# PI Parameter Influence on Underfloor Heating Energy Consumption and Setpoint Tracking in nZEBs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Building

^{2}were analyzed, except that one of them (Room 6 or R6) has two 4 m

^{2}windows facing south and west, while the windows of the other (Room 5 or R5) face north and west. The floor plan of the building is shown in Figure 1 with the two test rooms highlighted with red rectangles. Previously, the test house model in IDA ICE 4.8 software [28] was calibrated against measured air temperatures in the test room R5 during temperature setback cycles with varying durations [27]. As a result, the heat losses and thermal mass of the room structures are adequately defined in the model. This model was used for the simulations in the current work. In the simulations, all of the other rooms were heated constantly with ideal heaters to the setpoint of 21 °C.

#### 2.2. Outline of the Work

#### 2.3. Input Data

- A sunny week with moderate temperature (19–25.03);
- A cold week with almost no sun (29.01–04.02).

#### 2.4. Model Fitting

_{p}is the process gain (unitless), T is the time constant, and L is the time delay, both in seconds. The model fitting was performed in Matlab using System Identification Toolbox [30].

#### 2.5. Estimating PI Parameters

_{i}were estimated, where K is the proportional factor and T

_{i}is the integration time of the integral part of the PI in its ideal form:

- Optimized using GenOpt;
- Tuned in Matlab/Simulink;
- Calculated from an applicable simple method.

^{®}/Simulink for the previously fitted simplified models (described in Section 2.4). The tuning was performed aiming for a short rise time (speed) and overshoot of no more than 5% of the desired temperature increase.

Cohen–Coon (CC): | $K=0.9\xb7\left(1+0.092\xb7\frac{\tau}{1-\tau}\text{}\right)/$ | (3) | $Ti=\frac{3.3-3\tau}{1+1.2\tau}L$ | (4) |

Skogestad IMC (SIMC): | $K=T\text{}/\text{}\left(2{K}_{p}L\right)$ | (5) | $Ti=\mathrm{min}\left(T;8L\right)$ | (6) |

AMIGO: | $K=\frac{0.15}{{K}_{p}}+\left(0.35-\frac{L\xb7T}{{\left(L+T\right)}^{2}}\right)\text{}\xb7\frac{T}{{K}_{p}L}$ | (7) | $Ti=0.35L+\frac{13L{T}^{2}}{{T}^{2}+12LT+7{L}^{2}}$ | (8) |

#### 2.6. The Evaluation Tests

- The average absolute error (AAE) of the air temperature from the setpoint;
- The heating energy consumption per square meter of the floor area.

#### 2.7. Benchmarks

_{i}= 300 s were used for the benchmark simulations. Furthermore, on–off controls with four different dead-band widths were evaluated for the comparison. A modern one with a dead-band of 0.5 °C was used, but also close to ideal versions, with dead-bands of 0.16 °C and 0.05 °C and a conservative one with a 1 °C dead-band, were used as well.

#### 2.8. PI Implementation in IDA ICE and PI Mechanics

## 3. Results

#### 3.1. Found Simplified Models

_{p}) has two clearly different orders and altogether three different levels. The values were around 1 for all cases where the PRBS signal was used as the setpoint and were much larger for other cases. For the ideal step and measured setbacks, the K

_{p}value was around 20, for all other setback cases, around 40.

_{p}and L values, the models are divided into four groups, shown in Table 2. The setbacks and longer step groups are self-evident from above. The PRBS models are divided into models with a short L (PRBS sL) and a long L (PRBS lL). These groups will be used below for visualization.

#### 3.2. Identified PI Parameters

_{i}to be longer. For the optimal cases, the combinations closer to the blue ones are optimized for the variable setpoint, the lower values for the constant setpoint.

_{i}pairs are colored by the log10 (K/T

_{i}) value. This logarithm is further used for describing the pairs, as this is a clear indicator whether the pair is in the lower right or upper left corner of the log10-log10 graph.

