Visualized calculation of regional power grid power data based on multiple linear regression equation

The rapid development of society and the economy has led to a sharp increase in energy consumption, increasingly depleted fossil energy sources, and increased environmental pollution. This poses a severe threat to the ecological balance of the earth and the survival of humankind. Large-scale photovoltaic power plants are an effective way to meet human demand for energy. Because photovoltaic power generation is affected by solar radiation intensity, battery component temperature, weather, cloud cover, and some random factors, the system operation process is an unbalanced random process. Its power generation and power output are highly random, fluctuating, and uncontrollable. This is particularly prominent when the weather changes suddenly. This kind of power generation inevitably brings a series of problems to the safety and management of the power grid after it is connected to the power grid [1]. Therefore, it is imperative to predict the output of the photovoltaic system accurately. At present, there are several types of prediction models for the power generation of photovoltaic power generation systems, such as neural network models, radial basis function models, and multi-layer perception models. In addition, the photovoltaic system is a typical grey system with partly clear and partly unknown information [2]. Grey theory forecasting models have a wide range of applications in power planning and load forecasting. Because these models are relatively complex, this paper establishes a relatively simple multiple linear regression prediction model to predict the power generation of grid-connected photovoltaic power generation systems.

Photovoltaic power generation uses the photovoltaic effect of the P–N junction to convert light energy into electrical energy. The equivalent circuit of its components and system under ideal conditions is shown in Figure 1. The maximum output power is represented as follows: (1) Pm=UmIm=UocIscFF {P_m} = {U_m}{I_m} = {U_{oc}}{I_{sc}}FF

Fig. 1

Here, U_oc,I_sc,FF are the open-circuit voltage, short-circuit current, and fill factor of the battery. Silicon solar cells are n = 1,R_s → 0,R_sh → ∞ under ideal conditions [3]. Therefore, the open-circuit voltage, short-circuit current, and fill factor can be expressed as follows: (2) {Uoc=nsnkTqln(ILID+1)I=IL≈IscFF=uoc−ln(uoc+0.72)uoc+1(1−RsIscUoc) \left\{{\matrix{{{U_{oc}} = {{{n_s}nkT} \over q}\ln \left({{{{I_L}} \over {{I_D}}} + 1} \right)} \hfill \cr {I = {I_L} \approx {I_{sc}}} \hfill \cr {FF = {{{u_{oc}} - \ln ({u_{oc}} + 0.72)} \over {{u_{oc}} + 1}}\left({1 - {{{R_s}{I_{sc}}} \over {{U_{oc}}}}} \right)} \hfill \cr}} \right.

Here, n_s,n are the number of batteries in series in the module and the ideality factor of the diode, respectively; B is the normalized open-circuit voltage; u_oc = qU_oc/nkT ; are the load current, photogenerated current, and diode response saturation current.

The performance of monolithic silicon solar cell modules is mainly affected by radiation intensity and temperature. Battery’s output power and power generation also strongly depend on both these factors. The short-circuit current is proportional to the radiation intensity within several orders of magnitude and is proportional to the light-receiving area of the battery [4]. The open-circuit voltage tends to saturate quickly with radiation, regardless of the battery area. The fill factor does not have a simple, functional relationship with the radiation amount. The open-circuit voltage, fill factor, and output power all decrease with an increase of component temperature. The temperature coefficients are −2.3 mV/°C, −0.0015/°C and −(0.004–0.005) W/°C, respectively. As the temperature of the components increases, the short-circuit current also increases slightly. The temperature coefficient is +0.107 mA/°C. Therefore, the output power of the photovoltaic system can be expressed as a function of the radiation intensity and temperature: P_m = P_m(G,T). The relationship between the short-circuit current on the one hand and the photogenerated current, the incident light intensity and the temperature is shown as follows: (3) Isc≈IL=IscR(GGR)+KT(t−tR) {I_{sc}} \approx {I_L} = {I_{scR}}\left({{G \over {{G_R}}}} \right) + {K_T}(t - {t_R})

Here, I_scR is the component’s short-circuit current obtained under standard testing conditions; is the radiation intensity under the standard test (1000 W/m²); K_T is the temperature coefficient of the short-circuit current (amperes per degree Celsius [A/°C]); t,t_R are the temperatures of the battery under arbitrary conditions and standard conditions (25 °C), respectively. Because the temperature has less influence on the short-circuit current, Eq. (3) can be rewritten as follows: (4) Isc≈IL=IscR(GGR)=γG {I_{sc}} \approx {I_L} = {I_{scR}}\left({{G \over {{G_R}}}} \right) = \gamma G

