Identification of Predicted Load Cluster Pattern Power Generation Parameters Based on Descriptive Time Series Analysis

Abstract

This chapter describes the process of identifying a power generation system. This is important because in principle the system parameters as a whole are not linear and uncertain. For this reason, it is necessary to carry out an identification process using an experimental approach that is able to represent the system as a whole. The technique used in this identification process is Prediction Error Minimization (PEM) as a tool available in Matlab. Identification is done by simulating changes in the value of frequency, voltage and electrical power due to changes in load. The change in load over time is a characteristic of the time series pattern. Through descriptive analytic approach, the cluster load is patterned for each load operating condition. Through load clusters, the identification results of power generation systems are obtained based on their operating conditions. This chapter presents validated parameter estimates for each change in instantaneous load conditions. The simulation results obtained better performance between the actual output and the identification model, namely the calculation of the Intergal Absolute Error (IAE), with MAPE for the average frequency value of 73.95 percent, nominal voltage of 0.23 percent, and electric power of 23.46 percent.

1. Introduction

In reality, existing systems are a combination of linear and nonlinear models. Modeling with mathematical derivation is done with a lot of neglect of unmeasured parameters. The calculation is carried out around the linear area only, while the nonlinear area is not much heeded or neglected. Therefore, the results of the model derived based on the laws of physics or mathematical derivation, are still not very effective to be applied directly in the field.

Apart from the method of deriving the laws of physics or deriving it mathematically, there are also other methods for modeling, namely by using the identification method. This method will model a system as a whole, both linear and nonlinear parameters. All of these are considered to be one integrated system. Identification is an approach process by means of mathematical modeling of an unknown system through learning of data collected from previous experiments as well as input–output data from pre-existing controlled systems. The results obtained are parameter values ​​in the form of a mathematical model. The data is compared with the actual model, wherein the difference between the two systems is used as an objective function which will later be minimized. In principle, the identification process is an introduction to a system to be controlled. Identification is one of the important factors that support success in engineering a control system that is stable, robust and able to represent the actual model in the field.

Matlab has a very effective identification tool for simulations. Several studies on parameter identification have been carried out through one of them the System Identification Toolbox (SIT) package. Initial research which is the reference for this writing as has been done by Fruk et al. [1]. The model can be determined by adjusting the assumed process model settings, until the modeling is satisfactory and accurate according to the input data in the SIT package. Research has also been carried out to identify the dynamic model parameters of a permanent magnet direct current (PMDC) DC motor [2]. The process of identifying DC motor parameters in the last year has also been carried out and published. This study identifies DC motor parameters whose initial values ​​are estimated by MATLAB/ Simulink based on a Genetic Algorithm [3]. Identification research on large-scale power generation systems has been carried out to test the frequency value of the load shedding test on the 39 bus hydraulic turbogenerator system [4]. This identification process is carried out in the form of a Matlab simulation.

The identification in this paper is a variant of the single machine infinite bus (SMIB) parameters due to load changes. The fluctuation of electric power at the load center is very important to analyze so that the stability of the power generation system can be maintained optimally. Changes in load over time can be identified as minor disturbances in the generating system. Minor disturbances or so called small-signal stability studies occur in the operating conditions of the system after the first swing at which time the response of control equipment such as governors, AVR, and auxiliary devices has been taken into account [5, 6]. Load fluctuation results in changes in the value of the frequency and voltage in the power generation system so that the stability of the system will be disturbed.

The load characteristics of the period of use whether it is used by household, commercial, industrial and public loads are necessary so that fluctuations in the loading system can be analyzed. Changes in electric load form a continuous pattern and are time series in nature. Through a descriptive analytical statistical approach, the load pattern is analyzed so that the operating conditions of the power generation system are obtained in the form of a continuous load cluster pattern. To achieve the research objectives, steps were taken to identify any changes in electrical power at the load center. Changes in load are classified through the identification of SMIB parameters using the reduced-order model approach.

2. Research methods

This research aims to identify and validate the power generation system parameters. The identification process uses the prediction error minimization approach to obtain a more accurate mathematical model.

In this research, identification is carried out for all dynamic conditions due to changes in electric power on the load side. Namely, creating a load cluster pattern based on the analytical descriptive method in a time series approach.

The achievement that will be produced in this research is the identification model equation for each cluster in the form of a state space matrix equation. So that the results of this research will be useful in designing controls on a power generation system that is more efficient, optimal response and robust.

2.1 Proposed research

Mathematical models through physical analysis will not provide accurate identification results, because there are parameters that can only be obtained through an experimental process. So it is necessary to approach the system model. The system model approach can be carried out through the identification and validation process stages. System identification is defined as a method used to obtain an approach model from the actual system through evaluation of input–output measurement data [7]. In other words, the process of identifying a system is a combination of two efforts, namely the effort to form a mathematical model and estimate the optimal parameter value through experimental steps. System identification in general can be described as in Figure 1 below.

Figure 1.

Block diagram of the system identification process.