#### 3.3. Setpoint Temperature Tracking and PI Output Signal Behaviour

#### 3.4. Setpoint Shifting

#### 3.5. Energy Performance and Total Setpoint Tracking Accuracy

^{2}/year more energy than the PI cases for the variable setpoint. For the constant setpoint, the lowest PI results are up to 7 kWh/m

^{2}/year or 9% lower than for the on–off with a 0.5 K dead-band, which, for example, in R6 is at 81 kWh/m

^{2}/year. Omitting the extreme poorly performing cases, the total variation in energy consumption is more than 10 kWh/m

^{2}/year or 12% in the constant setpoint case.

^{2}/year or 5% higher on that edge compared with the optimal case. The parameters optimized for setpoint tracking are also close to an optimal energy consumption. The PRBS sL group performs well almost in all cases but not optimally, while in all other groups some combinations perform poorly. The optimal range of parameters is shown in detail in Table 3. Most of the optimal values were calculated using TRY climate data but the methods varied.

^{4}and a K higher than 10

^{0.5}. The error is around 0.2 K for all the simulations in R5, for R6 the error ranges from 0.25 to 0.6 K, and in extreme cases to 1 K.

## 4. Discussion

^{2}/year. The comparison of heat emitters and controllers in the European standard room shows similar results with 5% to 10% savings for the PI controlled UFH compared with the on–off control [20]. This does not compare to the 32% achieved for radiators in [23], however, the actual difference is difficult to compare as the baselines are different. The reduction of 7 kWh/m

^{2}/year here can be seen as highly significant as this can be achieved with only parameter correction, which does not require intensive computation when the simple tests are applied. Accounting for the more expensive thermostat head with variable parameters option, the payback time of this change is around 5 years. This saving can be achieved without setpoint reductions, which means no penalty on comfort. On the contrary, due to less fluctuation, comfort could even improve.

## 5. Conclusions

- For the first time in the scientific literature, it is shown that UFH can operate with determined PI parameters similar to ideal control;
- A performance close to optimal could also be achieved by parameters achieved from shorter tests, e.g., weekend pseudo-random setpoints, and 6- to 24-h setbacks which were shown to be suitable;
- The optimal PI parameters improved the room temperature control accuracy considerably, and that the results show that the UFH PI control with the correct parameters started to work in a predictive fashion and the resulting room temperature curves were practically ideal;
- The optimal PI parameters reduced the energy consumption for heating by up to 9% (7 kWh/m
^{2}/year) in comparison with the on–off control (at around 80 kWh/m^{2}/year) and by 5% in comparison with the default PI parameters; - The variation amplitude of the heating energy needed using different estimated (not random) parameters was more than 15 kWh/m
^{2}/year for the constant setpoint, which stresses the importance of having the correct PI parameters; - The optimal PI parameters included combinations with log10 (K/Ti) between −3 and −1, in these combinations, the proportional gain K ranged from 2 to 100 and the integration time Ti from 500 to 6700 s, and thus higher gain and longer integration time values than are conventionally used are recommended;
- For the variable setpoint, using the PI control had a similar effect to decreasing the dead-band and the variation in the PI parameters did not have a significant further effect on the energy consumption, except for when they were extremely poorly tuned;
- The average absolute error for the air temperatures from the setpoint was well below 0.5 K for the constant setpoints, but above for the variable setpoints.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**All obtained parameter values, which were not already shown in Table 3, sorted by log10 ratio from largest to smallest.

K | Ti (s) | Model | Method | Climate | Setpoint | Room | Total Length (Days) |
---|---|---|---|---|---|---|---|

