# Extended-Window Algorithms for Model Prediction Applied to Hybrid Power Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{TM}to build a hybrid model that enables us to obtain system responses without extensive experimentation. We also introduce model prediction that can improve system performance based on forecast system responses. Finally, we propose the extended-window algorithms, which can update the forecast models and further improve the system performance based on accumulated data.

#### 2.1. Hybrid Power System and Model

^{TM}to develop the hybrid power model shown in Figure 2. The model enables us to estimate system responses under different operational conditions (e.g., varying components and PMS parameters) without extensive experimentation. The simulation model consisted of six modules: PVs, load, battery, PEMFC, chemical hydrogen generation, and power management with a prediction model [24].

^{2}. Solar energy is the primary energy source in this hybrid power system. The load module calculates the load current ${I}_{Load}$ as follows:

^{−1}, with PEMFC currents of 20 A, 30 A, 40 A, and 50 A, respectively [25]. The chemical hydrogen generation module provides hydrogen to the PEMFC when hydrogen is insufficient [5]. The sodium borohydride (NaBH

_{4}) solution is batched to generate 150 L of hydrogen each time, using 400 mL of 15wt% NaBH

_{4}solution [14]. The power management module adjusts the PEMFC output current ${I}_{PEMFC}$ according to the battery SOC with the following PMS [14]:

- (a)
- When the battery SOC drops to $SO{C}_{low}$, the prediction model is activated to forecast the solar radiation and load demand to calculate the battery SOC in the next 24 h.
- (b)
- If the battery SOC in the next 24 h is higher than 20%, the PEMFC remains silent and returns to Step (a). Otherwise, the PEMFC is activated to provide a supplementary power supply with the following current:$$SO{C}_{\mathrm{min}}(T)+\frac{{\displaystyle {\int}_{0}^{{}_{T}}({\overline{I}}_{solar}(t)+{I}_{PEMFC}^{req}-{\overline{I}}_{Load}(t))dt}}{C}=20\%,$$
- (c)
- When the battery SOC reaches $SO{C}_{high}$, the PEMFC is turned off and returns to Step (a). Otherwise, it returns to Step (b).

#### 2.2. System Costs and Reliability

_{4}.

_{4}as ${\mathbb{Z}}_{o(b,s)}^{{\mathrm{NaBH}}_{4}}={C}_{o}^{{\mathrm{NaBH}}_{4}}\times {n}_{{\mathrm{NaBH}}_{4}}$, where ${C}_{o}^{{\mathrm{NaBH}}_{4}}$ is the unit cost for each batch that consumes 400 mL of 15wt% of NaBH

_{4}solution, and ${n}_{{\mathrm{NaBH}}_{4}}$ is the number of batches. The component costs and lifespans are shown in Table 1. Finally, battery, PEMFC, and hydrogen costs depend on the system responses.

#### 2.3. Model Prediction for the Hybrid Power System

^{2}and 1083.33 W/m

^{2}, and an average daily availability of 2.31 kWh and 2.39 kWh in the first and second years, respectively. Figure 3b shows the load profiles, with peaks of 6.83 kW and 6.32 kW, and average daily consumptions of 15.91 kWh and 18.41 kWh in the first and second years, respectively.

_{low}, SOC

_{high}) = (40%, 50%). We then applied these optimal settings to estimate system responses in 2015 without model prediction, where the system cost is USD 0.7889/kWh and the hydrogen consumption is 468,007.2 L, for a system reliability of LPSP = 0%. Second, Case-A applied the same optimal setting to estimate system responses in 2015 with XGBoost prediction models [14]. The system cost was reduced to USD 0.7389/kWh, and the hydrogen consumption decreased to 311,683.9 L, while maintaining system reliability LPSP = 0%. The results showed that model prediction can significantly reduce system costs and hydrogen consumption by 6.45% and 33.40%, respectively.