According to Law and Kelton, validation is the process of determining whether the simulation conceptual model is really an accurate representation of the real system being modeled [8]. Model validation can also be said as a step to test whether the identified model can represent the real system correctly. A model can be said to be valid when it does not have a significant difference with the real system which is observed either from its characteristics or from its behavior.

It is important to identify and validate the power generation system in determining the load change pattern. The dynamic load characteristics will be patterned in the form of clusters based on statistical analysis. Load every moment is a time series dynamic behavior. A time series is a series of observations carried out sequentially based on time [9]. The observation process is carried out at the same intervals, for example in hourly, daily, weekly, monthly, yearly intervals, or other intervals. There are two objectives of time series analysis, namely to model the stochastic mechanisms contained in the observations and to predict the value of future observations.

Electric load data can be viewed as a reality of stochastic processes [9]. Where statistical phenomena are arranged in time order based on the law of probability. If the time series observation is denoted by $Z_t$, where $t\in A$ with A the set of natural numbers. According to Wei, the stochastic process is a time-based data group composed of random variables $Z\left(\omega ,t\right)$ where $\omega$ is the sample space and t is the time index [10]. The distribution functions of the random variables $Z_{t_1},Z_{t_2},\dots ,Z_{t_n}$ are as follows:

$$F\left(z_{t_1},z_{t_2},\dots ,z_{t_n}\right)=p\left\{\omega :z\left(\omega ,t\right)\le z_{t_1},\dots ,z\left(\omega ,t_n\right)\le z_{t_n}\right\}$$E1

Observation of $Z_1,Z_2,\dots ,Z_n$ is a stochastic process, so the random variables $Z_{t_1},Z_{t_2},\dots ,Z_{t_n}$ are said to be stationary in the distribution if:

$$F\left(z_{t_1},z_{t_2},\dots ,z_{t_n}\right)=F\left(z_{t1+k},z_{t2+k},\dots ,z_{\mathit{tn}+k}\right)$$E2

A model like the one above is called a stochastic process, because of the sequential observations that are arranged in time. The variant of the electric load due to use in consumers can be viewed as an approach to the electrical load cluster pattern. The main purpose of this cluster pattern is to determine the operating conditions of the power generation system. Analytical descriptive method is a statistical approach that can describe a continuous load state. This method is concerned with collecting and presenting a data set so that it provides useful information [11]. Analysis of this cluster data processing into information containing a set of electrical load characteristics that can be summed up numerically.

This descriptive analysis includes several things, namely: first, frequency distribution, namely the arrangement of data according to certain categories in a systematically arranged list. Second, the measurement of central tension, which is a statistical analysis that specifically describes a representative score, includes data frequency figures such as mode, mean, median or arithmetic mean. Third, the measurement of variability, namely the degree of spread of variable values from a central tendency in a distribution. Variability is also known as dispersion. Variability can be measured through measurements: range, mean deviation, and standard deviation [12].

Cluster pattern groups in the same interval will make it easier to analyze load changes and be able to provide intervals of variants of the distribution of electric power from the power plant to the load. This analysis is important in achieving the planning goals and schedules of a more efficient and optimal generating system in maintaining the balance of the electric power system.

2.2 Physical model of the power generation system

The physical model used in this study is the SMIB model. The SMIB model refers to Park modeling, with the following criteria: negligible stator resistance, balanced system conditions and negligible core saturation of the generator, and the load is considered a static load [13]. This model refers to the synchronous machine that was introduced by De Mello and Concordia [14]. Recent studies still refer to this single engine model as has been done by [15, 16, 17].

The electric power produced by the power plant must be balanced with the electric power absorbed at the load center. By applying a variant of the electric load pattern and the load cluster approach, this research is able to maintain a balance between the electrical power supplied and the electric power consumed at the load center.

2.3 Identification power generation system

Parameter identification through derivation of the mathematical equation of the SMIB model. The mathematical model of a dynamic system is defined as a set of mathematical equations that represent the dynamics of a system accurately or at least close to the characteristics of the dynamic system. As a first step in analyzing a dynamic system is to derive the mathematical model. The mathematical model of each system will vary, it depends on the system components. Previous studies have derived mathematical models of generating systems to support the design of small signal stability control [18].

The identification approach uses a system identification toolbox with prediction error minimization (PEM) black-box identification techniques. PEM builds a measured input–output system mathematical model with the aim of updating the initial model as a reference model.

The PEM function also handles multiple-input-single-output structures in the form of polynomial representation of transfer functions

$$A\left(q\right)y\left(t\right)=\frac{B_1\left(q\right)}{F_1\left(q\right)}{u}_1\left(t-n{k}_1\right)+\dots +\frac{B_{\mathit{nu}}\left(q\right)}{F_{\mathit{nu}}\left(q\right)}{u}_{\mathit{nu}}\left(t-n{k}_{\mathit{nu}}\right)+\frac{C\left(q\right)}{D\left(q\right)}e\left(t\right)$$E3

Where A, B, F,C and D are polynomials in the operator delay. Here, the numbers $n_a$ and $n_b$ are the orders of the respective polynomials. The number nk is the number of delays from input to output. With $n_y$ output channels and nu input channels. An Output-Error structure is obtained as$e\left(t\right)$. In this study, using discrete time with a 3rd order approach.

