A Hybrid Genetic/Powell Algorithm for Wind Measurement in Doppler Lidar

Abstract

Doppler peaks extraction from massive raw data is a tricky part of coherent Doppler wind Lidar (CDWL) optimization. In this paper, a hybrid genetic/Powell algorithm (HGAP) is proposed to process the power spectrum of the measured signal from CDWL. The HGAP has excellent global exploration capability, which likes traditional genetic algorithms and fast convergence, which like the Powell method. Hence, the HGAP has advantages to find the center frequency of the Doppler peaks from massive raw data, especially to search multiple peaks in complex wind field measurement. Compared with other notable algorithms, the HGAP shows excellent performance in numerical optimization when we use it to solve 27 typical benchmark functions. Then, our algorithm is used to process the raw data in a field experiment of radial wind measurement. The results show that the HGAP can obtain wind speed components quickly and accurately and has value for application in complex wind field analysis.

1. Introduction

Coherent Doppler wind Lidar (CDWL), as a widespread wind remote sensing method, has been used in the fields of aerodynamic, aviation safety and wind power generation [1,2,3]. The CDWL recodes the Doppler peaks of the backscattered signal to retrieve the radial wind speed. To increase the identification of the Doppler peaks with low carrier-to-noise ratio (CNR), the raw data are transformed into the frequency domain and accumulated incoherently [4]. Then, the retrial algorithms are used to calculate the center frequency of the Doppler peaks from the power spectrum. With the development of hardware technology, the raw data recoded by the CDWL contain more details, which make complex wind field (e.g., wind shear and turbulence) analysis possible [5,6]. Several typical algorithms have been reported and evaluated: maximum, centroid, Gaussian fitting, maximum likelihood, etc., but they are not good at processing massive data or irregular signals [7,8,9]. Therefore, it is necessary to design more effective algorithms to extract the Doppler peaks in CDWL systems.

In recent years, machine learning has been widely used in wind measurement. Especially, when a mechanism is difficult to analyze, data training becomes a more preferred choice. Some new modes and algorithms based on machine learning are introduced for wind field forecasting and noise filtering, which achieved acceptable accuracy [10,11,12,13]. For wind field analysis, many researchers have proposed machine learning algorithms to measure important parameters, such as wind shear index and turbulence intensity [14,15,16]. Numerous studies have shown that machine learning technology has advantages in resolving complex mapping relationships in which data are nonlinear, massive and multivariable.

In this paper, a hybrid genetic/Powell algorithm (HGAP) is proposed to process the backscattered signals from CDWL. The algorithm has global search ability and fast convergence, which can output all wind speed components and their corresponding strength. The structure of this paper is as follows: in Section 2, the problem of the Doppler peaks extraction in the CDWL system is described; in Section 3, the HGAP is designed for finding the global numerical solution; in Section 4, we used our algorithm to solve the benchmark functions published by IEEE Congress on Evolutionary Computation (ICEC) and process the raw data measured by CDWL to show performance; in Section 5, we summarized our work.

2. The CDWL System

There are a lot of aerosols moving within the atmosphere, which are the tracers of wind. A laser with frequency $f$ emits into the atmosphere from the transmitting optics. When the light beam impacts the aerosols, a small fraction of light is backscattered into the receiving optics. The motion of the aerosols along the beam direction leads to a change $\Delta f$ in the frequency of light via the Doppler shift, given by:

$$\Delta f=2v/\lambda$$

(1)

where $\lambda$ is the laser wavelength. The essential features are readily seen in the simplified generic CDWL depicted in Figure 1. The local oscillator, serving as the reference beam, amplifies the backscattered signal via the beating process to allow operation at a sensitivity that approaches the shot-noise limit. Therefore, the CDWL system has the ability to be applied in complex wind field detection.

According to Formula (1), the frequency of the Doppler signal $\Delta f$ determines the retrieved result of the wind speed. The complex wind field contains inconsistent speed or direction components, and the backscattered signals received by CDWL are irregular, which makes it difficult to extract the Doppler peaks from the background noise reliably [17]. With the development of acquisition technology, a high sampling rate helps to record more details of wind fields. Hence, more efficient algorithms are needed for finding the Doppler shift from a large amount of raw data. In this paper, the proposed HGAP algorithm can output the numerical solution of global max/min values and local max/min values accurately and quickly. On wind measurement, our algorithm can obtain all speed components and their corresponding strength, respectively.

