(Part 1) Process of forecasting the output electricity for power plant working in combined cycle by the linear regression equation

September 27, 2019 admin Uncategorized 0

Today we find out how to create simple regression model to predict level of output electricity in power plant works in combined cycle.

Źródło bazy danych: https://archive.ics.uci.edu/ml/datasets/Combined+Cycle+Power+Plant

Registry contains 9568 measurements made in the period between 2006 and 2011. During the survey power plant was working in full capacity. Registry contains following of process characteristic:

  • Temperature (T) in the range 1.81°C and 37.11°C,
  • Ambient Pressure (AP) in the range 992.89-1033.30 milibar,
  • Relative Humidity (RH) in the range 25.56% to 100.16%
  • Exhaust Vacuum (V) in teh range 25.36-81.56 cm Hg
  • Net hourly electrical energy output (EP) 420.26-495.76 MW

A combined cycle power plant (CCPP) is composed of gas turbines (GT), steam turbines (ST) and heat recovery steam generators. In a CCPP, the electricity is generated by gas and steam turbines, which are combined in one cycle, and is transferred from one turbine to another. While the Vacuum is colected from and has effect on the Steam Turbine, he other three of the ambient variables effect the GT performance.

Process of forecasting by the linear regression model

We open database and useful libraries.

In [1]:
import pandas as pd
import numpy as np
import itertools
from itertools import chain, combinations
import statsmodels.formula.api as smf
import scipy.stats as scipystats
import statsmodels.api as sm
import statsmodels.stats.stattools as stools
import statsmodels.stats as stats
from statsmodels.graphics.regressionplots import *
import matplotlib.pyplot as plt
import seaborn as sns
import copy
import math

## https://archive.ics.uci.edu/ml/datasets/Combined+Cycle+Power+Plant

df = pd.read_excel('c:/1/Folds5x2_pp.xlsx')
df.sample(5)
Out[1]:
AT V AP RH PE
5844 20.76 59.21 1017.94 80.84 447.67
8410 7.54 38.56 1016.49 69.10 486.76
7916 14.73 58.20 1019.45 82.48 458.56
2253 14.56 40.69 1015.48 73.73 469.31
6735 14.57 37.86 1022.44 68.82 463.58

We analyze size and parameters of registry.

In [2]:
df.shape
Out[2]:
(9568, 5)
In [3]:
df.dtypes
Out[3]:
AT    float64
V     float64
AP    float64
RH    float64
PE    float64
dtype: object

For improve transparency of code we changed names of columns.

In [4]:
df.columns = ['Temperature', 'Exhaust_Vacuum', 'Ambient_Pressure', 'Relative_Humidity', 'Energy_output']
df.sample(5)
Out[4]:
Temperature Exhaust_Vacuum Ambient_Pressure Relative_Humidity Energy_output
9540 23.53 50.05 1005.63 78.40 443.71
9270 7.49 39.81 1018.20 85.96 483.71
163 21.87 61.45 1011.13 92.22 445.21
3479 21.63 50.16 1005.70 84.84 448.32
774 26.28 75.23 1011.44 68.35 435.28

Base conditions to realize linear regression model

Failure to comply of these rules lead to wrong, misleading result of model calculation.

  1. There are no correlation between descriptive variables (independent variables)
  2. There are exist linear relationship between predictive and result variables.
  3. The errors from the model should have normal distribution.
  4. Homogeneous of error variance
  5. Lack of correlation between errors. Errors from one observation should be independent to errors from other observation.

Analyze of relationship between dependent and independent variables

In [5]:
CORREL = df.corr().sort_values('Energy_output')
CORREL['Energy_output']
Out[5]:
Temperature         -0.948128
Exhaust_Vacuum      -0.869780
Relative_Humidity    0.389794
Ambient_Pressure     0.518429
Energy_output        1.000000
Name: Energy_output, dtype: float64

There are high level of correlation between result variable: energy output (EP) and exogenic factors: Temperature (T) and Exhaust Vacuum (V).

One factor regression model

In [6]:
lm = smf.ols(formula = 'Energy_output ~ Temperature', data = df).fit()
lm.summary()
Out[6]:
OLS Regression Results
Dep. Variable: Energy_output R-squared: 0.899
Model: OLS Adj. R-squared: 0.899
Method: Least Squares F-statistic: 8.510e+04
Date: Fri, 06 Sep 2019 Prob (F-statistic): 0.00
Time: 13:10:52 Log-Likelihood: -29756.
No. Observations: 9568 AIC: 5.952e+04
Df Residuals: 9566 BIC: 5.953e+04
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 497.0341 0.156 3177.280 0.000 496.727 497.341
Temperature -2.1713 0.007 -291.715 0.000 -2.186 -2.157
Omnibus: 417.457 Durbin-Watson: 2.033
Prob(Omnibus): 0.000 Jarque-Bera (JB): 1117.844
Skew: -0.209 Prob(JB): 1.83e-243
Kurtosis: 4.621 Cond. No. 59.4

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

One factor regression model has a very good predictive properties.

