1. Introduction

2. Experimental study of PVWPS

PVWPS design

To evaluate PVWPS standalone performance, a test bench (Figure 1) is developed. Five solar panels (SOLARWATT M18-36 GET EK; 18W_p) were used to power three DC diaphragm pumps (SHURflo, 12 VDC, P=30W). Solar converter (Victron Blue Solar 48/24/12V) was required between solar panels and pumps. These pumps mounted in parallel to maximize the amount of water pumped compared to installed PVWPS power and potential irradiance during irrigation.

Figure 1. Established photovoltaic DC pumping system in laboratory.

Figure 1. Established photovoltaic DC pumping system in laboratory.

Output currents of solar panel and converter were measured using current sensor (ACS712), The operating pressure was gauged with an analog sensor (Keller PR 21Y, maximum pressure = 5 bar) positioned at the pump outlet to assess their flow rates. This data collection occurred over a single day, spanning from 8:00 am to 6:00 pm. The data obtained through NI 6009 were meticulously processed to systematically pick five data points per second. This was done to evaluate the electrical and hydraulic efficiencies, which were influenced by the changes in the solar irradiance and the operating points of the pumps.

P_pv= V_pv×I_pv;

P_mppt = V_mppt×I_mppt

(1)

I_pv = (a₁×V_s1) + b₁;

I_mppt = (a₂×V_s2) + b₂

(2)

Where, P_pv, P_mppt, a₁, b₁, a₂, b₂, V_s1 and V_s2 are power of PV panel, power of MPPT converter, coefficients of current calibration curves and Voltages sensors outputs, respectively.

Pressure and flow at the pump outlet were manually recorded several times to verify the accuracy of calibration test (Figure 2).

Figure 2. Setup calibration of the pump; and the calibration curve (Pressure versus Voltage).

Figure 2. Setup calibration of the pump; and the calibration curve (Pressure versus Voltage).

Irradiance measurements were obtained from a meteorological station situated at the research center, precisely located at coordinates X: –7.624233 and Y: 32.953487. The irradiance data for March 20^th, served as the reference point (Figure 3).

Figure 3. Daily solar irradiance of March 20^th.

Figure 3. Daily solar irradiance of March 20^th.

3. Modeling of PVWPS

4. Results and Discussions

4.1. Results of Experimental Study

Standalone DC PVWPS performance was evaluated using a multi pumps approach. The system was monitored from 08:00 am to 06:00 pm showing its behavior. Taking into consideration irradiance and hydraulic power induced, three behaviors were obtained, the behavior of morning, mid-day and afternoon (Table 1)

Table 1. Daily performance of PVWPS according to irradiance change and portion of switched on pump (experimental test results).

Table 1. Daily performance of PVWPS according to irradiance change and portion of switched on pump (experimental test results).

