Estimation of Solar Radiation for Tomato Water Requirement Calculation in Chinese-Style Solar Greenhouses Based on Least Mean Squares Filter

Abstract

The area covered by Chinese-style solar greenhouses (CSGs) has been increasing rapidly. However, only a few pyranometers, which are fundamental for solar radiation sensing, have been installed inside CSGs. The lack of solar radiation sensing will bring negative effects in greenhouse cultivation such as over irrigation or under irrigation, and unnecessary power consumption. We aim to provide accurate and low-cost solar radiation estimation methods that are urgently needed. In this paper, a method of estimation of solar radiation inside CSGs based on a least mean squares (LMS) filter is proposed. The water required for tomato growth was also calculated based on the estimated solar radiation. Then, we compared the accuracy of this method to methods based on knowledge of astronomy and geometry for both solar radiation estimation and tomato water requirement. The results showed that the fitting function of estimation data based on the LMS filter and data collected from sensors inside the greenhouse was y = 0.7634x + 50.58, with the evaluation parameters of R² = 0.8384, rRMSE = 23.1%, RMSE = 37.6 Wm⁻², and MAE = 25.4 Wm⁻². The fitting function of the water requirement calculated according to the proposed method and data collected from sensors inside the greenhouse was y = 0.8550x + 99.10 with the evaluation parameters of R² = 0.9123, rRMSE = 8.8%, RMSE = 40.4 mL plant⁻¹, and MAE = 31.5 mL plant⁻¹. The results also indicate that this method is more effective. Additionally, its accuracy decreases as cloud cover increases. The performance is due to the LMS filter’s low pass characteristic that smooth the fluctuations. Furthermore, the LMS filter can be easily implemented on low cost processors. Therefore, the adoption of the proposed method is useful to improve the solar radiation sensing in CSGs with more accuracy and less expense.

1. Introduction

Since being introduced in the 1930s, the Chinese-style solar greenhouse (CSG) has gradually grown in Chinese agriculture. The greenhouse vegetable industry accounts for 20% of the total vegetable production area in China, but it produces 35% of the output and 60% of the economic value in 2013 [1]. In 2013, the CSG cultivation area amounted to 612,000 ha [2].

Solar radiation is an important factor affecting the calculations of water requirement [3,4] and environmental evaluation [5] in precision agriculture. However, few CSGs are equipped with enough sensors due to purchase price and subsequent maintenance [6,7]. Solar radiation sensors, among the sensors in solar greenhouse applications, are very expensive [8,9,10,11]. To produce quality crops in a sufficient quantity in greenhouses, the demand for solar radiation sensing in CSGs is increasing rapidly [12,13,14]. To address this need, some methods that estimate the solar radiation inside CSGs (H_i) with few or no sensors have been introduced. Ahamed et al. (2018) proposes the CSGHEAT model in their study and this model can estimate H_i based on outdoor solar radiation and cloud cover [15]. Tong et al. (2009), Ahamed (2018) et al. use outdoor solar radiation and film transmittance to estimate H_i [16,17]. In addition, some researches about indoor solar radiation estimation on other type greenhouses are also performed [18,19].

In studies of Tong (2009), Ahamed et al. (2018) and Sethi (2009), the horizontal solar radiation outside the greenhouse (H_out) is decomposed into beam radiation (H_b) and diffuse radiation (H_d) [15,16,18]. In the study of Gavilán (2015), decomposition is not conducted [19].

For decomposition, H_out is first divided into H_b and H_d [16,17,20]. Inman et al. (2013) proved that the ratio of the clearness index (K_t) of H_out to extraterrestrial solar radiation (H_o) is an important factor for decomposition [21]. According to K_t, the ratio of H_d to H_out is a fixed function relationship [22,23]. After decomposition, H_b and H_d are multiplied by film transmittance of beam radiation (τ_b) and film transmittance of diffuse radiation (τ_d), respectively. Finally, the sum of the two multiplications is the estimation of radiation inside a greenhouse. In our study, the estimation of H_i estimated by astronomy and geometry method is presented as H_c. In the other case, H_out is directly multiplied by greenhouse transmittance (τ_g) so decomposition is not needed [19]. In summary, the accuracy of these methods is related to K_t, τ_b, τ_d, or τ_g, and τ_b, τ_d, or τ_g are always constants. However, none of these parameters are constant in practice, reducing the accuracy of these methods [24,25,26].

At the experimental site, the low outside temperature in winter means ventilation is rare, so properly calculating the water requirement for the tomatoes is important because overestimating the requirement will lead to higher humidity, which is harmful to tomatoes [27].

Some models of crop water requirement have been formulated, such as the Penman–Monteith equation [28,29] and the Hargreaves equation [30], which are based on meteorological parameters including solar radiation, temperature, relative humidity, wind speed, etc. However, the Penman–Monteith equation was proven to be restricted because too many sensors, which are costly and require frequent maintenance, are needed in greenhouse applications [31]. Hargreaves and Allen (2003) proved the accuracy of the Hargreaves equation is related with the calculation period, which must be five days or more, so it is not suitable for high-frequency irrigation applications [32]. In many cases, tomatoes are cultivated in substrates with lower water-holding capacity, so high-frequency irrigation is required.

To address the restrictions of the Penman–Monteith and Hargreaves equations in greenhouse applications, an equation was proposed by Carmassi et al. (2007) [33]. To perform Carmassi’s equation, only solar radiation and temperature are needed.