In [7]:
plt.figure()
plt.scatter(df.Temperature, df.Energy_output, c = 'grey')
plt.plot(df.Temperature, lm.params[0] + lm.params[1] * df.Temperature, c = 'r')
plt.xlabel('Temperature')
plt.ylabel('Energy_output')
plt.title("Linear Regression Plot")
Out[7]:
Text(0.5, 1.0, 'Linear Regression Plot')

Multi factor regression model

In [8]:
lm = smf.ols(formula = 'Energy_output ~ Temperature + Exhaust_Vacuum + Relative_Humidity + Ambient_Pressure', data = df).fit()
print (lm.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:          Energy_output   R-squared:                       0.929
Model:                            OLS   Adj. R-squared:                  0.929
Method:                 Least Squares   F-statistic:                 3.114e+04
Date:                Fri, 06 Sep 2019   Prob (F-statistic):               0.00
Time:                        13:10:53   Log-Likelihood:                -28088.
No. Observations:                9568   AIC:                         5.619e+04
Df Residuals:                    9563   BIC:                         5.622e+04
Df Model:                           4                                         
Covariance Type:            nonrobust                                         
=====================================================================================
                        coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------------
Intercept           454.6093      9.749     46.634      0.000     435.500     473.718
Temperature          -1.9775      0.015   -129.342      0.000      -2.007      -1.948
Exhaust_Vacuum       -0.2339      0.007    -32.122      0.000      -0.248      -0.220
Relative_Humidity    -0.1581      0.004    -37.918      0.000      -0.166      -0.150
Ambient_Pressure      0.0621      0.009      6.564      0.000       0.044       0.081
==============================================================================
Omnibus:                      892.002   Durbin-Watson:                   2.033
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4086.777
Skew:                          -0.352   Prob(JB):                         0.00
Kurtosis:                       6.123   Cond. No.                     2.13e+05
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.13e+05. This might indicate that there are
strong multicollinearity or other numerical problems.

Multi factor regression model have very good descriptive features. It raises doubts.

Analyze of variables used in mode

In [9]:
df.describe()
Out[9]:
Temperature Exhaust_Vacuum Ambient_Pressure Relative_Humidity Energy_output
count 9568.000000 9568.000000 9568.000000 9568.000000 9568.000000
mean 19.651231 54.305804 1013.259078 73.308978 454.365009
std 7.452473 12.707893 5.938784 14.600269 17.066995
min 1.810000 25.360000 992.890000 25.560000 420.260000
25% 13.510000 41.740000 1009.100000 63.327500 439.750000
50% 20.345000 52.080000 1012.940000 74.975000 451.550000
75% 25.720000 66.540000 1017.260000 84.830000 468.430000
max 37.110000 81.560000 1033.300000 100.160000 495.760000

Analyze of distribution of result variable

In [10]:
plt.rcParams['figure.figsize'] = (5, 4)
sns.distplot(df['Energy_output'])
Out[10]:
<matplotlib.axes._subplots.AxesSubplot at 0xcb106d8>

Analysis of correlation between independent variables

In [11]:
plt.rcParams['figure.figsize'] = (5, 4)
sns.heatmap (df.corr (), cmap="YlGnBu")
Out[11]:
<matplotlib.axes._subplots.AxesSubplot at 0xcb03c18>

We can easily remark that between independent variables: Temperature (T) and Exhaust Vacuum (V) exist very high positive correlation.
Another form of correlation matrix.

In [12]:
sns.heatmap (df.corr (), cmap="coolwarm", annot=True, cbar=False)
Out[12]:
<matplotlib.axes._subplots.AxesSubplot at 0xd5ae438>

We check completeness of survey.

In [13]:
df.isnull().sum()
Out[13]:
Temperature          0
Exhaust_Vacuum       0
Ambient_Pressure     0
Relative_Humidity    0
Energy_output        0
dtype: int64

Graphical presentation of relationship between independent and dependent variables

We divide result variable into two categories: “High power” and “Low power”. We do it to remark what descriptive variables work.

In [14]:
Ewa = ['High power', 'Low power']

df['Power'] = pd.qcut(df['Energy_output'],2, labels=Ewa)
df.sample(4)
Out[14]:
Temperature Exhaust_Vacuum Ambient_Pressure Relative_Humidity Energy_output Power
2743 30.20 73.67 1006.31 62.14 428.72 High power
1095 30.33 68.67 1006.00 54.99 435.53 High power
4327 9.15 41.82 1032.88 75.11 477.78 Low power
3457 32.74 68.31 1010.23 41.29 441.66 High power

We create graphical relationship matrix.

In [15]:
sns.pairplot(data=df[['Temperature' ,'Exhaust_Vacuum','Ambient_Pressure', 'Relative_Humidity', 'Power']], hue='Power', dropna=True, height=2)
Out[15]:
<seaborn.axisgrid.PairGrid at 0xd633240>

Relationship among Temperature (T) and Exhaust Vacuum (V) ones again showed high level of correlation.

In [16]:
sns.jointplot(x='Temperature', y='Exhaust_Vacuum', data=df)
Out[16]:
<seaborn.axisgrid.JointGrid at 0xb9946d8>