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# Transient Uncertainty Analysis in Solar Thermal System Modeling

- Heejin Cho
^{1}Email authorView ORCID ID profile, - Aaron Smith
^{1}, - Rogelio Luck
^{1}and - Pedro J. Mago
^{1}

**5**:1

https://doi.org/10.1186/s40467-017-0055-6

© The Author(s). 2017

**Received:**21 September 2016**Accepted:**10 January 2017**Published:**19 January 2017

## Abstract

Complex, dynamic, computational models are routinely used to evaluate and optimize the design and performance of solar thermal systems. As models become more complex, performing uncertainty analysis on such models can be quite challenging and computationally expensive. This paper presents an effective approach to quantify uncertainties associated with transient simulation results from a dynamic solar thermal energy system model with uncertain parameters. The proposed method utilizes the concept of impulse response and convolution process to estimate the sensitivities to time-varying external inputs. Using this method, the number of simulations required to propagate uncertainties through dynamic models can be significantly reduced. An example is presented throughout the paper to demonstrate the procedure of the proposed uncertainty analysis approach.

## Keywords

- Solar thermal
- Dynamic model
- Uncertainty
- Sensitivity

## Introduction

A computational model of a complex energy system is often required to evaluate and optimize the design and performance of the actual system, e.g., [1–4]. When systems and their models are complex (i.e., containing large numbers of parameters and requiring extensive computational time to converge under time-varying condition), assuring the reliability and accuracy of models becomes very challenging and a methodical and efficient way to estimate uncertainty is necessary. The quantification of uncertainty is an essential feature in the verification and validation (V&V) procedures to validate simulation results against experimental measurements [5]. In addition, a long-term (e.g., a whole year) evaluation of system performance, which is often a necessary feature when the system performance depends on weather conditions or varying operational circumstances, makes uncertainty analysis even more difficult.

A variety of computational models have been developed to evaluate and optimize the design and performance of solar thermal systems [6–12]. Those models have been implemented in many engineering software tools such as TRNSYS [13], EnergyPlus [14], and Modelica [15]. While many studies have been done in this area, relatively few have considered the effects of uncertainty on the reliability of the results and conclusions. Xu et al. [11] presented a TRNSYS based optimization study of a solar thermal system with consideration of uncertainty. The Monte-Carlo method was used to analyze the uncertainty in the system. However, the study only considered a very limited number of uncertain parameters. Additionally, the simulation included dynamic elements, but since the study only considered cumulative effects the model was simplified to a regression that eliminated the dynamics. Dominguez-Munoz et al. [12] also presented an uncertainty analysis of the design of a solar thermal system that was based on a dynamic model. The study considered many uncertain parameters and inputs using the Monte-Carlo method for uncertainty propagation. A powerful method for design optimization under uncertainty was presented. However, this study only evaluated cumulative effects of the uncertainty over long periods of time rather than presenting the propagation of uncertainty for each time step.

This paper presents an approach to quantify uncertainties associated with transient simulation results from a dynamic solar thermal energy system model with uncertain input parameters. The uncertainty in the simulation result is composed of contributions from the errors due to modeling assumptions and approximations, numerical solution of the equations, and simulation inputs [5]. This study primarily focuses on determining uncertainties due to simulation inputs including model parameters, initial conditions, and transient external inputs. The sensitivity (i.e., partial derivative) to each model parameter and initial condition at each time step can be determined by perturbing each of the arguments at a nominal value. The sensitivity to the time-varying external inputs can be determined in a similar manner by calculating sensitivities at each time step. However, this numerical procedure can be greatly simplified using the principle of linearity and superposition. The proposed method utilizes the impulse response and the convolution process to estimate the sensitivities to time-varying external inputs. Finally, the total uncertainties on the final result due to the simulation input parameters are estimated based on the sensitivities and systematic/random uncertainties [5].

## Model Description

### Description of the System

### Energy Conservation of the Storage Tank

*ρ*and

*c*

_{p}are the density and specific heat of water,

*V*is the storage volume of the tank,

*T*

_{st}is the average temperature of the water in the tank,

*q*

_{c}is the energy collected in the solar collector,

*q*

_{LD}is the energy used to meet the building hot water load, and

*q*

_{LS}is the energy lost through the tank walls to the ambient environment. The energy transferred to the water in the solar collector can be determined by a heat transfer balance on the collector between energy absorbed from solar radiation and energy lost by convection to the ambient environment. The effective energy transferred to the water has been defined by Duffie and Beckman [10] as

where *A*
_{c} is the collector area, *F*
_{R} is the heat removal factor, *I*
_{T} is the total solar radiation incident on the collector surface, τα is the transmittance-absorptance product for the collector glazing, and *U*
_{c} is the loss coefficient for the collector. The variable Φ_{Va} is a Heaviside step function that represents the opening and closing of the valve in energy collection loop (i.e., location (a) in Fig. 1) to maximize energy collection. This step function is equal to one (i.e., the valve at (a) is open) when the heat transfer to the water in the collector is positive and equal to zero (i.e., the valve at (a) is closed) otherwise. The incident radiation on the collector surface can be determined from standard radiation measurements such as the diffuse and direct radiation on the horizontal. However, the relationship between these standard measurements and the radiation incident on the collector surface varies with the position of the sun in the sky. Therefore, the (“Definition of Radiation Incident on Collector Surface” section) gives the equations for angles of the sun as a function of time and location.

where *U*
_{st} is the loss coefficient for the storage tank, *A*
_{st} is the exposed surface area of the storage tank, and *T*
_{amb} is the ambient temperature.