#### 2.4. Extended-Window Algorithms for Model Prediction

- (a)
- Set i = 1.
- (b)
- Use the radiation and load data in $T=[0,{T}_{i-1}]$ to train the prediction models.
- (c)
- Apply the prediction models obtained in step (b) to forecast the solar current and load responses in the next interval $T=[{T}_{i-1},{T}_{i}]$, labeled as ${\overline{I}}_{solar}(t)\mathrm{and}{\overline{I}}_{Load}(t)$ for $t\in [{T}_{i-1},{T}_{i}]$.
- (d)
- Use ${\overline{I}}_{solar}(t)\mathrm{and}{\overline{I}}_{Load}(t)$ to optimize system components and PMS for the interval $t\in [{T}_{i-1},{T}_{i}]$, denoted as ${\left.(b,s)\right|}_{[{T}_{i-1},{T}_{i}]}$ and ${\left.(SO{C}_{low},SO{C}_{high})\right|}_{{[\mathrm{T}}_{i-1},{T}_{i}]}$.
- (e)
- Apply the optimal settings in step (d) to the interval $t\in [{T}_{i-1},{T}_{i}]$. Calculate the system costs and hydrogen consumption in this interval.
- (f)
- Complete the design when i = n. Otherwise, increase i by one and return to step (b).

## 3. Results

#### 3.1. Optimal Prediction Models

#### 3.2. Impacts of Window Sizes and Replacement Costs

## 4. Discussion

^{TM}to develop a hybrid power model that enables the estimation of system responses without extensive experimentation. We then applied five machine learning methods to develop prediction models for our hybrid power system. The results showed that the LightGBM and XGBoost models could forecast solar radiation and load profiles with a higher than 97% accuracy. Therefore, we integrated these two models into the hybrid power system to investigate the impacts of extended-window model prediction on system performance.

_{low}, SOC

_{high}) = (40%, 50%). With no extended-window model prediction, the optimal settings remained the same in Period II. With extended-window model prediction, the optimal settings became (b, s) = (10, 7) and (SOC

_{low}, SOC

_{high}) = (40%, 50%) in period II. The practical implementation consists of a PEMFC and a loadmeter. When the system needed supplementary power, the simulation model sent current commands ${I}_{PEMFC}$ to the PEMFC, which was then physically activated to provide the required current. The original period was ten days; we applied a scale factor of 1/600 to shorten the experimental time to 24 min, with the initial SOC = 30%. We measured system signals, such as the currents and voltages, to detect signs of potentially declining efficiency.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{TM}for their collaboration and technical support.

## Conflicts of Interest

## Appendix A

- (1)
- eXtreme Gradient Boosting (XGBoost)

- (2)
- Light Gradient Boosting Machine (LightGBM)

- (3)
- CatBoost

- (4)
- K-Nearest Neighbors (KNN)

- (5)
- Random Forest (RF)

## Appendix B

Period | (b, s) | (SOC_{low}, SOC_{high}) | System Cost (USD) | Hydrogen Consumption (L) |
---|---|---|---|---|