To the best of our knowledge, adaptive filters have rarely been applied in solar radiation estimation. Among digital filters used in applications, the adaptive filter [34,35] is advantageous compared with the finite impulse response (FIR) filter and infinite impulse response (IIR) filter due to its better performance in situations with a spectrum overlap between the signal and noise [36]. Adaptive filters are widely used in many fields such as noise canceling, system identification, and signal prediction [37,38].

Therefore, in our study, we adopted an adaptive filter to estimate H_i more accurately and less expensively. The objectives of this study were (1) to estimate H_i and compare the results conducting estimations with other methods (2) to calculate the water required by tomatoes using the H_i estimated with an adaptive filter and compare the results with the requirement calculated using other methods and (3) to analyze the performance of the proposed method.

2. Material and Methods

2.1. Experimental Materials, Measurement and Evaluation

2.1.1. CSG Architecture

As shown in Figure 1, the architecture of a CSG consists of a south roof, north roof, north wall, gables, and blanket. In many cases, a CSG is east-west oriented to intercept more solar energy. In the cold season, the blanket is rolled up during the day and the sunlight enters from the south roof. The crops, ground, north wall, and gables absorb energy. After the blanket is dropped at night, the gables and north wall release heat to maintain temperature [39].

CSGs extend the crop growing season in the cold areas in China between 34° and 41° N, where the temperature falls below −20 °C at night. CSG cultivation requires little auxiliary heating equipment; the consumption of energy and emissions of carbon dioxide are considerably reduced.

2.1.2. Experimental Site and Measurement Methods

We conducted this study in an east-west-oriented CSG in Shenyang, China (41°48′ N, 123°24′ E, 42 m a.s.l.). The greenhouse was 60 m long and 12 m wide. The height of the north wall and north roof were 3 m and 5.5 m, respectively. The south roof was covered by a single layer of 0.00012 m thick polyethylene film.

The cultivation area inside the greenhouse was 55 m long from east to west and 10 m wide. Tomatoes were grown with row spacing of 1 m, within-row spacing of 0.33 m, and plant density of 4 plants·m⁻². Tomatoes were grown in substrate and irrigated using a drip irrigation system. The tomatoes were sown on 5 August 2017. Then, the transplant was performed on 1 September 2017 and the cultivation finished on 27 December 2017. We conducted this study using H_out data collected from outer weather station and indoor pyranometers (H_s) from 1–26 December 2017. It was cold and nature ventilation was rare during the experimental days.

Six temperature sensors (SHT10, Sensirion, Zurich, Switzerland) and three pyranometers (MP200, Apogee Instruments, Logan, UT, USA) were installed in the experimental greenhouse. The temperature sensors were hung 1.5 m above the ground and pyranometers were placed horizontally at different heights above the ground (1.5, 2, and 2.5 m) according to the growth condition of the tomatoes. The placement of indoor sensors is shown in Figure S1.

The temperature sample interval was 15 min and the mean of the temperature recorded by the six installed sensors was considered the temperature inside the greenhouse. The sample interval of solar radiation was 15 min [40] and the mean of three installed pyranometers was taken as the true H_i value.

2.1.3. Evaluation Parameters

As evaluation parameters, we adopted the coefficient of determination (R²), percent error (PE), root mean square error (RMSE), relative root mean square error (rRMSE), and mean absolute error (MAE), and these parameters were calculated according to Equations (1)–(5), respectively [22,23,40]. In our study, R², RMSE, rRMSE, and MAE were firstly used for comparisons between the estimated solar radiation and measured solar radiation. Additionally, they were also used to compare the daily water requirements of tomatoes calculated by estimated data and sensor data. RE was used to compare the error rates of water requirements.

$${\mathrm{R}}^2=1-\frac{{\displaystyle \sum }{\left({\mathrm{y}}_{\mathrm{e}}-{\mathrm{y}}_{\mathrm{m}}\right)}^2}{{\displaystyle \sum }\left({\mathrm{y}}_{\mathrm{m}}-{\mathrm{y}}_{\mathrm{m}\_\mathrm{mean}}\right)}$$

(1)

$$\mathrm{Percent}\mathrm{Error}\left(\mathrm{PE}\right)=\frac{{\mathrm{y}}_{\mathrm{e}}-{\mathrm{y}}_{\mathrm{m}}}{{\mathrm{y}}_{\mathrm{m}}}\times 100\%$$

(2)

$$\mathrm{RMSE}=\sqrt{\frac{{\displaystyle \sum }{\left({\mathrm{y}}_{\mathrm{e}}-{\mathrm{y}}_{\mathrm{m}}\right)}^2}{\mathrm{n}}}$$

(3)

$$\mathrm{rRMSE}=\frac{100}{{\mathrm{y}}_{\mathrm{m}\_\mathrm{mean}}}\sqrt{\frac{{\displaystyle \sum }{\left({\mathrm{y}}_{\mathrm{e}}-{\mathrm{y}}_{\mathrm{m}}\right)}^2}{\mathrm{n}}}$$

(4)

$$\mathrm{MAE}=\frac{{\displaystyle \sum }\left|{\mathrm{y}}_{\mathrm{e}}-{\mathrm{y}}_{\mathrm{m}}\right|}{\mathrm{n}}$$

(5)

where y_e is the estimated value, y_m is measured value, y_{m_mean} is the mean of y_m, and n is the number of samples.