2400 | 42 | 14 | Cohen-Coon | TRY, Jan/Feb week | PRBS | R6 | 7 |

235 | 13 | 6 | Cohen-Coon | Const | 1-h setback | equal | 0.25 |

1300 | 100 | 14 | SIMC | TRY, Jan/Feb week | PRBS | R6 | 7 |

580 | 85 | 13 | Cohen-Coon | TRY, Jan/Feb weekend | PRBS | R6 | 2 |

930 | 170 | 14 | AMIGO | TRY, Jan/Feb week | PRBS | R6 | 7 |

130 | 31 | 6 | SIMC | Const | 1-h setback | equal | 0.25 |

77 | 32 | 5 | Cohen-Coon | Const | 3-h setback | equal | 0.75 |

91 | 52 | 6 | AMIGO | Const | 1-h setback | equal | 0.25 |

320 | 210 | 13 | SIMC | TRY, Jan/Feb weekend | PRBS | R6 | 2 |

230 | 340 | 13 | AMIGO | TRY, Jan/Feb weekend | PRBS | R6 | 2 |

43 | 77 | 5 | SIMC | Const | 3-h setback | equal | 0.75 |

30 | 130 | 5 | AMIGO | Const | 3-h setback | equal | 0.75 |

150 | 940 | 10 | Cohen-Coon | TRY, Jan/Feb week | PRBS | R5 | 7 |

110 | 710 | 9 | Cohen-Coon | TRY, Jan/Feb weekend | PRBS | R5 | 2 |

1.9 | 2000 | 4 | AMIGO | Const | 6-h setback | equal | 1.5 |

2.3 | 2500 | 8 | Cohen-Coon | Actual | 3-day measured | R5 | 3 |

2 | 2300 | 7 | Cohen-Coon | Actual | 2-day measured | R5 | 2 |

1.4 | 1800 | 2 | AMIGO | Const | 24-h setback | equal | 6 |

8.7 | 11,000 | 15 | Cohen-Coon | TRY, March weekend | PRBS | R6 | 2 |

11 | 16,000 | 11 | SIMC | TRY, March weekend | PRBS | R5 | 2 |

11 | 18,000 | 12 | SIMC | TRY, March week | PRBS | R5 | 7 |

7.2 | 18,000 | 11 | AMIGO | TRY, March weekend | PRBS | R5 | 2 |

7 | 20,000 | 12 | AMIGO | TRY, March week | PRBS | R5 | 7 |

1.3 | 3800 | 1 | SIMC | Const | Ideal step | equal | 60 |

0.9 | 3300 | 3 | SIMC | Const | 12-h setback | equal | 3 |

1.1 | 6200 | 8 | SIMC | Actual | 3-day measured | R5 | 3 |

1 | 5800 | 7 | SIMC | Actual | 2-day measured | R5 | 2 |

0.9 | 5300 | 1 | AMIGO | Const | Ideal step | equal | 60 |

4.8 | 32,000 | 15 | SIMC | TRY, March weekend | PRBS | R6 | 2 |

0.6 | 4700 | 3 | AMIGO | Const | 12-h setback | equal | 3 |

2.7 | 25000 | 15 | AMIGO | TRY, March weekend | PRBS | R6 | 2 |

0.7 | 8500 | 8 | AMIGO | Actual | 3-day measured | R5 | 3 |

0.6 | 7700 | 7 | AMIGO | Actual | 2-day measured | R5 | 2 |

0.81 | 16,000 | 1 | tuned in Matlab | Const | Ideal step | equal | 60 |

0.83 | 20,000 | 7 | tuned in Matlab | Actual | 2-day measured | R5 | 2 |

0.82 | 24,000 | 8 | tuned in Matlab | Actual | 3-day measured | R5 | 3 |

0.82 | 24,000 | 2 | tuned in Matlab | Const | 24-h setback | equal | 6 |

0.41 | 32,000 | 3 | tuned in Matlab | Const | 12-h setback | equal | 3 |

0.41 | 34,000 | 5 | tuned in Matlab | Const | 3-h setback | equal | 0.75 |

0.15 | 13,000 | 16 | Cohen-Coon | TRY, March week | PRBS | R6 | 7 |

0.43 | 40,000 | 6 | tuned in Matlab | Const | 1-h setback | equal | 0.25 |

0.41 | 39,000 | 4 | tuned in Matlab | Const | 6-h setback | equal | 1.5 |

0.043 | 4700 | 16 | SIMC | TRY, March week | PRBS | R6 | 7 |

0.16 | 18,000 | 16 | AMIGO | TRY, March week | PRBS | R6 | 7 |

0.012 | 15,000 | - | genopt | TRY, March week | variable | R6 | Inf |

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**Figure 2.**PI implementation in IDA ICE and example signals. In (