1st | (14, 9) | (25%, 30%) | 74.17 | 0 |

2nd | (14, 12) | (30%, 35%) | 88.31 | 862.4 |

3rd | (18, 9) | (25%, 30%) | 80.78 | 0 |

4th | (11, 9) | (25%, 30%) | 71.87 | 0 |

5th | (13, 12) | (25%, 30%) | 88.79 | 1479.2 |

6th | (16, 23) | (40%, 45%) | 191.2 | 21,628.4 |

7th | (7, 5) | (25%, 30%) | 59.49 | 0 |

8th | (8, 9) | (25%, 30%) | 68.4 | 0 |

9th | (11, 8) | (25%, 30%) | 68.41 | 0 |

10th | (11, 14) | (40%, 45%) | 92.55 | 0 |

11th | (11, 5) | (30%, 35%) | 59.83 | 123.2 |

12th | (13, 8) | (25%, 30%) | 71.4 | 0 |

13th | (20, 6) | (30%, 35%) | 72.48 | 123.2 |

14th | (12, 3) | (25%, 30%) | 51.77 | 0 |

15th | (26, 6) | (35%, 40%) | 81.88 | 369.6 |

16th | (7, 6) | (25%, 30%) | 57.59 | 0 |

17th | (7, 5) | (35%, 40%) | 52.6 | 0 |

18th | (13, 6) | (25%, 30%) | 63.12 | 0 |

19th | (30, 16) | (25%, 30%) | 127.7 | 0 |

20th | (14, 6) | (25%, 30%) | 69.17 | 0 |

21th | (17, 26) | (25%, 30%) | 149.5 | 0 |

22th | (17, 9) | (35%, 40%) | 84.15 | 0 |

23th | (12, 7) | (45%, 50%) | 68.36 | 0 |

24th | (20, 8) | (25%, 30%) | 80.25 | 0 |

25th | (19, 11) | (25%, 30%) | 89.74 | 0 |

26th | (16, 8) | (35%, 40%) | 76.25 | 0 |

27th | (16, 8) | (45%, 50%) | 75.1 | 0 |

28th | (26, 11) | (25%, 30%) | 99.8 | 0 |

29th | (28, 12) | (25%, 30%) | 103.2 | 0 |

30th | (16, 9) | (25%, 30%) | 79.67 | 0 |

31th | (19, 9) | (45%, 50%) | 81.55 | 0 |

32th | (21, 8) | (25%, 30%) | 80.48 | 0 |

33th | (17, 12) | (25%, 30%) | 91.15 | 0 |

34th | (18, 9) | (25%, 30%) | 80.29 | 0 |

35th | (27, 24) | (40%, 45%) | 152.7 | 0 |

36th | (18, 11) | (25%, 30%) | 93.36 | 0 |

37th | (17, 8) | (45%, 50%) | 76.76 | 0 |

38th | (16, 6) | (25%, 30%) | 66.32 | 0 |

39th | (19, 11) | (25%, 30%) | 90.78 | 0 |

40th | (10, 4) | (25%, 30%) | 56.18 | 0 |

41th | (13, 6) | (25%, 30%) | 62.98 | 0 |

42th | (10, 5) | (25%, 30%) | 56.02 | 0 |

43th | (11, 6) | (25%, 30%) | 61.49 | 0 |

44th | (12, 11) | (25%, 30%) | 81.88 | 0 |

45th | (6, 5) | (25%, 30%) | 53.99 | 0 |

46th | (12, 6) | (25%, 30%) | 61.9 | 0 |

47th | (12, 9) | (40%, 45%) | 73.36 | 739.2 |

48th | (11, 8) | (25%, 30%) | 68.08 | 0 |

49th | (12, 14) | (25%, 30%) | 95.05 | 0 |

50th | (11, 9) | (25%, 30%) | 72.43 | 0 |

51th | (14, 11) | (25%, 30%) | 83.61 | 0 |

52th | (12, 23) | (25%, 30%) | 147.8 | 0 |

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**Figure 3.**Solar and load data in two years. (