2.2. Classic Methods of Estimating H_i

Two methods have mainly been used for estimating H_i. Method 1 is based on knowledge of astronomy and geometry according to the following procedure:

Step 1: Calculate H₀ using Equation (6) [12]:

$${\mathrm{H}}_0=\frac{24\times 3600}{\mathsf{\pi }}{\mathrm{G}}_{\mathrm{sc}}(1+0.033\cos \frac{360{\mathrm{n}}_{\mathrm{day}}}{365})\times (\cos \mathsf{\phi }\cos \mathsf{\delta }\sin {\mathsf{\omega }}_{\mathrm{s}}+\frac{\mathsf{\pi }{\mathsf{\omega }}_{\mathrm{s}}}{180}\sin \mathsf{\phi }\sin \mathsf{\delta })$$

(6)

where G_sc is solar constant, G_sc = 1367 Wm⁻², ϕ is the latitude of the location and n_day is the day number of the year, counted from 1 January, and δ and ω_s are the daily solar declination and sunset hour angle, respectively [22]:

$$\mathsf{\delta }=23.45\sin [\frac{360\left({\mathrm{n}}_{\mathrm{day}}+284\right)}{365}]$$

(7)

$${\mathsf{\omega }}_{\mathrm{s}}={\cos }^{-1}\left(-\tan \mathsf{\phi }\tan \mathsf{\delta }\right)$$

(8)

Step 2: Calculate K_t according to H_o and H_out via Equation (9) [22]:

$${\mathrm{K}}_{\mathrm{t}}=\frac{{\mathrm{H}}_{\mathrm{out}}}{{\mathrm{H}}_0}$$

(9)

Step 3: Decompose H_out into H_b and H_d according to K_t [8,13] using Equation (10):

$$\frac{{\mathrm{H}}_{\mathrm{d}}}{{\mathrm{H}}_{\mathrm{out}}}=\{\begin{array}{cc}0.95& {\mathrm{K}}_{\mathrm{t}}<0.175\\ 0.9698+0.4353{\mathrm{K}}_{\mathrm{t}}-3.4499{\mathrm{K}}_{\mathrm{t}}^2+2.1888{\mathrm{K}}_{\mathrm{t}}^3& 0.175<{\mathrm{K}}_{\mathrm{t}}<0.775\\ 0.26& {\mathrm{K}}_{\mathrm{t}}>0.775.\end{array}$$

(10)

Step 4: Calculate H_c via Equation (11) [15,17]:

$${\mathrm{H}}_{\mathrm{c}}={\mathrm{H}}_{\mathrm{c}}{\mathsf{\tau }}_{\mathrm{b}}+{\mathrm{H}}_{\mathrm{d}}{\mathsf{\tau }}_{\mathrm{d}}=\left({\mathrm{H}}_{\mathrm{out}}-{\mathrm{H}}_{\mathrm{d}}\right){\mathsf{\tau }}_{\mathrm{b}}+{\mathrm{H}}_{\mathrm{d}}{\mathsf{\tau }}_{\mathrm{d}}$$

(11)

where τ_b is film transmittance to H_b and τ_d is film transmittance to H_d. The values of τ_b and τ_d are 0.88 and 0.65 in the experimental greenhouse, respectively.

Method 2 [19] is based on Equation (12):

$${\mathrm{H}}_{\mathrm{c}}={\mathrm{H}}_{\mathrm{out}}{\mathsf{\tau }}_{\mathrm{g}}$$

(12)

Method 2 is simpler than Method 1, but as shown in Figure S2, the mean and variance τ_g of the experimental greenhouse was different from that in another study [19], so Method 1 was adopted in this study for comparison.

In both methods above, film transmittance and global transmittance are constant, but this does not reflect the reality. Some studies proved global transmittance changed with the incidence angle of the sun [24,25] and film transmittance changed due to aging and deposition [26]. So, estimating solar radiation merely according to fixed transmittance is not reliable and is prone to error.

2.3. Tomato Water Requirement Calculation

Carmassi’s equation was dedicated to calculating the water requirement of tomatoes according to only two meteorological parameters: solar radiation and temperature. Carmassi’s equation is calculated as follows:

Step 1: Calculate leaf area index (LAI) according to Equation (13) [33]:

$$\{\begin{array}{c}\mathrm{LAI}=-\mathrm{a}+\frac{\mathrm{b}+\mathrm{a}}{1+\exp [\left(\mathrm{c}-\mathrm{GDD}\right)/\mathrm{d}]}\\ \mathrm{GDD}={\sum }_{\mathrm{Start}\_\mathrm{day}}^{\mathrm{Stop}\_\mathrm{day}}\left({\mathrm{T}}_{\mathrm{avg}}-8\right)\end{array}$$

(13)

where a, b, c, and d are regression constants, a = 0.335, b = 4.803, c = 755.3, and d = 134.7; GDD is tomato growing degree days; T_avg is indoor daily average temperature, °C; Start_day presents the sowing date and Stop_day presents the date when cultivation ends. Something to be pointed out is that GDD is taken as dimensionless in LAI’s computation. The GDD values are shown in Figure S3, and the values of GDD on 1 and 26 December were 1252 and 1340, respectively.