Time	Irradiance (W/m²)	Number of operating Pumps	Mean Pressure (bar) (CV%)	Mean Flowrate (L/min) (CV%)	Mean Hydraulic Power (CV%)	Mean Hydraulic Yield% (CV%)
08. am	178	1	01.18 (16)	02.46 (08)	04.65 (24)	30 (19)
	197	2	00.45 (20)	03.28 (11)	02.41 (28)	13 (31)
	213	3	00.26 (19)	03.61 (10)	01.51 (27)	07 (26)
09. am	331	1	01.24 (17)	02.43 (09)	04.74 (24)	33 (18)
	350	2	00.71 (20)	02.91 (10)	03.44 (28)	14 (23)
	368	3	00.45 (22)	03.29 (12)	02.39 (32)	07 (32)
10. am	515	1	02.63 (27)	03.19 (14)	02.51 (37)	06 (28)
	530	2	00.93 (37)	02.70 (19)	04.00 (56)	13 (38)
	544	3	00.60 (32)	03.08 (17)	02.96 (45)	06 (40)
11. am	662	2	00.84 (37)	02.83 (19)	03.39 (55)	12 (42)
	679	3	00.65 (35)	03.03 (20)	03.11 (47)	07 (38)
12. pm	749	2	00.76 (24)	02.91 (13)	03.39 (32)	12 (31)
	758	3	00.72 (26)	02.93 (15)	03.38 (36)	06 (38)
01. pm	777	3	00.78 (31)	02.87 (17)	3.55 (41)	34 (38)
02. pm	754	2	01.12 (33)	02.61 (18)	04.25 (46)	13 (37)
	748	3	00.95 (28)	02.74 (16)	03.90 (40)	06 (38)
03. pm	670	2	01.08 (32)	02.64 (18)	04.16 (45)	12 (38)
	508	3	00.94 (27)	02.71 (15)	03.96 (37)	06 (36)
04. pm	534	1	01.97 (29)	02.69 (16)	03.97 (40)	12 (35)
	517	2	01.00 (29)	02.67 (16)	04.08 (40)	09 (37)
	500	3	00.45 (38)	03.30 (22)	02.36 (53)	06 (39)
05. pm	344	1	00.80 (53)	03.01 (31)	03.16 (73)	24 (48)
	326	2	00.59 (32)	03.11 (19)	02.89 (44)	10 (38)
	312	3	00.36 (30)	03.45 (19)	01.95 (38)	06 (38)
06. pm	37	1	00.62 (28)	03.06 (16)	03.03 (38)	25 (35)
	34	2	00.20 (25)	03.74 (13)	01.16 (35)	09 (47)
	42	3	00.04 (34)	04.05 (19)	00.29 (46)	02 (55)

Table 1 presents the system’s response to different levels of irradiance. In the morning hours (8:00 am, 9:00 am, 10:00 am), three distinct flow patterns are described, each associated with a specific number of active pumps required to adjust the system’s operating point. This adjustment is essential for optimizing system performance. In essence, achieving better performance depends on adapting the number of active pumps to match the actual irradiance levels. When the pressure requirements are met, an improved performance is linked to a specific count of engaged pumps.

However, at 11:00 am, 12:00 pm, 2:00 pm and 3:00 pm, the increase of pressure was observed as irradiance were important. Consequently, an increase of hydraulic power was obtained. At 11:00 am, 12:00 pm, 2:00 pm and 3:00 pm, there is no need to operate with one small pump, since using the two pumps meet maximum pressure (1.12 bar). The results obtained showed efficiency arranged from 6% to 13%. According to mid-day (1:00 pm) with high irradiance (777 W/m²), the system showed interesting performance (Around 34%). Indeed, the abundant irradiance at 777 W/m² enabled the system to run with the maximum number of pumps without causing any decline in pump efficiency, as previously demonstrated during the morning incident irradiance. One system behavior using three pumps was obtained, since use of a single small pump increases pressure and risks damaging the system.

At the afternoon, the system performance showed a behavior similar to the morning. This can be attributed to the insufficient incident irradiance in the range of 38-516 W/m². The findings reveal an efficiency range between 2% and 25%. When the pump is not operating under low levels of sunlight, adjustments in flow and pressure become necessary to optimize pump efficiency, aligning with the hydraulic equivalence model of the connected pumps. The system’s performance is noticeably influenced by the presence of low incident irradiance at 38 W/m², subsequently affecting pump efficiencies.

As per the pumping system’s response for each configuration of the number of active pumps, Table 1 demonstrates a significant variance in terms of efficiency. This table provides a summary of the independent system’s behavior throughout the day. Under low irradiance conditions, the system operates in normal mode (with all pumps active), resulting in lower yields. However, activating the pumps shifts the system’s operating point to an optimal position, compelling it to perform more efficiently (yielding higher results).

In contrast, during periods of high irradiance, the system operates in normal mode, maintaining consistent performance. It is worth noting that changes in irradiance negatively impact the system’s efficiency. This observed variability in efficiency is attributed to the standalone system’s behavior, as it lacks capacitors to regulate and stabilize the instantaneous relationship between flow and pressure, which is influenced by fluctuations in solar irradiance.

Hydraulic performance of the PVWPS was tested by three cases: The first case using one equivalent pump (30W), the second using two equivalent pumps (2*30W) and the last using three equivalent pumps (3*30W).

