# Fault Coverage-Aware Metrics for Evaluating the Reliability Factor of Solar Tracking Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Weibull Distribution Model for Calculating the Reliability of Dual-Axis Solar Tracking Systems

#### 3.1. Failure Frequency (Hazard Rate)

_{i}the corresponding time failures for each of the N products (industrial components), and the Mean Time to Failure (MTF) represents the average time to failure. The reliability equation will be written as in (3) [20]:

_{1}from the initial MTF relation and by dividing the result by the average time to failure, the expression (9) will be obtained as follows:

_{Di}denotes failures or maintenance times, N

_{F}the total number of defects, and T, the total functioning time of N products. Accordantly, the operating time of the service products can be expressed as in the mathematical rule (11):

_{F}and MDT having the same meaning as in Equation (10). Consequently, at this point the Mean Time Between Failures (MTBF) formula can be defined, as in Equation (12) [21]:

#### 3.2. Availability (Efficiency)

_{F}= 40. The total time is calculated for N

_{T}= 30 $\cdot $90 $\cdot $24 = 64800 h = 2700 days. Accordingly, we can compute the MDT parameter with the relation (19):

#### 3.3. Typical Variation Form of the Hazard Rate (Bathtub Curve)

_{0}is the initial moment, and exp represents the exponential form of Euler’s constant named value.

#### 3.4. Applications

**we**want to determine the reliability, respectively the probability of failure for 1, 2, and 3 years. This problem can be solved by just relying on the equation set (30) as follows:

_{0}= 0, η = 625 days and β = 1.5. It is considered that for the given input parameters, a test was performed on 1000 random components that were mass-produced to construct solar trackers. The goal here is to determine the survival (operating) probability of these components after t = 10

^{4}h, by applying the Weibull equation, as presented in (31):

_{0}= 1000 components, moment t

_{1}will correspond to the subtraction of 100 components from the initial stack, in other words, t

_{1}= 1000 – 100 = 900 remaining components. The associated reliability R(t

_{0}) = 1 (as a general rule) and R(t

_{1}) = 0.9. The expression of R(t

_{1}) is expanded according to Formula (32):

_{0}= 0 will be computed as in (33):

## 4. Proposed Fault Coverage-Aware Model for Calculating the Solar Reliability Factor

#### 4.1. Fault-Coverage Aware Metric for Determining the Reliability of Electronic Components with the Help of BIST Devices

_{V}represents the number of executed test vectors, N

_{E}the number of errors per test case, T

_{P}the total number of test patterns, and D the number of identical devices used for detecting errors (which in our case are given by the FF units). Since the generalized STF formula can be adapted to various test scenarios, several STF properties will be listed as follows:

- (1)
- The number of errors N
_{E}is always equal, lower, or greater than T_{V}in any test scenario regardless of the noise factors that are considered, and can be expressed mathematically as in (35):$${N}_{E}\ge {T}_{V}\_\_{N}_{E}\le {T}_{V}\_\_{N}_{E},{T}_{V}\in \mathbb{N}$$ - (2)
- The number of test vectors T
_{V}is always equal to or lower than the total number of test patterns T_{P}given by the expression (36):$${T}_{V}\le {T}_{P}\_\_{T}_{V},{T}_{P}\in \mathbb{N}$$ - (3)
- The total number of test patterns is calculated with the Equation (37):$${T}_{P}={2}^{D}-1$$
- (4)
- If no errors are detected during the testing routines, the STF parameter will be 0 resulting in 100% reliability and availability of the tested equipment, expressed in Equation (38):$$STF=0\leftrightarrow R=A=1$$

_{V}= 7 (test cases) the test system has successfully identified N

_{E}= 10 bit-flip errors. It should be noted, according to property (1) that the number of errors detected may be greater than the number of test cases because multiple errors (burst errors) may occur within a single test vector. Additionally, for (b) a number D = 4 FFs are used in the structure of the MISR in this paper. According to property (3), the total number of test cases T

_{P}is calculated based on the Formula (39):

_{V}= 7 and T

_{P}= 10 property (2) is also satisfied. Finally, by having this hypothetical data the variables from Equation (34) can be adequately substituted as in (40):

_{E}will be reduced to the value 0. The corresponding equation will be expressed as in (41):

#### 4.2. Fault-Coverage Aware Metric for Determining the Reliability of Electronic Components with the Help of WBST Methods

_{E}represents the number of errors per test vector, T

_{V}denotes the number of considered test vectors, T

_{P}provides the total number of test patterns and B designates the number of breakpoints/software functions that are implemented across the algorithm debugging stage.