3. HGAP

It is well known that genetic algorithms have outstanding global search ability, and the Powell method has fast convergence [18,19]. The HGAP is a hybrid algorithm, which is the combination of a genetic algorithm and Powell method, so it can find global numerical solutions in massive data accurately and quickly. In order to be consistent with the wind measurement application, we use the HGAP to locate the maximum value point in a rectangular coordinated system, as shown in Figure 2.

(1) Initial population. A chaos map is used to generate random numbers in the interested range of the x-axis. In order to simplify the expression, it is assumed that the generated random numbers are $x_1$, $x_2$, $x_3$ and $x_4$, which are the first population. We use a logistic map to initialize populations with random distributions as Formulas (2) and (3).

$$x_{n+1}=r{x}_n(1-x_n) \left(\begin{array}{cc}0\le r\le 4,& 0<x_n<1\end{array}\right)$$

(2)

$$x_i=x_{n+1}(U-L)+L \left(0\le i\le \begin{array}{cc}population& size\end{array}\right)$$

(3)

where $U$ is the upper bound and $L$ is the lower bound. In this paper, numbers of population $r=4$ and iterations $n=100$ are chosen. The initial value $x_0$ is a random number and $x_0\notin \{0.25,0.5,0.75\}$.

(2) First Powell method. The Powell method is always known as a directional acceleration algorithm. From any initial point, the method can finally obtain the global max/min value by using the conjugate direction as the search direction [20]. At first, the population will converge at the local optimal solutions, and we need to apply the Powell method iteratively to find the global optimal solution.

(3) Crossover and mutation. New populations $x_1^{\prime }$, $x_2^{\prime }$, $x_3^{\prime }$ and $x_4^{\prime }$ are generated by crossover and mutation to maintain the diversity of population, as well as avoid premature convergence. The float encoding is selected for crossover operators as Formulas (4) and (5):

$$x_A^{i+1}=\alpha x_B^i+(1-\alpha )x_A^i$$

(4)

$$x_B^{i+1}=\alpha x_A^i+(1-\alpha )x_B^i$$

(5)

where $x_A^i$ and $x_B^i$ are two individuals in the population. $x_A^{i+1}$ and $x_B^{i+1}$ are new individuals after crossover. $\alpha$ is a random number and $0<\alpha \le 1$. We still choose the logistic map for mutation, and the new genes will be obtained by Formulas (2) and (3).

(4) Second Powell method. The population will converge at global optimal solutions by using the Powell method again. In order to prevent linear correlation of direction sets, the Powell method is used twice in the HGAP. It not only provides excellent populations but also a complementary measure to refine search.

(5) Selection. In this step, we use roulette selection method. A new population will be generated according to $Fitness$, which is defined by Formula (6).

$$Fitness(x_i)=Max(F(x))-F(x_i)+\epsilon$$

(6)

where $F(x)$ is the objective function, and $Fitness(x_i)$ is the fitness of each individual in the population. Note that $F(x)$ is fitted by the sampling data in consideration of real time and accuracy. To prevent $Fitness(x_i)$ from being zero, we added a very small number $\epsilon ={10}^{-3}$. If there are $N$ individuals in the population, the probability of each individual being selected is calculated by Formula (7).

$$P_i=\frac{Fitness(x_i)}{{\displaystyle {\sum }_{i=1}^{N}Fitness(x_i)}}$$

(7)

According to $P_i$, excellent individuals will be selected. The survival of the fittest is achieved by choosing the population with high probability.

According to the number of wind speed components needed to be extracted, the HGAP can set the number of individuals and the times of iteration. Usually, one time for each iteration is enough when the wind speed components we want are fewer than ten.

4. Test and Discussion

In order to test performance, the HGAP was compared with other notable algorithms to test the benchmark functions published by ICEC. Then, our algorithm was used to process the raw data from a CDWL, and the result is analyzed.