At low irradiance, first case results (Figure 4a–c) showed that the equivalent pump (30W) operated efficiently in low irradiance. In normal condition, results showed that the system could not operate efficiently (7%). But when we changed three pumps (90W) with one equivalent pump (30W), the efficiency of the system was reached (30%), with an average daily pumped water obtained was about 164 (L/h). This case (Use of one equivalent pump) allowed the system to operate 5 hours per day since, at high irradiance, use of a single equivalent pump increases pressure and risks damaging the system. (Figure 4a/ Case 2) showed that the system perform using two pumps (60W) with irradiance increasing. Consequently, the daily pumped water was increased and reached a maximum of 349 (L/h) (Figure 4c/ Case 2). Operating time of the system using two equivalent pumps (60W) was 5 hours (From 10:00 am 03:00 pm excluding 01:00 pm). At low irradiance the system showed a worse efficiency using three equivalent pumps (90W). Except a significant increase in yield (46%) was obtained at 01:00 pm when irradiance was at its maximum (777 W/m²) (Figure 4a/ Case 3). This case (use of three equivalent pumps (90W)) allowed the system to operate 1 hour. At high irradiance,the daily pumped water was about 516(L/h) (Figure 4c/ Case 3).

Figure 4. PVWPS behavior using: One equivalent Pump (Case 1); Two equivalent Pumps (Case 2); Three equivalent Pumps (Case 3): a). Hydraulic Efficiency (HE-G); (b). Flowrate (Q-G); (c). Pumped Volume (VP-G).

Figure 4. PVWPS behavior using: One equivalent Pump (Case 1); Two equivalent Pumps (Case 2); Three equivalent Pumps (Case 3): a). Hydraulic Efficiency (HE-G); (b). Flowrate (Q-G); (c). Pumped Volume (VP-G).

4.2. Modeling of PVWP System Using Experimental Data

PVWPS model

Simulations of the standalone PVWPS’s performance are conducted on an hourly basis. Growth curves were generated through the application of two non-linear models—the exponential model and the power model—utilizing two distinct software platforms: Python and R. These models were evaluated based on the coefficient of determination R²(Table 2)

Table 2. Coefficients values of the generated models linking all parameters with time.

Table 2. Coefficients values of the generated models linking all parameters with time.

Increasing Irradiance Trend
	Model	Software Tools	a	b	R²
Irradiance Vs Time	Exponential	Python	62,093	0,202	0,86
	Exponential	R	62,092	0,202	0,86
	Power	Python	2,874	2,219	0,9
	Power	R	2,874	2,22	0,9
Electrical Power Vs Time	Exponential	Python	5,185	0,203	0,88
	Exponential	R	5,185	0,203	0,88
	Power	Python	0,262	2,188	0,88
	Power	R	0,262	2,188	0,88
Hydraulic Power Vs Time	Exponential	Python	0,293	0,358	0,82
	Exponential	R	0,293	0,358	0,82
	Power	Python	0,001	3,917	0,62
	Power	R	0,001	3,917	0,62
Decreasing Irradiance Trend
	Model	Software Tools	a	b	R²
Irradiance Vs Time	Exponential	Python	22539,772	-0,251	0,82
	Exponential	R	22539,772	-0,251	0,82
	Power	Python	10939913,381	-3,679	0,78
	Power	R	3378000	-3,245	0,77
Electrical Power Vs Time	Exponential	Python	1328,821	-0,218	0,85
	Exponential	R	1328,793	-0,218	0,84
	Power	Python	295836,902	-3,209	0,81
	Power	R	298100	-3,21	0,81
Hydraulic Power Vs Time	Exponential	Python	8165,452	-0,423	0,95
	Exponential	R	8165,454	-0,423	0,95
	Power	Python	243140608,495	-6,158	0,95
	Power	R	12560000	-5,038	0,92

After an analysis of the generated models, it was observed that their outcomes exhibited a high degree of similarity. Nevertheless, the exponential model emerged as the top choice due to its superior performance according to the evaluation metrics R² (Figure 5).