- (1)
- The number of errors N
_{E}is always equal, lower, or greater than T_{V}in any test scenario regardless of the number of considered breakpoints, and can be expressed mathematically as in statement (47):$${N}_{E}\ge {T}_{V}\_\_{N}_{E}\le {T}_{V}\_\_{N}_{E},{T}_{V}\in \mathbb{N}$$ - (2)
- The number of test vectors T
_{V}is always equal to or lower than the total number of test patterns T_{P}given by the expression (48):$${T}_{V}\le {T}_{P}\_\_{T}_{V},{T}_{P}\in \mathbb{N}$$ - (3)
- If no errors are detected during the testing routines the STF parameter will be 0 resulting in 100% reliability and availability of the tested equipment, expressed as in Equation (49):$$STF=0\leftrightarrow R=A=1$$

_{P}and the number of breakpoints B. At this point, the validity of the remaining three properties can be demonstrated in a real-life scenario. Regarding the STF, the following considerations are made: (a) for a total number T

_{V}= 7 (test cases), the test system has successfully identified N

_{E}= 10 calculation errors. Additionally, for (b) a number of T

_{P}= 10 test patterns and a number B = 10 breakpoints in the software code, meaning that the test system successfully identified all calculation errors with the deployed software functions. Because properties (1) and (2) are already satisfied from the above statement, let the STF parameter will be calculated as in relation (50):

#### 4.3. Fault-Coverage Aware Metric for Determining the Reliability of Electronic Components with the Help of ICT Methods

_{E}represents the number of errors per test round, T

_{R}denotes the number of considered test rounds, N

_{R}provides the total number of test routines and P designates the number of probes (nails) that are equipped to the FPICT device.

- (1)
- The number of errors N
_{E}is always equal, lower, or greater than T_{R}in any test scenario regardless of the number of probes that are used during testing, and can be expressed mathematically as in expression (54):$${N}_{E}\ge {T}_{R}\_\_{N}_{E}\le {T}_{R}\_\_{N}_{E},{T}_{R}\in \mathbb{N}$$ - (2)
- The number of test rounds T
_{R}is always equal to or lower than the total number of test routines N_{R}, given by the expression (55):$${T}_{R}\le {N}_{R}\_\_{T}_{R},{N}_{R}\in \mathbb{N}$$ - (3)
- If no errors are detected during the testing routines the STF parameter will be 0 resulting in 100% reliability and availability of the tested equipment, expressed as in relation (56):$$STF=0\leftrightarrow R=A=1$$

_{R}= 10 rounds for each test point, a total number of N

_{R}= 100 test routines, and a number of P = 2 probes, to identify N

_{E}= 12 voltage deviations. The STF parameter will be computed in relation (57), based on the previous configuration:

## 5. Experimental Setup and Results

#### 5.1. Fault Coverage-Aware Metrics Equation Reduction for Computing the Reliability Test Factor

_{E}$\cdot $ T

_{V}will be denoted with E, representing the hardware error factor. According to property (3), Equation (34) can be rewritten as in relation (61):

_{P}the number of software test patterns, and B the number of breakpoints used during the debugging stage.

_{R}the total number of test routines, and P the total number of test probes. Thus, by reducing the equations to the simplest form, we can proceed with the description of the simulation environment and the reliability graph generation of the tested solar tracking system.

#### 5.2. Fault Coverage-Aware Metrics for BIST, WBST, and ICT Test Scenarios

_{P}is a constant value (see column 5) and the error factor E is a variable parameter (see column 6). Thus, to generate the graphical representation of the STF parameter, the following equation system (64) must be computed:

_{1}, STF

_{2}, STF

_{3}, STF

_{4,}and STF

_{5}were previously calculated in the equation set (64).