4.1. Test the Benchmark Functions

Four notable global optimization algorithms: particle swarm optimization algorithm (PSO), sparrow search algorithm (SSA), gray wolf optimization algorithm (GWO) and whale optimization algorithm (WOA) are selected for comparison with the HGAP to solve the benchmark functions. The parameters we used are the same as the original paper [21,22,23,24]. We chose three criteria (Success Rate, Standard Deviation and Iteration) to evaluate the processing results.

The twenty-seven benchmark functions published by ICEC 2017 are summarized in

Table 1. Benchmark functions published by ICEC 2017.

Function ID	Function Name	Dimension	Globe Minimum
f1	Ackley	10	0
f2	Beale	2	0
f3	Bent Cigar	10	0
f4	Booth	2	0
f5	Bukin	2	0
f6	Cross-in-Tray	2	−2.06
f7	Discus	10	0
f8	Easom	2	−1
f9	Goldstein-Price	2	−3
f10	Griewank’s	10	0
f11	High Conditioned Elliptic	10	0
f12	Himmelblau’s	2	0
f13	Hölder table	2	−19.21
f14	Levy	10	0
f15	Matyas	2	0
f16	McCormick	2	−1.91
f17	Michalewicz	5	−4.69
f18	Rastrigin’s	10	0
f19	Rosenbrock’s	10	0
f20	Schaffer’s F7	10	0
f21	Schaffer N.2	2	0
f22	Schaffer N.4	2	0.29
f23	Sphere	10	0
f24	Styblinski-Tang	5	−195.83
f25	Sum of Different Power	10	0
f26	Three-hump camel	2	0
f27	Zakharov	10	0

. In reference [25], these functions can be divided into five classifications, which are many Local Minima ($f_1$, $f_5$, $f_6$, $f_{10}$, $f_{13}$, $f_{14}$, $f_{18}$, $f_{20}$, $f_{21}$ and $f_{22}$), bowl-shaped ($f_{23}$ and $f_{25}$), valley-shaped ($f_{19}$ and $f_{26}$), steep-ridges ($f_8$ and $f_{17}$), plate-shaped ($f_4$, $f_{15}$, $f_{16}$ and $f_{27}$) and others ($f_2$, $f_3$, $f_7$, $f_9$, $f_{11}$, $f_{12}$ and $f_{24}$).

To be fair, each benchmark function test was repeated 50 times, and the results are shown in

Table 2. The results of the Benchmark functions processing.