In the equation, γ is a constant, which is related to the battery area. We substitute Eq. (4) into Eq. (2) to obtain the open-circuit voltage: (5) Uoc=nsnkTqln(γG/ID+1)≈nsnkTqln(γG/ID)=nsnkTq[ln(γ/ID)+lnG]=a+blnG {U_{oc}} = {{{n_s}nkT} \over q}\ln (\gamma G/{I_D} + 1) \approx {{{n_s}nkT} \over q}\ln (\gamma G/{I_D}) = {{{n_s}nkT} \over q}[\ln (\gamma /{I_D}) + \ln G] = a + b\ln G

Among them, a=nsnkTqln(γID) a = {{{n_s}nkT} \over q}\ln \left({{\gamma \over {{I_D}}}} \right) , b=nsnkTq b = {{{n_s}nkT} \over q} . We substitute Eqs (4) and (5) into Eq. (2) to get the fill factor, as follows: (6) FF≈uoc−ln(uoc+0.72)uoc+1(1−RsγGa+blnG)=Uocns/b−ln[(Uocns/b)+0.72](Uocns/b)+1×[1−RsγGa(1+(lnγ−lnID)−1×lnG]≈Uoc−bln(Uoc/b)Uoc×(1−RsγGa) \matrix{{FF \approx {{{u_{oc}} - \ln ({u_{oc}} + 0.72)} \over {{u_{oc}} + 1}}\left({1 - {{{R_s}\gamma G} \over {a + b\ln G}}} \right) = {{{U_{oc}}{n_s}/b - \ln [({U_{oc}}{n_s}/b) + 0.72]} \over {({U_{oc}}{n_s}/b) + 1}}} \hfill \cr {\times \left[ {1 - {{{R_s}\gamma G} \over {a(1 + {{(\ln \gamma - \ln {I_D})}^{- 1}} \times \ln G}}} \right] \approx {{{U_{oc}} - b\ln ({U_{oc}}/b)} \over {{U_{oc}}}} \times \left({1 - {{{R_s}{\gamma _G}} \over a}} \right)} \hfill \cr}

Therefore, the output power can be expressed as (7) Pm=Uoc×Isc×FF×KT,P {P_m} = {U_{oc}} \times {I_{sc}} \times FF \times {K_{T,P}}

Here, K_T,P = 1 + ε(t − t_R) is the weight of the relationship between output power and temperature; ε is the temperature coefficient of output power [5]. The output power on the silicon solar cell monolithic module can be obtained by actually measuring the radiation intensity on the solar cell surface and the temperature of the module. The hourly power generation capacity of the solar cell can be calculated by adding up within hours.

The general form of the multiple linear regression model is yi=β0+β1x1i+β2x2i+⋯+βjxjii=1,2,⋯,n;j=1,2,⋯,k \matrix{{{y_i} = {\beta _0} + {\beta _1}{x_{1i}} + {\beta _2}{x_{2i}} + \cdots + {\beta _j}{x_{ji}}} \hfill \cr {i = 1,2, \cdots ,n;j = 1,2, \cdots ,k} \hfill \cr} , where j is the number of explanatory variables, β_j is the regression coefficient, and x_ji is a variable. In the photovoltaic power generation system, the radiation amount and temperature greatly influence the battery output [6]. Therefore, the article builds a multiple linear regression model of a photovoltaic power generation system around these two factors. We substitute Eqs (4)–(6) into Eq. (7) to obtain the following expressions: (8) Pm=γG(a+blnG)KT,P{(1−cG)a+blnG−bln[(a+blnG)/b])a+blnG {P_m} = \gamma G(a + b\ln G){K_{T,P}}\{(1 - cG){{a + b\ln G - b\ln [(a + b\ln G)/b])} \over {a + b\ln G}}

c = R_sγ/a. We can simplify the above formula to get (9) Pm=(γG−γcG2)[a−bln(a/b)+blnG−bln(1+balnG)]KT,P {P_m} = (\gamma G - \gamma c{G^2})[a - b\ln (a/b) + b\ln G - b\ln \left({1 + {b \over a}\ln G} \right)]{K_{T,P}}

We perform Taylor series expansion on the term ln(1+balnG) \ln \left({1 + {b \over a}\ln G} \right) in Eq. (9) and take the first three terms to obtain (10) ln(1+balnG)≈c0+c1lnG+c2(lnG)2 \ln \left({1 + {b \over a}\ln G} \right) \approx {c^0} + {c^1}\ln G + {c^2}{(\ln G)^2}