_{P}= 4334 (computed by adding all test vectors together from column 6), and the variable number of errors E is calculated for each data batch. By using the simplified Equation (62), the equation set for calculating the STF parameter in regards to the WBST routines will be constructed as in equation system (68):

_{R}is a constant value (see column 3) and FC is the variable fault coverage determined for each test batch (see column 4). Following, the calculated values from the equation system (70) will be substituted in the STF equation set (71):

#### 5.3. Fault Coverage-Aware Metrics Comparison with Other Related Works

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Reliability Curve Linear Function at moment 0 and moment T for a chain of solar tracking systems [20].

**Figure 3.**Failure and Maintenance Periods for N solar tracking components [20].

**Figure 4.**Bathtub Curve representation of Run-In period (I), Operating Period (II), and Catastrophic Failure Period (III) [21].

**Figure 5.**The number of malfunctioning products from a total of 1000 components on a fixed time scale (10

^{3}h) [20].

**Figure 7.**Conventional on-line testing mechanism composed of a software TPG (STPG), an application under test (AUT), and an ORA.

**Figure 9.**Fault Coverage Aware Metrics for Hardware Test Scenarios: (

**a**) Hardware STF (

**Y**-axis) generated according to the total number of faults (

**X**-axis); (

**b**) Hardware SRF (

**Y**-axis) generated according to the Hardware STF parameter (

**X**-axis).

**Figure 10.**Fault Coverage Aware Metrics for Software Test Scenarios: (

**a**) Software STF (

**Y**-axis) generated according to the total number of errors (

**X**-axis); (

**b**) Software SRF (

**Y**-axis) generated according to the Software STF parameter (

**X**-axis).

**Figure 11.**Fault Coverage Aware Metrics for ICT Test Scenarios: (

**a**) ICT STF (

**Y**-axis) generated according to the total number of defects (

**X**-axis); (

**b**) ICT SRF (

**Y**-axis) generated according to the ICT STF parameter (

**X**-axis).

Failure Moments (T_{i}) | T_{i} (Days) |
---|---|

T_{1} | 125 |

T_{2} | 170.83 |

T_{3} | 220.83 |

T_{4} | 283.3 |

T_{5} | 291.6 |

T_{6} | 341.6 |

T_{7} | 375 |

T_{8} | 512.5 |

T_{9} | 541.6 |

T_{10} | 583.3 |

T_{11} | 625 |

T_{12} | 666.6 |

T_{13} | 687.5 |

T_{14} | 741.6 |

T_{15} | 795.83 |

**Table 2.**Malfunctioning (Repairing Service Times) Collected During 3 Months (expressed in days units).

Batches (B) | B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 |
---|---|---|---|---|---|---|---|---|

Maintenance Operation Days | 2 | 4 | 3 | 2 | 7 | 4 | 1 | 6 |

3 | 2 | 4 | 3 | 8 | 6 | 5 | 5 | |

1 | 5 | 1 | 5 | 1 | 1 | 2 | 1 | |

5 | 1 | 1 | 4 | 3 | 2 | 3 | 2 | |

6 | 7 | 2 | 1 | 2 | 1 | 4 | 3 |

**Table 3.**Extended Fault Analysis of Single Bit-Flip Errors As Well As Single Stuck-At-Faults [7].

Crt. No. | Initial Seed (HEX) | Fault Coverage | Total Test Patterns | Total Detected Errors | |
---|---|---|---|---|---|

Single Bit Flip Errors | Single Stuck-at-Faults | ||||

Last 8 Bits (Mutant) | |||||

1 | FFFF | 93.95% | 100 % | 65,535 | 61,570 |

2 | 8FFF | 93.93% | 61,557 | ||

3 | 8CFF | 93.92% | 61,550 | ||

4 | 8C9F | 93.91% | 61,544 | ||

5 | 8C94 | 93.94% | 61,563 |

**Table 4.**WBST Fault Coverage for Control Flow, Communication, Calculation, and Error Handling Errors [6].

Crt. No. | Total Detected Errors | Total Test Patterns | Error Coverage (%) | |||
---|---|---|---|---|---|---|

Control Flow Errors | Communication Errors | Calculation Errors | Error Handling Faults | |||