Step 2: Calculate extinction factor k according to Equation (14) [33]:

$$\frac{{\mathrm{H}}_{\mathrm{up}}}{{\mathrm{H}}_{\mathrm{down}}}=\exp \left(-\mathrm{k}\times \mathrm{LAI}\right)$$

(14)

where H_up and H_down are solar radiation above and below the canopy, respectively; H_up and H_down were measured using two pyranometers placed horizontally at 2.5 and 1.5 m above the ground. During the experimental days, k was 0.69.

Step 3: Calculate the water requirement of tomatoes according to Equation (15) [33]:

$${\mathrm{V}}_{\mathrm{u}}=\mathrm{A}\left[1-\exp \left(-\mathrm{k}\times \mathrm{LAI}\right)\right]\frac{{\mathrm{R}}_{\mathrm{i}}}{{\mathsf{\lambda }}^{*}}$$

(15)

where A = 0.946, B = 0.188, λ* is the latent heat of vaporization (2.45 MJ kg⁻¹), and R_i is the energy intercepted by canopy (MJ m⁻² day⁻¹). R_i is calculated as [33]:

$${\mathrm{R}}_{\mathrm{i}}={\displaystyle \sum _{{\mathrm{T}}_{\mathrm{start}}}^{{\mathrm{T}}_{\mathrm{stop}}}}{\mathrm{H}}_{\mathrm{i}}{\mathrm{T}}_{\mathrm{s}}$$

(16)

where T_s is the sample interval, and T_start and T_stop are start time and stop time of water requirement calculation, respectively. T_start was confirmed by the time when H_s first rose above 10 Wm⁻² and T_stop was confirmed by the time when the H_s first fell to 0 Wm⁻². On sunny days during the experiment, T_start was 08:00 and T_stop was 16:00.

2.4. LMS Filter

According to the filter refresh algorithm, some kinds of adaptive filters can be used [41], such as the least mean squares (LMS) filter [35], recursive least squares (RLS) filter [42], least mean p-norm (LMP) filter [43], normalized LMP (NLMP) filter [44], least mean absolute deviation (LMAD) filter [44], and normalized LMAD (NLMAD) filter [44]. The basic diagram of adaptive filter is shown in Figure S4. Among these, the LMS filter’s resource consumption is low, making it suitable for applications in resource-constrained systems such as microcontrollers, which have only smaller RAM and run at a lower speed [45]. So, we only focused on the LMS filter’s performance.

The LMS filter updates its filter coefficients according to least mean squares algorithm; computation proceeds according to Equations (17)–(19) [36]:

$$\mathrm{y}\left(\mathrm{n}\right)={\mathrm{x}}^{\mathrm{T}}\left(\mathrm{n}\right)\mathrm{w}\left(\mathrm{n}\right)$$

(17)

$$\mathrm{e}\left(\mathrm{n}\right)=\mathrm{x}\left(\mathrm{n}\right)-\mathrm{d}\left(\mathrm{n}\right)$$

(18)

$$\mathrm{w}\left(\mathrm{n}+1\right)=\mathrm{w}\left(\mathrm{n}\right)+2\mathsf{\mu }\mathrm{e}\left(\mathrm{n}\right)\mathrm{x}\left(\mathrm{n}\right)$$

(19)

where μ is the convergence factor of LMS filter, and w(n + 1) is the filter coefficient in the next iteration.

μ, which is related to convergence speed and approximate precision, is an important factor in the LMS filter. In addition, a smaller value of μ leads to higher approximate precision and lower convergence speed, and vice versa. The LMS filter is fully analyzed according to Equations (20)–(22) [37,38]:

$$\mathrm{E}\left[\mathrm{w}\left(\mathrm{n}+1\right)\right]=\mathrm{E}\left[\mathrm{w}\left(\mathrm{n}\right)\right]+2\mathsf{\mu }\mathrm{E}\left[\mathrm{e}\left(\mathrm{n}\right)\mathrm{x}\left(\mathrm{n}\right)\right]=\mathrm{E}\left[\mathrm{w}\left(\mathrm{n}\right)\right]+2\mathsf{\mu }{\mathrm{R}}_{\mathrm{dx}}-2\mathsf{\mu }{\mathrm{R}}_{\mathrm{xx}}\mathrm{E}\left[\mathrm{w}\left(\mathrm{n}\right)\right]$$

(20)

where R_dx = E[d(n)x(n)] is the cross correlation matrix of the input and desired signals, R_xx = E[x(n)x^T(n)] is the autocorrelation matrix of the input signal, and R_xx is at least a positive semi-definite matrix, so a normalized orthogonal matrix Q sets up Equation (21) [37,38]:

$${\mathrm{R}}_{\mathrm{xx}}=\mathrm{Q}\mathsf{\Lambda }{\mathrm{Q}}^{-1}=\mathrm{Q}\mathsf{\Lambda }{\mathrm{Q}}^{\mathrm{T}}$$

(21)

where the modal matrix Q is orthonormal. The columns of Q, which are the eigenvectors of R_xx, are mutually orthogonal and normalized. Notice that Q⁻¹ = Q^T, Λ is the spectral matrix and all its elements are zero except for the main diagonal, whose elements are the set of eigenvalues of R_xx, which are presented as, λ₁, λ₂, λ₃, …, λ_L. According to Equation (22), Λ has the following form [37,38]:

$$\mathsf{\Lambda }=\left[\begin{array}{ccccccc}{\mathsf{\lambda }}_1& & 0& & 0& & 0\\ & & & \cdots & & & \\ 0& & {\mathsf{\lambda }}_2& & 0& & 0\\ & ⋮& & \ddots & & ⋮& \\ 0& & 0& & {\mathsf{\lambda }}_{\mathrm{L}-1}& & 0\\ & & & \cdots & & & \\ 0& & 0& & 0& & {\mathsf{\lambda }}_{\mathrm{L}}\end{array}\right]$$

(22)

The eigenvalues of R_xx are all real and greater or equal to zero, and μ can be calculated according to [37,38]:

$$0<\mathsf{\mu }<\frac{1}{{\mathsf{\lambda }}_{\max }}$$

(23)

where λ_max is the maximum eigenvalue of R_xx.