**A**), variables in the script are colored after each line it is referred where the example signals are visualized. Lines in (

**B**–

**D**) graphs use the same color-coding.

**Figure 3.**Log-value of all model parameters shown in Table 2.

**Figure 4.**All PI parameter value pairs (K, Ti) on log-valued axes colored based on the method, in (

**a**) the default black circle with cross is the PI parameters pair used in IDA ICE by default, in (

**b**) the black points are the ones that would not result in acceptable temperatures for R6 without setpoint shifting.

**Figure 5.**Graph (

**a**) shows the underlying climate data and graph (

**b**) shows the log-ratio values of all the PI parameter pairs. In (

**a**), the constant climate is at 0 °C with no solar radiation, HP stands for heating period and all the dates are covered in Section 2.

**Figure 6.**Air temperatures and PI output signals for the constant setpoint case during one week in January/February (

**left**), and in a week in March (

**right**) for the chosen four pairs of parameters.

**Figure 7.**Air temperatures and PI output signals for the variable setpoint case during one week in January/February (

**left**), and in a week in March (

**right**) for the chosen four pairs of parameters.

**Figure 8.**Duration curves over the heating period for temperatures on the left and PI output signals on the right. The purple indicates the temperatures below the setpoint and the black dashed line shows the results of the IDA ICE default parameters.

**Figure 9.**(

**Left**): Zoom-in on Figure 8’s lower temperature end of the constant temperature graph of R6; (

**right**): Zoom-in on the temperature end of the same graph.

**Figure 10.**Influence of the log10 of the PI parameters ratio K/Ti on energy consumption in the 10.4 m

^{2}rooms; color-scale shows the setpoint shift; the grey values are below 0 which means that the temperature setpoints have been decreased. The horizontal lines depict the setpoint shifts and energy consumption of the on–off cases with different dead-bands.

**Figure 11.**Influence of the log10 of the PI parameters ratio K/Ti on energy consumption; colors visualize the underlying model group.

**Table 1.**Overview of the input data for the model calculation (grey area) and optimization as well as the methods for getting the proportional–integral (PI) parameters.

Climate | Setpoint | Room | Source | Estimation Basis | Method |
---|---|---|---|---|---|

Actual | 2-3-day (long) setbacks | R5 | Measured | Simplified model | Calculation methods + tuning in Matlab |

Constant | Shorter setbacks | R5/R6 (equal) | Simulated | Simplified model | Calculation methods + tuning in Matlab |

Constant | Infinite/ideal step | R5/R6 (equal) | Simulated | Simplified model | Calculation methods + tuning in Matlab |

Estonian TRY | PRBS | R5 and R6 | Simulated | Simplified model | Calculation methods |

Estonian TRY | Constant | R5 and R6 | Simulated | Optimization | GenOpt |

Estonian TRY | Variable (price-based) | R5 and R6 | Simulated | Optimization | GenOpt |

Based on Room | Model Number | Model Group | Model Source | K_{p} | L (Seconds) | T (Seconds) |
---|---|---|---|---|---|---|