1 | 395 | 2 | 568 | 736 | 2277 | 74.70 |

2 | 175 | 1 | 242 | 299 | 1087 | 65.96 |

3 | 118 | 1 | 166 | 214 | 840 | 59.40 |

4 | 78 | 0 | 2 | 42 | 130 | 93.84 |

**Table 5.**Extended ICT Fault Coverage for Multiple Point Testing Results [8].

Crt. No. | Test Type | No. of Test Samples (Parameters) | Fault Coverage (%) |
---|---|---|---|

1 | Multiple Point Testing | 250 | 88.70 |

2 | 90 | ||

3 | 91.40 | ||

4 | 95.70 |

Crt. No. | Reliability Evaluation Methodologies for Fielded PV Systems | Time (Hours/ Days/ Years) | Availability/ Reliability Factor (%) | |||
---|---|---|---|---|---|---|

Major Metrics Models | Tested Components | Particular Metrics/ Distribution Model | Number of Failures/Errors | |||

1 | LCOE based Metrics [14] | PV modules | >LCOE | Not specified | 15 Years | 0.5–3 |

2 | Weibull Distribution Model [19] | PV 150 Inverter | Not specified | 125 | 0–1095 Days | 100–0 |

PV module | Lognormal 2-RRX | 29 | ||||

AC Disconnect | Weibull 2-RRX | 22 | ||||

Lightning 208/480 | Weibull 2-RRX | 16 | ||||

Transformer | Lognormal 2-RRX | 4 | ||||

Row Box | Lognormal 2-RRX | 34 | ||||

Marshalling Box | Weibull 2-RRX | 2 | ||||

480VAC/ 34.5KV Transformer | Weibull 2-RRX | >5 | ||||

3 | Proposed Fault Coverage-Aware Metrics | Optocoupler LTV-847 IC | OBIST | 15,392.5 | 0.00138 h | 0.3919 |

Arduino UNO Development Board | WBST | 4334 | 6.63 h | 0.7912 | ||

OBIST | 15,387.5 | 0.00083 h | 0.3920 | |||

FPICT | 957 | 5.178 h | 0.8922 | |||

L298N Motor Drivers | OBIST | 30,772 | 0.0016 h | 0.3921 |

**Table 7.**Comparison between the Proposed Fault Coverage-aware Metrics and the Weibull Distribution Model regarding Calculation Steps and Execution Time.

Crt. Nr. | Weibull Distribution Model | Fault Coverage Aware Metrics | Runtime Execution for Weibull | Runtime Execution for Our Metrics | ||
---|---|---|---|---|---|---|

Speed (ms) | Steps (n) | Speed (ms) | Steps (n) | |||

1 | Weibull Execution Cycles | Metrics Execution Cycles | 0.1 | 274 | 0.03 | 69 |

2 | 0.1 | 274 | 0.02 | 69 | ||

3 | 0.09 | 274 | 0.02 | 69 | ||

4 | 0.09 | 274 | 0.02 | 69 | ||

5 | 0.08 | 274 | 0.02 | 69 | ||

6 | 0.08 | 274 | 0.02 | 69 | ||

7 | 0.09 | 274 | 0.02 | 69 | ||

8 | 0.08 | 274 | 0.02 | 69 | ||

Average Values | 0.08875 | 274 | 0.02125 | 69 |

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**MDPI and ACS Style**

Rotar, R.; Jurj, S.L.; Opritoiu, F.; Vladutiu, M.
Fault Coverage-Aware Metrics for Evaluating the Reliability Factor of Solar Tracking Systems. *Energies* **2021**, *14*, 1074.
https://doi.org/10.3390/en14041074

**AMA Style**

Rotar R, Jurj SL, Opritoiu F, Vladutiu M.
Fault Coverage-Aware Metrics for Evaluating the Reliability Factor of Solar Tracking Systems. *Energies*. 2021; 14(4):1074.
https://doi.org/10.3390/en14041074

**Chicago/Turabian Style**

Rotar, Raul, Sorin Liviu Jurj, Flavius Opritoiu, and Mircea Vladutiu.
2021. "Fault Coverage-Aware Metrics for Evaluating the Reliability Factor of Solar Tracking Systems" *Energies* 14, no. 4: 1074.
https://doi.org/10.3390/en14041074