In this expression, c₀,c₁.c₂ are the expanded constant terms respectively. Eq. (10) is inserted into Eq. (9), and after finishing, we get (11) Pm={[γa−γbln(a/b)−γbc0]G+(γb−γbc1)GlnG+(−γbc2)[G(lnG)2]{N1=γa−γbln(a/b)−γbc0N2=γb−γbc1N4=γbcln(a/b)−γac+γbcc0N3=γbc2N5=γbcc1−γbcN6=γbcc2{x1=KT,PGx4=KT,PG2x2=KT,PGlnGx5=KT,PG2lnGx3=KT,PG(lnG)2x6=KT,PG2(lnG)2 \matrix{{{P_m} = \{[\gamma a - \gamma b\ln (a/b) - \gamma b{c^0}]} \hfill \cr {G + (\gamma b - \gamma b{c^1})G\ln G + (- \gamma b{c^2})[G{{(\ln G)}^2}]} \hfill \cr {\left\{{\matrix{{{N_1} = \gamma a - \gamma b\ln (a/b) - \gamma b{c_0}} & {{N_2} = \gamma b - \gamma b{c^1}} \cr {{N_4} = \gamma bc\ln (a/b) - \gamma ac + \gamma bc{c_0}} & {{N_3} = \gamma b{c^2}} \cr {{N_5} = \gamma bc{c_1} - \gamma bc} & {{N_6} = \gamma bc{c^2}} \cr}} \right.} \hfill \cr {\left\{{\matrix{{{x_1} = {K_{T,P}}G} & {{x_4} = {K_{T,P}}{G^2}} \cr {{x_2} = {K_{T,P}}G\ln G} & {{x_5} = {K_{T,P}}{G^2}\ln G} \cr {{x_3} = {K_{T,P}}G{{(\ln G)}^2}} & {{x_6} = {K_{T,P}}{G^2}{{(\ln G)}^2}} \cr}} \right.} \hfill \cr}

Then, Eq. (11) can be changed to (12) Pm=N1x1+N2x2+N3x3+N4x4+N5x5+N6x6 {P_m} = {N_1}{x_1} + {N_2}{x_2} + {N_3}{x_3} + {N_4}{x_4} + {N_5}{x_5} + {N_6}{x_6}

Here, N₁–N₆ are constants, x₁–x₆ are six unknowns about temperature and radiation intensity. In the grid-connected photovoltaic system, the photovoltaic array is composed of many photovoltaic modules connected in series (N_s) and parallel (N_P). The conversion efficiency of each part of the entire system is not 100%. The overall efficiency of the system η is η = η₁η₂η₃, where η₁,η₂,η₃ are the photovoltaic array efficiency, inverter efficiency, and alternating current (AC) grid-connected efficiency [7]. Then, the output power of the photovoltaic array is (13) PA=PmNsNPη {P_A} = {P_m}{N_s}{N_P}\eta

We substitute Eq. (12) into Eq. (13) to get (14) PA=N′1x1+N′2x2+N′3x3+N′4x4+N′5x5+N′6x6 {P_A} = N{{'}_1}{x_1} + N{{'}_2}{x_2} + N{{'}_3}{x_3} + N{{'}_4}{x_4} + N{{'}_5}{x_5} + N{{'}_6}{x_6}

Here, – N1′N6′ \mathop N\nolimits_1^{'} \,\mathop N\nolimits_6^{'} are constants. It can be seen that there is a multivariate linear relationship between the output power and the unknown variables. They can collect a lot of radiation intensity and temperature data. We use regression methods to obtain the six constants in Eq. (14). The predictive regression equation can be obtained. In a short period, we consider the test volume to be approximately constant [8]. At the same time, we can predict the power and output of the photovoltaic system by using the radiation and temperature measured at the last moment.

We apply the prediction model to the 2 kW grid-connected photovoltaic system. Among them, the weather of sunny (I), cloudy (II), sunny to cloudy (or overcast to cloudy) (III) adopts the P_A,I,P_A,II,P_A,IV forecast models, respectively [12]. We predict the average power generation (in kilowatts [kW]) and power generation (in kilowatt hour [kWh]) per hour for 2 consecutive days under different weather types. The forecasted values for sunny days (I) are higher than the measured values, the forecasted values for sunny to cloudy (or overcast to cloudy) (III) days are lower than the measured values, and cloudy days (II) have a better forecasting effect. The average residuals under sunny (I), cloudy (II), and sunny to cloudy (or overcast to cloudy) (III) weather are about 0.095, 0.066, and 0.126, respectively. The comparison result of the prediction residuals is shown in Figure 4.

Fig. 4

We have carried out theoretical calculations on the monolithic components of silicon crystal cells. On this basis, we establish a multiple linear regression model of photovoltaic system power generation and power generation with radiation and module working temperature as variables. Multiple linear regression analyses were conducted on the data under weather classification (I, II, and III) and unclassified conditions. The model has a good fit, and its accuracy is high.