R5 & R6 | 1 | Longer step | Ideal step | 21.842 | 476 | 27,892 |

2 | Setbacks | 24-h setback at 0 ℃ | 41.063 | 141.12 | 23,652 | |

3 | Setbacks | 12-h setback at 0 ℃ | 42.649 | 410.58 | 30,141 | |

4 | Setbacks | 6-h setback at 0 ℃ | 44.717 | 156.96 | 38,648 | |

5 | Setbacks | 3-h setback at 0 ℃ | 42.664 | 9.66 | 35,191 | |

6 | Setbacks | 1-h setback at 0 ℃ | 41.446 | 3.9 | 42,130 | |

R5 | 7 | Longer step | 2-day measured setback | 24.256 | 720 | 33,720 |

8 | Longer step | 3-day measured setback | 21.472 | 780 | 41,820 | |

9 | PRBS sL | 2-day PRBS in February | 1.0123 | 218.4 | 27,152 | |

10 | PRBS sL | 1-week PRBS in February | 1.03 | 286.8 | 50,122 | |

11 | PRBS lL | 2-day PRBS in March | 1.0555 | 2034 | 48,950 | |

12 | PRBS lL | 1-week PRBS in March | 1.0599 | 2226.6 | 51,845 | |

R6 | 13 | PRBS sL | 2-day PRBS in February | 1.03 | 25.8 | 17,237 |

14 | PRBS sL | 1-week PRBS in February | 1.042 | 12.6 | 34,930 | |

15 | PRBS lL | 2-day PRBS in March | 1.0973 | 3996 | 41,990 | |

16 | PRBS lL | 1-week PRBS in March | 1.1035 | 50,084 | 4737 |

**Table 3.**Optimal parameter combinations from log10 ratio from −3 (excluded) to −1. Ordered in increasing energy consumption values for the R6 constant setpoint.

K | Ti | Model | Method | Climate | Setpoint | Room | Total Length |
---|---|---|---|---|---|---|---|

18 | 2300 | - | GenOpt | TRY, Jan/Feb week | variable | R6 | Inf |

13 | 1500 | - | GenOpt | TRY, March week | constant | R5 | Inf |

28 | 2800 | - | GenOpt | TRY, March week | variable | R5 | Inf |

21 | 6200 | 11 | Cohen-Coon | TRY, March weekend | PRBS | R5 | 2 days |

20 | 6700 | 12 | Cohen-Coon | TRY, March week | PRBS | R5 | 7 days |

27 | 1500 | - | GenOpt | TRY, Jan/Feb week | constant | R5 | Inf |

16 | 820 | - | GenOpt | TRY, Jan/Feb week | constant | R6 | Inf |

32 | 1700 | - | GenOpt | TRY, Jan/Feb week | variable | R5 | Inf |

5.2 | 510 | 4 | Cohen-Coon | Const | 6-h setback | equal | 1.5 days |

3.7 | 460 | 2 | Cohen-Coon | Const | 24-h setback | equal | 6 days |

42 | 2700 | 9 | AMIGO | TRY, Jan/Feb weekend | PRBS | equal | 2 days |

27 | 650 | - | GenOpt | TRY, March week | constant | R6 | Inf |

54 | 1900 | - | GenOpt | TRY, heating period | variable | R5 | Inf |

2.8 | 1300 | 4 | SIMC | Const | 6-h setback | equal | 1.5 days |

59 | 3600 | 10 | AMIGO | TRY, Jan/Feb week | PRBS | R5 | 7 days |

61 | 1800 | 9 | SIMC | TRY, Jan/Feb weekend | PRBS | R5 | 2 days |

41 | 930 | - | GenOpt | TRY, heating period | constant | R6 | Inf |

2.4 | 1500 | 1 | Cohen-Coon | Const | Ideal step | equal | 60 days |

2.0 | 1100 | 2 | SIMC | Const | 24-h setback | equal | 6 days |

85 | 2300 | 10 | SIMC | TRY, Jan/Feb week | PRBS | R5 | 7 days |

55 | 800 | - | GenOpt | TRY, heating period | constant | R5 | Inf |

98 | 3200 | - | GenOpt | TRY, heating period | variable | R6 | Inf |

1.6 | 1300 | 3 | Cohen-Coon | Const | 12-h setback | equal | 3 days |

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## Share and Cite

**MDPI and ACS Style**

Kull, T.M.; Thalfeldt, M.; Kurnitski, J.
PI Parameter Influence on Underfloor Heating Energy Consumption and Setpoint Tracking in nZEBs. *Energies* **2020**, *13*, 2068.
https://doi.org/10.3390/en13082068

**AMA Style**

Kull TM, Thalfeldt M, Kurnitski J.
PI Parameter Influence on Underfloor Heating Energy Consumption and Setpoint Tracking in nZEBs. *Energies*. 2020; 13(8):2068.
https://doi.org/10.3390/en13082068

**Chicago/Turabian Style**

Kull, Tuule Mall, Martin Thalfeldt, and Jarek Kurnitski.
2020. "PI Parameter Influence on Underfloor Heating Energy Consumption and Setpoint Tracking in nZEBs" *Energies* 13, no. 8: 2068.
https://doi.org/10.3390/en13082068