^{1}A typical load for a day is illustrated in Fig. 2.

### Definition of Radiation Incident on Collector Surface

*I*

_{s,beam}is the component on the collector surface due to beam radiation,

*I*

_{s,diff}is the component due to diffuse radiation, and

*I*

_{s,GR}is the component due to ground reflected radiation. Radiation measurements are typically reported as beam and diffuse radiation on a horizontal surface. Therefore, these components must be adjusted for the slope of the collector surface with respect to horizontal and the position of the sun. The diffuse radiation on a surface tilted from the horizontal at angle,

*β*, is defined as

*I*

_{d}is the diffuse irradiation on a horizontal surface. The beam radiation on the tilted surface can be defined as

*I*

_{b}is the beam radiation on a horizontal surface and

*θ*is the angle of incidence (i.e., the angle between the direct sun beam and the normal to the collector surface). The radiation on the collector surface from ground reflected radiation can be defined as

*ρ*

_{g}is the reflectivity of the ground and

*θ*

_{z}is the solar zenith angle (i.e., the incidence angle for a horizontal surface). The solar zenith angle is illustrated in Fig. 3a. For this simulation, the ground reflectance is assumed to be 0.2, which is the commonly used value in the building energy simulations [16].

*γ*

_{ s }is the solar azimuth angle (i.e., the angle between south and the projection of the beam onto the horizontal, see Fig. 3b) and

*γ*is the tilted surface azimuth angle. The solar zenith angle and the solar azimuth angle can be found using Eqs. (9) and (10), respectively, as

*ϕ*is the latitude of the site,

*δ*

_{ s }is the solar declination, and

*ω*

_{ s }is the hour angle. The function sign returns the sign of the argument. Therefore, if

*ω*

_{ s }is positive, sign(

*ω*

_{ s }) = 1; otherwise, sign(

*ω*

_{ s }) = − 1. The solar declination refers to the angle of the sun relative to the equatorial plane of the earth. The solar declination for a given hour in the year can be obtained as

*h*

_{solar}corresponds to the hour in the year in terms of solar time. Solar time only corresponds to local time if the site is located along the standard meridian for the local time zone. Otherwise, the time must be corrected proportional to the amount the location varies from the standard meridian. The time correction can be defined as

*h*

_{local}is the hour in local time,

*h*

_{solar}is the hour in solar time,

*L*

_{local}is the longitude at the site location,

*L*

_{st}is the longitude of the standard meridian for the time zone, and

*E*is a parameter of time correction that can be determined from the following empirical relations [17]:

## Uncertainty Analysis

### Nominal Tank Solution

In this section, an example case of a flat plate solar thermal system located in San Diego, CA, USA, is used to illustrate the uncertainty analysis process for a day long simulation. Equation (1) is solved for the storage temperature in the tank by using a standard Runge-Kutta numerical solver. The storage tank temperature is calculated hourly. The storage tank temperature could be calculated for a variety of design conditions to determine if the design meets the requirements or could be implemented in an algorithm as part of an effort to optimize the operation method under uncertainty.

*k*, can be defined as a function of the parameters and inputs to the system.

*P*variables represent the

*N*parameters of the system. The parameters for the model developed above are listed in Table 1. The other variables in Eq. (15) are the inputs to the system at each time step,

*k*. The inputs are the hourly building load,

*q*

_{LD}, the diffuse radiation on the horizontal,

*I*

_{d}, the beam radiation on the horizontal,

*I*

_{b}, and ambient temperature,

*T*

_{amb}. An example of the storage tank temperature for a typical day is given in Fig. 4.

Parameters used to determine the storage temperature in San Diego, CA, USA

Parameter | Nominal value | Estimated uncertainty |
---|---|---|

Local longitude ( | 117.16 (deg) | 0 |

Location latitude ( | 32.733 (deg) | 10% |

Longitude of the standard meridian for the time zone ( | 120 (deg) | 0 |

Collector area ( | 5 (m | 1% |

Heat removal factor times the transmittance-absorptance product ( | 0.753 | 2% |

Heat removal factor times the collector loss coefficient ( | 3.79 | 2% |

Reflectivity of the ground ( | 0.2 | 0.1 |

Collector slope ( | 32.733 (deg) | 2° |

Collector azimuth angle ( | 0 (deg) | 2° |

Loss coefficient for storage tank (UA | 1.7 | 1% |

Ambient temperature ( | 25 ( °C) | 1 °C |

Specific heat of water ( | 4.1813 (kJ/kg K) | 0 |

Density of water ( | 974 (kg/m | 0.1 |

Storage tank volume ( | 0.19 (m | 0 |

*q*

_{c}, for the nominal conditions. The figure shows that positive heat transfer occurs from hours 9 through 17. Therefore, the valve will be opened at 9:00 AM and closed at 6:00 PM. At this point of the development of the uncertainty analysis procedure, it is important to point out that the valve system affects the linearity and time invariance of the system with respect to the time varying inputs.

