most useful in preliminary feasibility analyses. To this end, the regression model

presented in this paper has been incorporated into NREL’s Solar Advisor Model (SAM) tool [9] for both

the Power Tower and Physical Trough model.

Though the regression model reproduces the detailed model performance map with relatively low error,

space for improvement still exists. In particular, the accuracy of the power output regression is worst

near the design-point value for each independent variable. Improvement in estimating the interaction

effects could mitigate this problem, and future work will focus on this aspect of the model.

References

[1] M. Erbes, J. Rheinl ̈ander, and S. Bergmann. Technical and economic performance of parabolic trough

solar power plants - a computational tool for plant feasibility studies. Las Vegas, Nevada, USA, March

2008. Proceedings of the SolarPACES International Symposium.

[2] P. Griffin, K. Huschka, and G. Morin. Software for design, simulation, and cost estimation of solar

thermal power and heat cycles. Berlin, Germany, 2009. Proceedings of the SolarPACES International

Symposium.

[3] A. Patnode. Simulation and performance evaluation of parabolic trough solar power plants. Master’s

thesis, University of Wisconsin, Madison, Wisconsin, USA, 2006.

[4] N. Blair, M. Mehos, and C. Christensen. Sensitivity of concentrating solar power trough performance,

cost, and financing with the Solar Advisor Model. Las Vegas, Nevada, USA, March 2008. Proceedings

of the SolarPACES International Symposium.

[5] G. Kolb and V. Hassani. Performance analysis of thermocline energy storage proposed for the 1 MW

Saguaro solar trough plant. Denver, Colorado, USA, 2006. Proceedings of the ISEC 2006 ASME

International Solar Energy Conference.

[6] Paul Gauch ́e and Wen Wei. Linearized superposition using CFD for thermal and power characterization of electronic equipment with significant thermal radiation and natural convection. Denver,

Colorado (USA), September 2002. IMAPS.

[7] M. Wagner. Simulation and predictive performance modeling of utility-scale central receiver system

power plants. Master’s thesis, University of Wisconsin, Madison, Wisconsin, USA, 2008.

[8] C.F.J. Wu and M. Hamada.	 Experiments: Planning, Analysis, and Parameter Design Optimization.

Wiley, 2000.

[9] P. Gilman. Solar Advisor Model User Manual. Technical report, National Renewable Energy Laboratory, Golden, Colorado (USA), April 2010.

8
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14. ABSTRACT (Maximum 200 Words)

The inherent variability of the solar resource presents a unique challenge for CSP systems. Incident solar irradiation 

can fluctuate widely over a short time scale, but plant performance must be assessed for long time periods. As a 

result, annual simulations with hourly (or sub-hourly) timesteps are the norm in CSP analysis. A highly detailed power 

cycle model provides accuracy but tends to suffer from prohibitively long run-times; alternatively, simplified empirical 

models can run quickly but don’t always provide enough information, accuracy, or flexibility for the modeler. The ideal 

model for feasibility-level analysis incorporates both the detail and accuracy of a first-principle model with the low 

computational load of a regression model. The work presented in this paper proposes a methodology for organizing 

and extracting information from the performance output of a detailed model, then using it to develop a flexible 

reduced-order regression model in a systematic and structured way. A similar but less generalized approach for 

characterizing power cycle performance and a reduced-order modeling methodology for CFD analysis of heat 

transfer from electronic devices have been presented. This paper builds on these publications and the nondimensional approach originally described.

15. SUB

CSP; power block; power cycle; performance; CFD

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