Function Id	Criteria	HGAP	PSO	SSA	WOA	GWO
f1	Success rate	100%	12%	100%	100%	100%
	STD	4.97 × 10−13	9.49	0	6.90 × 10−5	2.69 × 10−9
	Iteration	1.9	96.83	5.12	71	26.72
f2	Success rate	100%	100%	100%	100%	98%
	STD	3.92 × 10−32	8.92 × 10−11	1.09 × 10−10	5.19 × 10−7	0.11
	Iteration	1.00	16.62	9.02	14.84	10.24
f3	Success rate	100%	0%	100%	100%	100%
	STD	6.21 × 10−22	39.04	5.19 × 10−40	5.56 × 10−6	6.35 × 10−13
	Iteration	1.00	-	7.24	75.6	30.28
f4	Success rate	100%	100%	100%	100%	100%
	STD	0	8.20 × 10−11	1.19 × 10−5	6.06 × 10−6	1.45 × 10−6
	Iteration	1.00	25.02	39.86	31.32	17.78
f5	Success rate	100%	0%	0%	0%	0%
	STD	2.54 × 10−6	0.07	2.96 × 10−16	0.13	0.15
	Iteration	72.86	-	-	-	-
f6	Success rate	100%	100%	100%	100%	100%
	STD	0	1.05 × 10−10	9.27 × 10−12	7.16 × 10−8	3.41 × 10−8
	Iteration	1.00	9.54	1.26	5.58	3.58
f7	Success rate	100%	0%	100%	100%	100%
	STD	2.95 × 10−26	6.12	1.41 × 10−56	1.10 × 10−9	4.65 × 10−18
	Iteration	1.00	-	7.22	54.74	17.72
f8	Success rate	100%	100%	100%	100%	100%
	STD	0	1.53 × 10−9	2.48 × 10−8	5.87 × 10−6	1.80 × 10−6
	Iteration	1.00	38.14	11.94	40.82	29.48
f9	Success rate	100%	100%	100%	100%	100%
	STD	3.14 × 10−16	1.86 × 10−10	4.98 × 10−8	2.33 × 10−5	1.02 × 10−4
	Iteration	1.00	27.18	30.32	35.04	21.34
f10	Success rate	100%	0%	100%	38%	4%
	STD	0	0.07	0	0.02	0.04
	Iteration	82.12	-	1.38	68.63	32.50
f11	Success rate	100%	0%	100%	100%	100%
	STD	4.96 × 10−20	5.17 × 10−4	1.69 × 10−38	2.36 × 10−8	1.72 × 10−15
	Iteration	1.00	-	6.02	65.90	24.82
f12	Success rate	100%	100%	100%	100%	98%
	STD	0	3.60 × 10−10	4.51 × 10−5	8.97 × 10−5	4.85 × 10−4
	Iteration	1.00	27.76	40.60	62.76	74.22
f13	Success rate	100%	100%	100%	100%	98%
	STD	3.55 × 10−15	3.55 × 10−10	5.25 × 10−6	2.30 × 10−4	5.00 × 10−3
	Iteration	1.00	25.22	36.00	73.78	86.43
f14	Success rate	100%	100%	100%	4%	44%
	STD	0	0.44	1.44 × 10−6	0.10	0.07
	Iteration	2.64	52.94	2.50	98.50	94.59
f15	Success rate	100%	100%	100%	100%	100%
	STD	2.31 × 10−32	7.79 × 10−12	1.03 × 10−45	6.70 × 10−46	1.47 × 10−43
	Iteration	1.00	11.72	1.00	8.40	3.16
f16	Success rate	100%	78%	90%	100%	100%
	STD	2.22 × 10−16	0.02	0.02	3.02 × 10−7	1.79 × 10−7
	Iteration	1.00	16.74	8.66	15.96	6.12
f17	Success rate	100%	18%	0%	0%	2%
	STD	3.97 × 10−16	0.37	0.46	0.46	0.26
	Iteration	81.68	61.11	-	-	100
f18	Success rate	100%	0%	100%	74%	22%
	STD	0	8.71	0	1.50	3.06
	Iteration	1.90	-	4.72	7.15	3.50
f19	Success rate	100%	0%	100%	0%	0%
	STD	8.23 × 10−24	390.47	2.26 × 10−7	97.87	6.15
	Iteration	1.00	-	6.00	-	-
f20	Success rate	100%	64%	100%	48%	26%
	STD	5.46 × 10−32	4.39 × 10−3	6.67 × 10−16	0.01	0.06
	Iteration	1.00	33.03	1.02	88.17	89.31
f21	Success rate	100%	100%	100%	100%	100%
	STD	0	$4.24\times {10}^{-13}$	0	0	0
	Iteration	17.28	18.32	1.00	17.68	5.26
f22	Success rate	100%	100%	100%	100%	100%
	STD	1.76 × 10−7	1.17 × 10−8	1.20 × 10−4	1.33 × 10−6	1.39 × 10−6
	Iteration	75.50	20.92	26.02	19.46	5.94
f23	Success rate	100%	100%	100%	100%	100%
	STD	1.78 × 10−26	8.42 × 10−7	6.03 × 10−42	1.40 × 10−8	9.80 × 10−17
	Iteration	1.00	71.06	1.80	60.06	20.94
f24	Success rate	100%	34%	98%	6%	32%
	STD	0	11.94	8.56 × 10−4	6.31	4.24
	Iteration	50.42	47.11	25.67	87.65	95.60
f25	Success rate	100%	28%	100%	100%	100%
	STD	1.15 × 10−32	1.40 × 10−9	1.13 × 10−38	1.02 × 10−13	2.94 × 10−30
	Iteration	1.00	83.50	1.40	56.98	18.14
f26	Success rate	100%	100%	100%	100%	100%
	STD	2.25 × 10−31	2.76 × 10−12	4.44 × 10−39	6.08 × 10−44	2.91 × 10−70
	Iteration	1.00	15.08	1.54	9.26	3.20
f27	Success rate	100%	0%	100%	100%	100%
	STD	1.15 × 10−25	9.28 × 10−9	2.19 × 10−52	6.19 × 10−9	1.17 × 10−17
	Iteration	1.00	50.16	2.76	54.26	16.08

.