Figure 5. a/ Fitting Curves of two models of Irradiance versus Time, b/ . Fitting Curves of two models of EP versus Time, c/ Fitting Curves of two models of HP versus Time.

Figure 5. a/ Fitting Curves of two models of Irradiance versus Time, b/ . Fitting Curves of two models of EP versus Time, c/ Fitting Curves of two models of HP versus Time.

Based on the fitting curves, the results have demonstrated comparable behavior between the two models, with a minor distinction observed primarily in relation to the exponential model (Table 3). The analytical model’s validation involved utilizing RMSE and MAE methods. The outcomes of the error assessments concerning irradiance, electrical power, and hydraulic power are consolidated in Table 3.

From the exponential models presented herein, the derivations of transfer functions become feasible, establishing interrelationships among various parameters such as hydraulic power versus electrical power, and electrical power concerning irradiance. The mathematical representation of these transfer functions is as follows:

• Electrical Power & Irradiance

  EP= 5,185*EXP(0,203*t)

  (7)

  G= 62,093*EXP(0,202*t)

  (8)

  G= EP* 11,975* EXP(-0,001*t)

  (9)
• Hydraulic Power & Electrical Power

  HP = 8165,452*EXP(-0,423*t)

  (10)

  EP= 5,185*EXP(0,203*t)

  (11)

  EP= HP*0.0006332 * EXP (0.626 * t)

  (12)

Where: EP, G, HP, t are Electrical Power, Irradiance, Hydraulic Power and time, respectively.

Table 3. Validation Model using RMSE and MAE Methods.

Table 3. Validation Model using RMSE and MAE Methods.

Increasing Irradiance Trend
	Time	Real Data	PP Exponential Model	Power Model
Irradiance Vs Time	08:00	196	313	290	Exponential Model
	09:00	350	382	377	RMSE	79,09
	10:00	530	468	476	MAE	73,83
	11:00	671	573	588	Power Model
	12:00	754	701	713	RMSE	66,68
	01:00	777	858	852	MAE	62,33
Electrical Power Vs Time	08:00	20,38	26,31	24,79	Exponential Model
	09:00	32,85	32,23	32,08	RMSE	05,97
	10:00	50,90	39,48	40,39	MAE	04,30
	11:00	41,72	48,36	49,76	Power Model
	12:00	59,84	59,25	60,19	RMSE	05,92
	01:00	72,71	72,59	71,71	MAE	04,25
Hydraulic Power Vs Time	08:00	04,84	05,14	03,45	Exponential Model
	09:00	05,01	07,35	05,47	RMSE	04,00
	10:00	16,69	10,51	08,26	MAE	3,15
	11:00	15,75	15,04	12,00	Power Model
	12:00	14,81	21,51	16,87	RMSE	5,75
	01:00	33,43	30,77	23,08	MAE	4,41
Decreasing Irradiance Trend
	Time	Real Data	PP Exponential Model	Power Model
Irradiance Vs Time	01:00	777	863	873	Exponential Model
	02:00	751	671	664	RMSE	111,05
	03:00	589	522	515	MAE	93,83
	04:00	517	406	406	Power Model
	05:00	327	316	325	RMSE	119,49
	06:00	38	2446	264	MAE	99,33
Electrical Power Vs Time	01:00	72,71	78,10	78,78	Exponential Model
	02:00	62,85	62,81	62,11	RMSE	07,94
	03:00	63,42	50,50	49,77	MAE	06,29
	04:00	46,40	40,61	40,46	Power Model
	05:00	31,37	32,66	33,31	RMSE	08,98
	06:00	13,18	26,26	27,73	MAE	07,24
Hydraulic Power Vs Time	01:00	33,43	33,40	33,59	Exponential Model
	02:00	19,49	21,88	21,28	RMSE	02,33
	03:00	18,95	14,33	13,92	MAE	01,77
	04:00	08,84	09,39	09,35	Power Model
	05:00	04,02	06,15	06,44	RMSE	02,47
	06:00	03,15	04,03	04,53	MAE	01,88

5. Conclusion

Abbreviations

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