### Sensitivities of the Tank Temperature

*k*, to the

*i*th parameter,

*P*

_{i}, can be defined as

*T*

_{st_N}represents the nominal solution which is found by simulating the system with all parameters and inputs at their nominal values. Notice that when finding the sensitivity of the solution to a given parameter,

*P*

_{i}, only this parameter is perturbed while all other parameters and all inputs are held constant. For example, the sensitivity of the solution at time,

*k*, to the collector area can be defined as

*k*, must be calculated for the entire history of the time varying inputs as shown in the following equations. Note: The index

*j*is used to indicate the time that the input perturbation occurred.

However, this approach would require a large number of numerical simulations. For instance, for a given time step *k*, the sensitivity must be determined for the current input as well as for the entire history of inputs. This would require *k* additional simulations for each time varying input. In this model, there are four time varying inputs. Therefore, for a single day simulation (i.e., 24 time steps), it would take 96 simulations to calculate the sensitivities for the inputs alone. This would become especially cumbersome if there was a need for multiple day simulations. An alternative approach is taken in this work.

Assuming a linear time invariant system, the principle of superposition can be used to greatly reduce the number of simulations required. In this case, discrete convolution (i.e., a discretized version of Duhamel’s integral) can be used to determine the response of the storage temperature to small changes in the input loads. Then, the impulse response can be found for each load. This impulse response can simply be multiplied by the load perturbation magnitude and shifted to the time of the load to determine the response to all perturbations. It follows that only one additional simulation will be required for each time varying input. That eliminates 92 simulations. In general using the principle of convolution reduces the number of simulations by *I* * *N* − *I*, where *N* is the number of time steps, and *I* represents the number of inputs (i.e., four in this case). For a 1-week simulation, this method would eliminate the need for 668 numerical ODE simulations. For a 1-year simulation, this method would eliminate 35,036 simulations, requiring only four simulations for calculating the sensitivities to the inputs.

The previous discussion assumes that the model is linear and time-invariant. However, the model developed in this work has fairly strong nonlinearities due to changes in the valve states. To address this issue, the simulation was split into three zones: before collector valve is open, during collector operation, after collector valve is closed. For the ambient temperature and hot water load, a simulation is required for each of the three zones, while the diffuse and beam radiation only affect the solution during the time when the collector is being used.

At zone transitions, the uncertainties from the previous zone are interpreted as an uncertainty in the initial temperature for the next zone. This requires additional simulations for each zone transition and for each time-varying input that is effective leading up to the zone transition. This leads to six additional simulations for a single day simulation.

*j*= 9 can be determined as

*I*

_{b}.

### Total Uncertainty of the Tank Temperature

*P*

_{i}, the list of parameters including the respective uncertainties is given in Table 1. In general, the uncertainties are estimated based on many factors such as expected measurement errors in experimental variables (obtained from instrument manufacturers), approximation errors in model parameters, conceptual errors in model equations, and engineering judgment. Additional information for estimating uncertainties can be found in references [18–20]. The nominal values of model parameters used in the present uncertainty analysis and their estimated uncertainties are listed in Table 1. Nominal values in this table refer to the model parameter values assuming zero uncertainty. The total uncertainties for the hourly building load,

*q*

_{LD}, the diffuse radiation on the horizontal,

*I*

_{d}, the beam radiation on the horizontal,

*I*

_{b}, and ambient temperature,

*T*

_{a}, can be determined using their sensitivities at time

*k*and estimated uncertainties as defined below

### Uncertainties for External Input Variables

*ρ*) is estimated to be 0.1 because many ground surfaces have the ground reflectance values between 0.1 and 0.3 based on their material [8]. The estimated uncertainties for the input variables are listed in Table 2.

Estimated uncertainties for input variables

Input variable | Source | Standard random uncertainty | Standard systematic uncertainty |
---|---|---|---|

| Hourly values given in TMY3 | 1% of maximum daily | 10% |

| Hourly values given in TMY3 | 1% of maximum daily | 16% |

| Typical uncertainty in temperature sensors | 1 °C | 1 °C |

| Estimation (engineering judgment) | 5 W | 5 W |

### Results of Uncertainty Propagation

## Conclusion

An approach was presented to quantify uncertainties associated with transient simulation results from a dynamic solar thermal energy system model with uncertain input parameters. The approach greatly reduced the number of simulations required by using the impulse response and convolution integral to estimate the sensitivities to time-varying external inputs. The results from the selected example indicated that the uncertainty in the time-varying temperature of the storage tank can vary as much as ±2.2 °C. This method can be helpful for validating models for system design and potentially for developing operation algorithms that take time varying uncertainties into account.

## Declarations

### Authors’ Contributions

All authors have made substantial contributions to the conception, analysis, and interpretation of the data and have been involved in drafting the manuscript and revising it critically for important intellectual content. All authors read and approved the final manuscript.

### Competing Interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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