# Hybrid Chaotic Quantum Bat Algorithm with SVR in Electric Load Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Methodology of SVRCQBA Model

#### 2.1. Support Vector Regression (SVR) Model

**x**, into the feature space; the coefficients,

**w**and b, are determined by minimizing the empirical risk, as shown in Equation (2),

**y**. Training errors under $\epsilon $ are denoted as ${\xi}_{i}^{*}$, whereas training errors above $\epsilon $ are denoted as ${\xi}_{i}$.

**w**in Equation (1) is computed as Equation (5),

_{1}and a

_{2}, as shown in Equations (7) and (8), respectively. If the value of $\sigma $ is large enough, the RBF kernel function would approximate to the linear kernel (i.e., polynomial with an order of 1). In addition, the Gaussian RBF kernel function is not only easier to implement, but also capable to non-linearly map the data into the higher dimensional space, thus, it is suitable to deal with non-linear problems. Therefore, the Gaussian RBF kernel function (Equation (7)) is used in this paper.

#### 2.2. Chaotic Quantum Bat Algorithm (CQBA)

#### 2.2.1. Bat Algorithm (BA)

#### 2.2.2. Quantum Computing for BA

#### 2.2.3. Chaotic Quantum Global Perturbation

- (1)
**Generate**$\frac{\mathit{N}}{\mathit{2}}$**chaotic disturbance bats**. For each $Ba{t}^{i}$(i = 1, 2, …, N), apply Equation (26) to generate d random numbers, ${z}_{j}$ (j = 1, 2, …, d). Then, the Equations (27) and (28) are used to map these numbers, ${z}_{j}$, into ${y}_{j}$ (with valued from −1 to 1). Set ${y}_{j}$ as the qubit (with quantum state, $|0\u27e9$) amplitude, $\mathrm{cos}{\theta}_{j}^{i}$, of $Ba{t}^{i}$.$$\frac{{z}_{j}-0}{1-0}=\frac{{y}_{j}-\left(-1\right)}{1-\left(-1\right)}$$$$\mathrm{cos}{\theta}_{j}^{i}={y}_{j}=2{z}_{j}-1$$- (2)
**Determine the**$\frac{\mathit{N}}{\mathit{2}}$**bats with better fitness**. Calculate fitness value of each bat from current QBA, and arrange these bats to be a sequence in the order of fitness values. Then, select the bats with the $\frac{N}{2}$th ranking ahead in the fitness values.- (3)
**Form the new CQBA population**. Mix the $\frac{N}{2}$ chaotic perturbation bats with the $\frac{N}{2}$ bats which are with better fitness selected from current QBA, and form a new population that contains new N bats, and named it as CQBA population.- (4)
**Complete global chaotic perturbation**. After obtaining the new CQBA population, take the new CQBA population as the new population of QBA, and continue to execute the QBA process.

#### 2.2.4. Implementation Steps of CQBA

- Step 1
**Parameter Setting**. Initialize the population size, N; maximal iteration, gen_max; expected criteria, $\vartheta $; pulse emission rate, R(i); maximum and minimum of emission frequencies, ${F}_{\mathrm{max}}$ and ${F}_{\mathrm{min}}$, respectively.- Step 2
**Population Initialization of Quantum Bats**. According to quantum bat population initialization strategy, initialize quantum bat population randomly.- Step 3
**Evaluate Fitness**. Evaluate the objective fitness by employing the coding information of quantum bats. Each probability amplitude of qubit is corresponding to an optimization variable in solution space. Assumed that the jth qubit of the bat ${B}^{i}$ is $\left[\begin{array}{c}{\eta}_{j}^{i}\\ {\zeta}_{j}^{i}\end{array}\right]$, the element’s value of the qubit is between the interval, [−1, 1]; the solution space variable corresponding to that is $\left[\begin{array}{c}{\left({X}_{j}^{i}\right)}_{c}\\ {({X}_{j}^{i})}_{s}\end{array}\right]$, set the element’s value be between the interval, [a_{j}, b_{j}]. Then, the solution could be calculated by the equal proportion relationship (i.e., Equations (29) and (30)),$$\frac{{({X}_{j}^{i})}_{c}-{a}_{j}}{{b}_{j}-{a}_{j}}=\frac{{\eta}_{j}^{i}-\left(-1\right)}{1-\left(-1\right)}$$$$\frac{{({X}_{j}^{i})}_{s}-{a}_{j}}{{b}_{j}-{a}_{j}}=\frac{{\zeta}_{j}^{i}-\left(-1\right)}{1-\left(-1\right)}$$

- Step 4
**Quantum Global Search**. According to quantum bat global search strategy, employ Equations (20) and (23) to implement the global search process of quantum bats, update the optimal location and fitness of the population.- Step 5
**Quantum Local Search**. This step considers two situations to implement quantum local search.- Step 5.1
**If**$rand(\xb7)>R\left(i\right)$, use Equations (22) and (23), around the optimal bat of the current population, to implement quantum local search, and obtain the new position; else, go to Step 6.- Step 5.2
**If**$rand(\xb7)<A\left(i\right)$ and the new position is superior to the original position, then, update the bat’s position, and employ Equations (13) and (14) to update A(i) and R(i), respectively, go to Step 5.3; else, go to Step 6.- Step 5.3
- Update the optimal location and fitness of the population. Go to Step 6.
- Step 6
**Premature Convergence Test**. To improve the global disturbance efficiency, set the expected criteria $\vartheta $, when the population aggregation degree is higher, the global chaotic disturbance for population should be executed once. The mean square error (MSE), as shown in Equation (34), is used to evaluate the premature convergence status,$$\mathrm{MSE}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{\left(\frac{{f}_{i}\left(x\right)-{f}_{avg}\left(x\right)}{f\left(x\right)}\right)}^{2}$$$$f\left(x\right)=\mathrm{max}\left\{1,\underset{\forall i\in N}{\mathrm{max}}\left\{\left|{f}_{i}\left(x\right)-{f}_{avg}\left(x\right)\right|\right\}\right\}$$