A tightly-constrained equation about μ is [37,38]:

$$0<\mathsf{\mu }<\frac{1}{\mathrm{tr}({\mathrm{R}}_{\mathrm{xx}})}$$

(24)

where tr(R_xx) is the trace of R_xx. And tr(R_xx) is calculated as [37,38]:

$$\mathrm{tr}\left({\mathrm{R}}_{\mathrm{xx}}\right)=\mathrm{L}{\mathrm{R}}_{\mathrm{xx}}\left(0\right)=\mathrm{L}\mathrm{E}\left[{\mathrm{x}}^2\left(\mathrm{n}\right)\right]$$

(25)

where L is the number of taps of the LMS filter.

Equations (24) and (25) prove that the upper bound of μ is the power of the input signal, which can be calculated easily in applications.

2.5. Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT), and Pass Band Characteristics of Filters

2.5.1. DFT and FFT

DFT is a fundamental tool in signal processing applications, and FFT [46] is the fast algorithm of DFT. The characteristics in the frequency domain of the interested signals are obtained by DFT and FFT. DFT is calculated according to [46]:

$$\mathrm{X}\left(\mathrm{k}\right)={\displaystyle \sum _{\mathrm{n}=0}^{\mathrm{N}-1}}\mathrm{x}\left(\mathrm{n}\right){\mathrm{e}}^{-\mathrm{j}\frac{2\mathsf{\pi }}{\mathrm{N}}\mathrm{nk}}$$

(26)

where X(k) is the frequency domain values of time series x(n), N is the calculation length, and both n and k range from 0 to N − 1.

2.5.2. Filter Pass Band Characteristic

The pass band characteristic of a filter is the response effect of the amplitude-frequency characteristic of a filter to the signal of a different frequency. It can be calculated using Equation (26). Filters are classified as high pass, low pass, band pass, notch, and all pass according to the band characteristics. For example, a low pass filter passes low-frequency components and attenuates high-frequency components, so the output signal is always smoother than the input signal.

2.6. Proposal Methods and Evaluation Procedures

We focused on estimating H_i using H_out recorded from a weather station, which is basic equipment in many growing areas in China.

The flow chart of data processing is shown in Figure 2. Firstly, H_out and H_s were obtained from a weather station and the sensors inside the greenhouse. H_c was calculated according Equations (6)–(11). Secondly, H_out and H_c were used as the x(n) and d(n) input signals for the LMS filter, respectively; therefore, the output signal of the LMS filter was the estimation of H_i and is presented as H_f. The required water volume for tomato, according to H_c, H_f, and H_s, which are presented as V_c, V_f, and V_s, respectively, were calculated via Equations (13)–(16).

The performance of curve fitting, including H_f–H_s and H_c–H_s, were evaluated to analyze each solar radiation estimation method. The performance of curve fitting, including V_f–V_s. and V_c–V_s, were evaluated to analyze water requirement according to each solar radiation estimation method. Both of the evaluations were based on the equations proposed in Section 2.1.3. All data in this study were processed and all figures were drawn using Python 3.7 (Python Software Foundation, Wilmington, DE, USA). The data of H_s and H_out is available in Data file S1 and Data file S2 respectively. And the program is available in Program file S1.

3. Results and Discussion

3.1. Determination of μ and L

The length (L) of the LMS filter varies in different applications and ordinary lengths are 8, 9, 64, and 128, but the sample interval in greenhouse applications always ranges from 1 min to 1 h or more. Given a sample interval of 15 min, L = 128, introduces a time delay at more than 24 h, so a smaller number of taps is preferred. In this study, L was 9.

According to Equations (24) and (25), the upper bound of μ was computed. In experimental days, the maximum of E[x²(n)] was 145. To avoid computational overflow in the program, each value of H_out was multiplied by 0.01. So, the upper bound of μ was 0.0007.

In the range of 10⁻⁵ to 5 × 10⁻⁴, six values were chosen for evaluation to determine the exact value of μ. According to the procedure proposed in Section 2.6, H_f and H_c were computed according to H_out collected between 07:00 and 17:00 on experimental days. Then, the performance was evaluated; notably, in the three taps of the left shift of H_f for compensation of computational delay. In the latter parts of this study, the length and direction of shift were constant unless otherwise mentioned.

As shown in

Table 1. Evaluation parameters computed by different values of μ.

Evaluation Parameter	M
	10−5	2 × 10−5	5 × 10−5	10−4	2 × 10−4	5 × 10−4
R2	0.3959	0.8031	0.8384	0.8254	0.7554	0.6010
RMSE (Wm−2)	72.7	41.5	37.6	39.1	46.2	59.0
rRMSE (%)	44.6	25.5	23.1	23.4	28.4	36.2
MAE (Wm−2)	58.7	30.2	25.3	25.8	33.6	45.9

, the distribution of evaluation parameters showed a single peak, and R², RMSE, rRMSE, and MAE reached their minimum when μ was 5 × 10⁻⁵; so, in this study, μ was determined.