**a**) Solar responses. (

**b**) Load profiles. (

**c**) Average monthly solar responses. (

**d**) Average monthly load profiles.

**Figure 6.**Combined effects of interval length M and replacement cost ratio $r\%$. (

**a**) System costs. (

**b**) Hydrogen consumption.

**Figure 9.**The energy module and experiments. (

**a**) Module architecture. (

**b**) ANCHI connector [29]. (

**c**) Current responses. (

**d**) Main battery SOC.

Component | Life | Price (US$) |
---|---|---|

PV cells (1 kW) | 15 years | 1833 |

Chemical hydrogen generator | 15 years | 10,666 |

Power electronic devices | 15 years | 1666 |

PEMFC (3 kW) | 8000 h | 6000 |

NaBH_{4} (60 g/Batch) | -- | 0.33 |

Lead–acid battery (48 V–100 Ah) | -- | 866 |

Case a | Case A | |
---|---|---|

Without Model Prediction | With Model Prediction | |

Battery (Ah) | 1300 | 1300 |

PV (kW) | 11 | 11 |

(SOC_{low}, SOC_{high}) (%) | (40, 50) | (40, 50) |

Unit cost (US$/kWh) | 0.7863 | 0.7353 |

Final SOC (%) | 43.87 | 40.22 |

Modified unit cost (US$/kWh) | 0.7889 | 0.7380 |

H_{2} (L) | 468,007.20 | 311,683.90 |

Model | NRMSE (%) | |
---|---|---|

Solar Radiation | Load Profile | |

XGBoost | 19.55 | 2.51 |

LightGBM | 2.75 | 9.94 |

Catboost | 7.19 | 16.9 |

KNN | 33.71 | 74.68 |

RF | 88.95 | 63.29 |

Case | Annual System Cost | Hydrogen Consumption | ||
---|---|---|---|---|

Cost (USD) | Reduction (%) | Consumption (L) | Reduction (%) | |

Case A | 4958.35 | -- | 311,683.9 | -- |

${\mathrm{Case}}_{6m}^{1\%}$ | 4954.17 | 0.08 | 251,403.8 | 19.34 |

${\mathrm{Case}}_{4m}^{1\%}$ | 4871.59 | 1.75 | 263,216.5 | 15.55 |

${\mathrm{Case}}_{3m}^{1\%}$ | 4863.26 | 1.92 | 198,582.3 | 36.29 |

${\mathrm{Case}}_{2m}^{1\%}$ | 4751.90 | 4.16 | 182,649.1 | 41.40 |

${\mathrm{Case}}_{1m}^{1\%}$ | 4599.35 | 7.24 | 108,248.7 | 65.27 |

${\mathrm{Case}}_{3w}^{1\%}$ | 4368.13 | 11.90 | 33,454.8 | 89.27 |

${\mathrm{Case}}_{2w}^{1\%}$ | 4382.65 | 11.61 | 42,319.6 | 86.42 |

${\mathrm{Case}}_{1w}^{1\%}$ | 4285.69 | 13.57 | 26,064.4 | 91.64 |

${\mathrm{Case}}_{5d}^{1\%}$ | 4381.26 | 11.64 | 50,538.0 | 83.79 |

Case | System Cost | Hydrogen Consumption | ||
---|---|---|---|---|

Cost (USD) | Reduction (%) | Consumption (L) | Reduction (%) | |

Case A | 4958.35 | -- | 311,683.9 | -- |

${\mathrm{Case}}_{1m}^{1\%}$ | 4599.35 | 7.24 | 108,248.7 | 65.27 |

${\mathrm{Case}}_{1m}^{2\%}$ | 4703.60 | 5.14 | 130,013.6 | 58.29 |

${\mathrm{Case}}_{1m}^{3\%}$ | 4764.19 | 3.92 | 131,100.5 | 57.94 |

${\mathrm{Case}}_{1m}^{4\%}$ | 4801.02 | 3.17 | 145,930.3 | 53.18 |

${\mathrm{Case}}_{1m}^{5\%}$ | 4866.32 | 1.86 | 145,930.3 | 53.18 |

**Table 6.**Statistical data of Figure 8.

Period I | ||

Optimal settings | $\begin{array}{l}(b,s)=(8,5)\\ (SO{C}_{low},SO{C}_{high})=(40\%,50\%)\end{array}$ | |

Results | $\begin{array}{l}\mathrm{system}\mathrm{cost}=\mathrm{US}\$37.66\\ \mathrm{Final}\mathrm{SOC}=40.98\%\\ \Delta \mathrm{SOC}=10.98\%\\ {\mathrm{H}}_{2}\mathrm{consumption}=0\end{array}$ | |

Period II | ||

without extended-window | with extended-window | |

Optimal settings | $\begin{array}{l}(b,s)=(8,5)\\ (SO{C}_{low},SO{C}_{high})=(40\%,50\%)\end{array}$ | $\begin{array}{l}(b,s)=(10,7)\\ (SO{C}_{low},SO{C}_{high})=(40\%,50\%)\end{array}$ |

Results | $\begin{array}{l}\mathrm{system}\mathrm{cost}=\mathrm{US}\$81.06\\ \mathrm{Final}\mathrm{SOC}=41.04\%\\ \Delta \mathrm{SOC}=0.06\%\\ {\mathrm{H}}_{2}\mathrm{consumption}=17740.8\mathrm{L}\end{array}$ | $\begin{array}{l}\mathrm{system}\mathrm{cost}=\mathrm{US}\$54.04\\ \mathrm{Final}\mathrm{SOC}=60.22\%\\ \Delta \mathrm{SOC}=19.24\%\\ {\mathrm{H}}_{2}\mathrm{consumption}=3942.4\mathrm{L}\end{array}$ |

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**MDPI and ACS Style**

Wang, F.-C.; Huang, H.-T.
Extended-Window Algorithms for Model Prediction Applied to Hybrid Power Systems. *Technologies* **2024**, *12*, 6.
https://doi.org/10.3390/technologies12010006

**AMA Style**

Wang F-C, Huang H-T.
Extended-Window Algorithms for Model Prediction Applied to Hybrid Power Systems. *Technologies*. 2024; 12(1):6.
https://doi.org/10.3390/technologies12010006

**Chicago/Turabian Style**

Wang, Fu-Cheng, and Hsiao-Tzu Huang.
2024. "Extended-Window Algorithms for Model Prediction Applied to Hybrid Power Systems" *Technologies* 12, no. 1: 6.
https://doi.org/10.3390/technologies12010006