From the results, the HGAP has a 100% success rate for the 27 tested functions, while the other four algorithms all fail on $f_5$. In addition, the standard deviation of the HGAP is the smallest in 18 tests, and the convergence speed of the HGAP is the fastest in 18 tests. The number of winning test criteria (NWTC) for each algorithm is counted and shown in Figure 3. A higher NWTC means a better algorithm performance. Note that different algorithms may win the same test criteria. From the results, the HGAP wins all of the test criteria, especially in minimum number of iterations. Most function tests output the final solutions with only a single iteration.

The results show that the HGAP has excellent exploration capabilities with a fewer number of iterations. In addition, since the HGAP does not require gradient information to guide the search, it can avoid falling into the local optimum trap and outputs the globe min/max value more reliably. Hence, the HGAP has more advantage to extract Doppler peaks (max value) from massive raw data in wind measurement.

4.2. Test the Raw Data from the CDWL

Radial wind measurement was carried out to collect complex wind field data by using a CDWL at Zhong Chuan Airport (36.51 N, 103.62 E), as illustrated in Figure 4. In the experiment, the laser beam emitted by the CDWL was pointed at the runway horizontally.

The key parameters of the CDWL are summarized in

Table 3. Key parameters of the CDWL.

Item	Parameters	Value
Laser	Wavelength	1547 nm
	Line width	4 kHz @ 1547 nm
	Pulse energy	150 uJ
	Pulse width	200 ns
	Pulse repetition frequency	10 kHz
Telescope	Effective aperture	150 mm
Balanced detector	Responsiveness	0.95 A/W
	3-dB bandwidth	250 MHz
A/D converter	Sample rate	500 MHz

. The detection distance is set at one kilometer with a distance gate of 60 m. In the raw data, it can be seen that some spectra of backscattered signals contain several Doppler peaks. After removing the noise floor, we converted frequency to wind speed according to Formula (1), and the typical spectra are shown in Figure 5.

We used the HGAP to process the raw data in Figure 5 and obtained the results as shown in Figure 6. The threshold is set to filter the local maximum value whose strength is not stronger than the noise floor. Figure 6a,b both have four wind speed components, in which the blue dot is the main one and the green dot is the minor. However, traditional Gaussian fitting (GF) just finds the peak of the main component and misses the others. As shown in

Table 4. The measurement result of Figure 6.

Figure	Component	Speed (m/s)	Strength (a.u.)	Speed Measured by GF (m/s)	Speed Measured by HGAP(m/s)
Figure 6a	Main	3.81	6.13	3.79	5.56
	Minor 1	5.15	1.52
	Minor 2	9.82	1.06
	Minor 3	10.35	1.43
Figure 6b	Main	3.94	4.76	3.91	6.71
	Minor 1	5.20	1.88
	Minor 2	9.28	3.45
	Minor 3	10.40	1.93

, the HGAP not only finds multiple wind speed components quickly but also records the corresponding strength of the backscattered signals. Hence, we can use the spectral centroid method to retrieve the wind speed conveniently, which is more reasonable than the average or the main component. It can be seen that the HGAP is helpful for the analysis of complex wind fields.

5. Conclusions

In this paper, we propose the HGAP, which is the combination of a genetic algorithm and the Powell method, for wind measurement in CDWL. The HGAP has strong ability in global search and fast convergence, which is suitable for massive data analysis. We compared the HGAP with four notable algorithms to solve 27 benchmark functions published by ICEC, and the result shows that our algorithm has great advantages in finding max/min values. In the wind measurement experiments, the HGAP is used to process the raw data from a CDWL, and it also obtained the Doppler peaks efficiently and reliably. The results show that our algorithm not only output wind speed components quickly but also recorded the signal strength of each component. Therefore, the HGAP is helpful for complex wind field analysis and can be further studied for application in more practical models.