3.2. Estimation of H_i and Tomato Water Requirement Calculation under Sunny, Partly Cloudy and Overcast Conditions

3.2.1. Estimation of H_i

Computations were performed using data from six days according to the procedure in Section 2.6. The curves of H_out, H_c, H_f, and H_s are shown in Figure 3. Two days were sunny, 11 and 12 December. In Figure 3c, the curve of H_s fluctuated obviously near its peak value, whereas the curve of H_f, which tended to have a half-wave sinusoidal shape, was smooth. In other words, the fluctuation range of H_f was very narrow near noon. The fluctuations of H_out and H_c ranged between H_s and H_f. We observed quick changes in the curves of H_out and H_c near their peak value at about 11:00 each day and the quick changes were introduced by a metal bar installed nearby the weather station. So, the weather station measurements were temporarily disturbed by the shadow of the bar. In contrast, the curve of H_f proved to be immune to this temporary disturbance.

The weather was partly cloudy on 5 and 10 December. According to the curves of H_out, H_c, H_f, and H_s in Figure 3b, as the cloud cover increased after 14:00 on 5 December, the curves of H_out, H_c, and H_s fluctuated considerably, and these fluctuations made the curves rougher than the H_f curve. The overall trend of the curves was H_f > H_c > H_s during this time. The fluctuations of the curves of H_out, H_c, and H_s on 10 December were more obvious than on 5 December. However, the H_f curve was smoother than the other curves and fluctuation range was narrow on 10 December. In addition, the shape of H_f on both days distorted gradually.

The day was overcast on 2 and 8 December. Due to the lower outer solar radiation in the morning on these days, the blanket was rolled up later than usual. So a distinguishing rising edge, after which H_s curve was close to the H_c and H_f curves, rapidly appeared on the H_s curve in Figure 3a. The operation of the blanket resulted in a T_start at 11:00 and 10:00 on 2 and 8 December, respectively. According to the H_out, H_c, H_f, and H_s curves in Figure 3c, as the cloud cover increased after 13:00 on 2 December, the curves of H_out, H_c, and H_s fluctuated obviously, making the curves rougher than the H_f curve. The fluctuations of H_out, H_c, and H_s curves on 8 December proved to be more obvious than on 2 December. However, the curve of H_f was smoother than other curves and the fluctuation range was narrow on 8 December. The shapes of H_f on both days were distorted and were no longer half-wave sinusoidal.

3.2.2. Tomato Water Requirement

According to Equation (16), R_i was computed using H_c, H_f, and H_s, which are presented as R_c, R_f, and R_s, respectively. T_start values were 08:15, 08:00, 08:00, 08:00, 11:00, 10:00 and T_stop values were 16:00, 16:15, 15:45, 16:15, 15:45, 16:00 on 11, 12, 5, 10, 2, 8 December, respectively. Then, V_c, V_f, and V_s. were calculated using Equation (15) as shown in

Table 2. Evaluation parameters of tomato water requirements calculated according to different estimation methods of H_i under different conditions.

Date	Calculated Value
	Vc (mL·Plant−1)	Vs (mL·Plant−1)	Vf (mL·Plant−1)	PEfs (%)	PEcs (%)
2 December	301.6	248.6	307.2	23.5	21.3
8 December	280.2	234.0	288.9	23.3	23.5
5 December	587.1	525.9	557.5	6.0	11.6
10 December	492.0	446.3	486.3	9.0	10.2
11 December	617.7	574.9	585.4	1.8	7.4
12 December	645.1	596.7	602.8	1.0	8.1

. The PE of V_f–V_s. and V_c–V_s, presented as PE_fs and PE_cs, respectively, calculated via Equations (27) and (28), respectively, are also shown in Table 2.

$${\mathrm{PE}}_{\mathrm{fs}}=\frac{{\mathrm{V}}_{\mathrm{f}}-{\mathrm{V}}_{\mathrm{s}}}{{\mathrm{V}}_{\mathrm{s}}}\times 100\%$$

(27)

$${\mathrm{PE}}_{\mathrm{cs}}=\frac{{\mathrm{V}}_{\mathrm{c}}-{\mathrm{V}}_{\mathrm{s}}}{{\mathrm{V}}_{\mathrm{s}}}\times 100\%.$$

(28)

The data on 11 and 12 December in Table 2 show that V_s. < V_f < V_c and PE_fs < PE_cs. The PE_fs values on both days were smaller than 2% but the values of PE_cs tended to be more than 7%. The data show PE_fs < PE_cs on 5 December and PE_fs > PE_cs on 10 December. The difference in PE_fs and PE_cs was 5.6% on 5 December and 1.2% on 10 December. The data on 2 and 8 December show that PE_fs = PE_cs.

3.3. Overall Performance of Estimation of H_i and Tomato Water Requirement Calculation

3.3.1. Overall Performance of Estimation of H_i

The scatter plot of H_f and H_s is shown in Figure 4a and the fitting function of H_f and H_s was y = 0.7634x + 50.58. The scatter plot of H_c and H_f is shown in Figure 4b and the fitting function was y = 0.9376x + 33.04. The latter analysis focuses on the performance of each method under sunny conditions, which dominated during the experimental period.