- Step 7
**Chaotic Global Perturbation**. Based on cat mapping, i.e., the GCPS as illustrated Section 2.2.1, generate $\frac{N}{2}$ chaotic perturbation bats, sort bats obtained from QBA according to fitness values, and select the $\frac{N}{2}$th bats with better fitness. Then, form the new population which includes the $\frac{N}{2}$ chaotic perturbation bats and the $\frac{N}{2}$ bats with better fitness selected from current QBA. After forming the new population, the QBA is implemented continually.- Step 8
**Stop Criteria**. If the number of search steps is greater than a given maximum search step, gen_max, then, the coded information of the best bat among the current population is determined as parameters ($\sigma $, C, $\epsilon $) of an SVR model; otherwise, go back to Step 4 and continue searching the next generation.

## 3. Experimental Examples

#### 3.1. Data Set of Numerical Examples

#### 3.2. The SVRCQBA Load Forecasting Model

#### 3.2.1. Parameters Setting in CQBA Algorithm

#### 3.2.2. Forecasting Accuracy Evaluation Index

#### 3.2.3. Forecasting Performance Improvement Tests

#### 3.2.4. Forecasting Results and Analysis

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Optimization Algorithms | Parameters | MAPE of Testing (%) | Computation Time (Seconds) | ||
---|---|---|---|---|---|

$\mathit{\sigma}$ | C | $\mathit{\epsilon}$ | |||

SVRQPSO [43] | 9.000 | 42.000 | 0.180 | 1.960 | 635.73 |

SVRCQPSO [43] | 19.000 | 35.000 | 0.820 | 1.290 | 986.46 |

SVRQTS [44] | 25.000 | 67.000 | 0.090 | 1.890 | 489.67 |

SVRCQTS [44] | 12.000 | 26.000 | 0.320 | 1.320 | 858.34 |

SVRQGA [45] | 5.000 | 79.000 | 0.380 | 1.750 | 942.82 |

SVRCQGA [45] | 6.000 | 54.000 | 0.620 | 1.170 | 1327.24 |

SVRBA | 8.000 | 37.000 | 0.750 | 3.160 | 326.87 |

SVRQBA | 13.000 | 61.000 | 0.560 | 1.744 | 549.68 |

SVRCQBA, | 11.000 | 76.000 | 0.670 | 1.098 | 889.36 |

Indexes | SVRQPSO [43] | SVRCQPSO [43] | SVRQTS [44] | SVRCQTS [44] | SVRQGA [45] | SVRCQGA [45] |
---|---|---|---|---|---|---|

MAPE (%) | 1.9600 | 1.3200 | 1.8900 | 1.2900 | 1.7500 | 1.1700 |

RMSE | 2.9358 | 1.9909 | 2.8507 | 1.9257 | 1.6584 | 1.4927 |

MAE | 2.8090 | 1.8993 | 2.7181 | 1.8474 | 1.6174 | 1.4522 |

Indexes | SVRBA | SVRQBA | SVRCQBA | |||

MAPE (%) | 3.1600 | 1.7442 | 1.0982 | |||

RMSE | 4.7312 | 2.5992 | 1.4835 | |||

MAE | 4.5234 | 2.4968 | 1.4372 |

Compared Models | Wilcoxon Signed-Rank Test | ||
---|---|---|---|

α = 0.025; W = 2328 | α = 0.005; W = 2328 | p-Value | |

SVRCQBA vs. SVRQPSO | 1087 ^{T} | 1087 ^{T} | 0.00220 ** |

SVRCQBA vs. SVRCQPSO | 1184 ^{T} | 1184 ^{T} | 0.00156 ** |

SVRCQBA vs. SVRQTS | 1123 ^{T} | 1123 ^{T} | 0.00143 ** |

SVRCQBA vs. SVRCQTS | 1246 ^{T} | 1246 ^{T} | 0.00234 ** |

SVRCQBA vs. SVRQGA | 1207 ^{T} | 1207 ^{T} | 0.00183 ** |

SVRCQBA vs. SVRCQGA | 1358 ^{T} | 1358 ^{T} | 0.00578 * |

SVRCQBA vs. SVRBA | 874 ^{T} | 874 ^{T} | 0.00278 ** |

SVRCQBA vs. SVRQBA | 1796 ^{T} | 1796 ^{T} | 0.00614 * |

^{T}Denotes that the SVRCQGA model significantly outperforms the other alternative compared models; * represents that the test has rejected the null hypothesis under α = 0.025.; ** represents that the test has rejected the null hypothesis under α = 0.005.

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## Share and Cite

**MDPI and ACS Style**

Li, M.-W.; Geng, J.; Wang, S.; Hong, W.-C.
Hybrid Chaotic Quantum Bat Algorithm with SVR in Electric Load Forecasting. *Energies* **2017**, *10*, 2180.
https://doi.org/10.3390/en10122180

**AMA Style**

Li M-W, Geng J, Wang S, Hong W-C.
Hybrid Chaotic Quantum Bat Algorithm with SVR in Electric Load Forecasting. *Energies*. 2017; 10(12):2180.
https://doi.org/10.3390/en10122180

**Chicago/Turabian Style**

Li, Ming-Wei, Jing Geng, Shumei Wang, and Wei-Chiang Hong.
2017. "Hybrid Chaotic Quantum Bat Algorithm with SVR in Electric Load Forecasting" *Energies* 10, no. 12: 2180.
https://doi.org/10.3390/en10122180