In Figure 4a,b, when some points of H_f increased above 30 Wm⁻², H_s was nearly 0 Wm⁻². These larger values are mainly attributed to the postponed blanket operations. Differences of H_s and estimated values including H_f and H_c appeared to be large because H_f and H_c had reached higher values when H_s was still near 0 Wm⁻².

As shown in Figure 4a, H_s increased to about 200 Wm⁻² during 09:00–10:00 and 14:00–15:00 when H_s stayed below H_f and H_c.

In Figure 4a, as H_s rose above 200 Wm⁻², the number of points of H_f < H_s also increased gradually. When H_s rose above 300 Wm⁻², the number of points of H_f < H_s was greater than the number of points of H_f > H_s. The details of the curves during 10:00–14:00 on 11 and 12 December indicates as H_s rose above 200 Wm⁻², the increasing speed rose so the curve of H_s gradually stayed above the curve of H_f. Hence, the increasing number of points of H_f < H_s in Figure 3c contributed to the increasing rising speed of H_s. The peaks in the H_s curve on these two days in Figure 4a reached more than 300 Wm⁻²; so, when H_s > 300 Wm⁻², the number of points of H_f < H_s dominated. Oscillation was observed when H_s rose above 200 Wm⁻², so some H_f > H_s points were introduced.

In contrast, in Figure 4b, the number of points of H_c < H_s changed within a small variation range, and H_s stayed above H_c only near its peak value. When H_s rose above 250 Wm⁻² (Figure 4b), some points of H_c < H_s occurred due to the disturbance caused by the metal bar near the outer weather station.

In summary, the overall trend in Figure 4a was H_c > H_f > H_s when H_s < 200 Wm⁻², and H_c > H_s > H_f when H_s > 200 Wm⁻². According to Figure 4a,b, the fluctuation range of H_f was the smallest among the four curves.

The pass band characteristics of LMS filters in Section 3.2 are shown in Figure 5. And the FFTs of H_out in Section 3.2 are also shown in this figure. The pass band characteristics of LMS filters in these six days were all low pass. The low pass characteristic made H_f smoother than H_out, which was the smoothest among H_out, H_c, and H_s. The low pass characteristic also made the LMS filter immune to temporary disturbances, which were common in many greenhouse applications.

We found the LMS filter is not applicable if research focuses on the fluctuations in solar radiation due to its low pass characteristic.

3.3.2. Overall Performance of Tomato Water Requirement Calculation

According to the procedure in Section 3.2.2, we calculated the V_c, V_f, and V_s. of each day during the experimental period, as shown in Figure 6. The overall trend of these three values was the same: they all increased when cloud cover decreased. V_f was close to V_s. when cloud cover was lower, and to V_c when cloud cover was higher.

The scatter plots of V_f–V_s. and V_c–V_s. are shown in Figure 7, and the fitting functions of V_f–V_s. and V_c–V_s. are y = 0.8470x + 102.2 and y = 0.9656x + 74.6, respectively. As shown in Figure 7, V_f was close to V_c when V_s. < 400 mL and V_f was close to V_s. when V_s. > 400 mL. The difference between V_f and V_s. decreased as V_s. increased but, conversely, the trend in the difference between V_c and V_s. was not the same as V_f–V_s. When V_s. rose above 600 mL plant⁻¹, the values of V_f and V_s. were almost the same; when V_s. dropped below 400 mL plant⁻¹, the values of V_f and V_c were almost the same.

PE_fs, PE_cs, and K_t are shown in Figure 8. We found an opposite trend between PEs and K_t. The lowest value of PE_fs was lower than that of PE_cs. Among 2, 8, 14, 22, and 24 December, in which K_t was greater than 40%, PE_fs was close to PE_cs and PE_fs < PE_cs on other days. Among 1, 2, 11, and 12 December, PE_fs was almost 0.

4. Discussions

The four evaluation parameters of H_f–H_s and H_c–H_s on days in Section 3.2 were computed according to Equations (1) and (3)–(5), as shown in

Table 3. Evaluation parameters of solar radiation estimation under different weather conditions.

Date		Evaluation Parameter
		R2	RMSE (Wm−2)	rRMSE (%)	MAE (Wm−2)
2 December	Hf–Hs	0.4123	57.3	47.5	37.2
	Hc–Hs	0.6535	44.0	36.4	29.9
8 December	Hf–Hs	0.1153	55.5	59.6	38.6
	Hc–Hs	0.7767	27.9	30.0	11.8
5 December	Hf–Hs	0.9056	33.5	19.6	27.4
	Hc–Hs	0.8972	34.8	20.4	28.0
10 December	Hf–Hs	0.7909	43.9	31.5	34.4
	Hc–Hs	0.9393	23.7	16.7	18.5
11 December	Hf–Hs	0.9630	18.1	9.8	14.3
	Hc–Hs	0.6231	50.3	31.6	50.3
12 December	Hf–Hs	0.9525	21.1	10.7	15.5
	Hc–Hs	0.6147	51.0	30.6	51.0

. Studies proved that estimation of solar radiation in hourly intervals was good enough if rRMSE ranged from 34% to 41% [16,47,48], so conclusions can be drawn as follows. Under sunny conditions, the estimation of H_f–H_s was more accurate than H_c–H_s and both of the methods were good enough on 11 and 12 December. Under partly cloudy conditions, the estimations of H_f–H_s and H_c–H_s were all good enough, with rRMSE in both cases below 41% on 5 December. Due to the rRMSE of H_f–H_s being above 41%, only the estimation of H_c–H_s was good enough on 10 December. Under overcast conditions, the estimations of H_f–H_s and H_c–H_s were all poor, with rRMSE above 41% on 2 December and rRMSE of H_f–H_s above 41%. Only estimation of H_c–H_s was acceptable on 8 December.

Badescu et al. (2013, 2014) and Son et al. (2018) prove the estimation accuracy of solar radiation decreases with increasing cloud cover [47,48,49]. Additionally, the decreasing of estimation accuracy lies in the increasing portion of diffuse solar radiation, which is always measured at a lower accuracy [48]. Since outer solar radiation is decomposed into beam and diffuse parts in our study, estimation accuracy is also affected if cloud cover increases. However, with the LMS filter being introduced, estimation accuracy of ours is affected less than in other methods because of the LMS filter’s low pass characteristic. In contrast, the estimation accuracy of H_c–H_s proved to be the best on partly cloudy days, the worst on sunny days, and medium on overcast days. In the study of Huang et al. (2019) and Tong et al. (2017) the fluctuation of H_i tends to be larger at noon [14,25], and the same trend was found in our study. It is a common phenomenon that large fluctuations appear in the data on a sunny noon, but to our knowledge the mechanism of this phenomenon needs further analysis.

The evaluation parameters of H_f–H_s, H_c–H_s, V_f–V_s. and V_c–V_s. during experimental days computed according to Equations (1) and (3)–(5) are shown in

Table 4. Evaluation parameters of solar radiation estimation and tomato water requirement calculation.

Evaluation Parameter	Hf–Hs	Hc–Hs	Evaluation Parameter	Vf–Vs	Vc–Vs
R2	0.8384	0.8084	R2	0.9123	0.7598
RMSE (Wm−2)	37.6	40.9	RMSE (mL·plant−1)	40.4	64.8
rRMSE (%)	23.1	25.1	rRMSE (%)	8.8	14.1
MAE (Wm−2)	25.4	29.6	MAE (mL·plant−1)	31.5	58.6

. The results indicate that estimation of H_f–H_s was more accurate than H_c–H_s, so the proposed method proves to be more accurate than astronomy and geometry method [16,17]. Additionally, rRMSE of H_f–H_s is within the range between 34% and 41% [16,48], so this method is acceptable in solar radiation estimation. In the study of Ahamed et al. (2018), the evaluation parameters of solar radiation estimation proved to be R² = 0.71, RMSE = 68.34 Wm⁻², and rRMSE = 30.54% in contrast [16]. Moreover, film transmittances are still important factors in our method and their values are not constant in applications, therefore, the accuracy of our model is affected by transmittances variations.

In the studies of Gueymard and Myers (2008), data filtering is considered as an important factor to improve solar radiation sensing accuracy [50]. In our study, the LMS filter performs low pass filtering and the operation of Equation (16) is also low pass filtering. So, the calculation of V_f performs a two-stage low pass filtering and the evaluation parameters of V_f–V_s. tend to be even better. The results of our study show the impacts introduced by fluctuations, especially in cloudy and overcast weather conditions. The results of our study have also proved the importance of filtering in both solar radiation estimation and water requirement calculation.

However, the following points need to be fully improved. Firstly, the accuracy of the model is affected by cloud cover. For better performance, the new methods in the measure/estimate diffuse of solar radiation with more accuracy are to be developed. Secondly, real-time transmittance computation should be an important part of the model for better accuracy. Thirdly, we conduct research supposing the CSG is at east-west orientation with no incline. However, some CSGs are built with an incline due to terrain restrictions. Finally, long term analysis is to be conducted for better use of our model.

In our study, with no indoor sensors needed, the cost of solar radiation sensing is little. In the literature of Villarrubia et al. (2017), the cost of an irrigation system is acceptable when cost amounts to 100€/250 m² [51]. Therefore, our research reduces the expense of solar radiation sensing and is contributing to the promotion of precision agriculture in CSGs.

5. Conclusions

In this study, solar radiation inside a CSG was estimated based on an LMS filter, and then the tomato water requirement was calculated according to the estimation data. The performance of both solar radiation estimation and water requirement calculation were compared to the corresponding methods based on knowledge of astronomy and geometry.

The results showed that the fitting function of estimation data based on the LMS filter and data collected from sensors inside the greenhouse was y = 0.7634x + 50.58, with the evaluation parameters of R² = 0.8384, rRMSE = 23.1%, RMSE = 37.6 Wm⁻², and MAE = 25.4 Wm⁻². The fitting function of the water requirement calculated according to the proposed method and data collected from sensors inside the greenhouse was y = 0.8550x + 99.10 with the evaluation parameters of R² = 0.9123, rRMSE = 8.8%, RMSE = 40.4 mL plant⁻¹, and MAE = 31.5 mL plant⁻¹.

The low pass characteristic of the LMS filter leads to the following two results. First, the performance of the proposed method is more accurate than that of the contrastive method. Second, the proposed method performs well on sunny days but performs worse on party cloudy and overcast days. In addition, LMS is easy to be performed in microcontrollers. Therefore, the method is proved to be efficient and low cost in both solar radiation estimation and tomato water requirement calculation. However, it is not applicable if focusing on the fluctuations of solar radiation inside a greenhouse.