3. Results of the parameter identification

To reach the results stage of this research, the steps taken in this study are simulations of the identification of the power generation system parameters, the results of time series analysis, validation, and simulation of the validation parameters compared to the actual model.

3.1 Parameter modeling

Based on Figure 1, the mathematical model of the SMIB system is represented in the form of the following state space Eq. [14]:

$$\dot{x}=Ax+Bu\text{and}y=Cx$$E4

Where, $x$ is the state variable, $n\times 1$, $u$ is the input variable, $m\times 1$, $y$ is the output variable, $r\times 1$, A is the system matrix, $n\times n$, B is the control matrix, $n\times m$, and C is the measurement matrix, $n\times m$.

The state space equation, where the state variable x is defined as

$$x={\left[∆Y∆T_m∆\delta ∆\omega ∆E_q^{\prime }∆v_F\right]}^T$$E5

and the output variable y as

$$y={\left[\begin{array}{cccccc}∆Y& ∆T_m& ∆P& ∆\omega & ∆v& ∆v_F\end{array}\right]}^T$$E6

where,

$$\mathrm{Matrix}{A}_{\mathit{ii}}=\left[\begin{array}{cccccc}-\frac{1}{T_{\mathit{gu}}}& 0& 0& -\frac{K_{\mathit{gu}}}{T_{\mathit{gu}}R}& 0& 0\\ \frac{1}{T_{\mathit{tu}}}& -\frac{1}{T_{\mathit{tu}}}& 0& 0& 0& 0\\ 0& 0& 0& {\omega }_o& 0& 0\\ 0& 0& -\frac{K_1}{M}& -\frac{D}{M}& -\frac{K_2}{M}& 0\\ 0& 0& -\frac{K_4}{T_{\mathit{do}}^{\prime }}& 0& -\frac{1}{K_3{T}_{\mathit{do}}^{\prime }}& \frac{1}{T_{\mathit{do}}^{\prime }}\\ 0& 0& -\frac{K_A{K}_5}{T_A}& 0& -\frac{K_A{K}_6}{T_A}& -\frac{1}{T_A}\end{array}\right]$$

$$\mathrm{Matrix}{B}_{\mathit{ii}}=\left[\begin{array}{cc}\frac{K_{\mathit{gu}}}{T_{\mathit{gu}}}& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& \frac{K_A}{T_A}\end{array}\right]\text{and}\text{Matrix}{C}_{\mathit{ii}}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& K_1& D& K_2& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& K_5& 0& K_6& 0\\ 0& 0& 0& 0& 0& 1.\end{array}\right]$$

And for the output state variable of the equation that satisfies:

$$∆P=K_1∆\delta +K_2∆E_q^{\prime }+D∆\omega$$E7

$$∆v=K_5∆\delta +K_6∆E_q^{\prime }$$E8

Where, $∆Y$ is the change in turbi valve height, $∆T_m$ is the change in mechanical torque, $∆\delta$ is the change in rotor angle, $∆\omega$ is the change in rotor angular velocity, $∆E_q^{\prime }$ is the change in generator transient voltage, $∆v_F$ is the change in excitation output voltage, $∆P$ is the change in generator electrical power, and $∆v$ is the change in terminal voltage.

3.2 Time series analysis based load cluster

The dynamics of the electrical load is a series of data calculated every half hour on the SMIB that transmits power to the load center. The operating points were selected based on load cluster modeling using descriptive analytical statistical methods. The cluster interval range is simply implemented between the minimum value of electrical load, quartile 1, middle value, quartile 3, and maximum value as the operating point under load conditions. To model the cluster pattern based on the distribution of electrical load data as shown in Figure 2, this study uses the Minitab software.

Figure 2.

(a) The distribution pattern of the cluster model data, (b) the level of data density.

Load variation is defined as a disturbance mechanism that occurs in the system due to changes in electrical power at the load center represented by changes in load groups. While the input from the turbine side and constant excitation. This modeling is represented by the input signal (u) which is on the turbine side ($∆u_{\mathit{gu}}$) and the excitation side ($∆u_E$) with a reference a signal of 0.5 sin (0.1 t) + 0.5 pu. Output (y) which represents the signal change in frequency value ($∆\omega$), change in terminal voltage on the generator bus, ($∆v$) and generator electrical power ($∆P$) in the form of 1 pu.

The input change load is in the form of a cluster pattern which represents the cluster load model in the form of a sinusoidal signal equation. As shown in Figure 3, the cluster pattern in the form of a variation in the load model is set manually. The cluster charge signal is a reference input with a signal pattern: cluster 1 represents a 0.125 sin (0.5 t) +0.125 signal, cluster 2 is 0.125 sin (0.5 t) +0.375, cluster 3 is 0.125 sin (0.5 t) +0.625, and cluster 4 of 0.125 sin (0.5 t) +0.875 in 1 pu.

Figure 3.

Block diagram of the SMIB validation.

In terms of the discrete state model equation, the identification results are expressed by:

$$x\left(\mathit{kT}+1\right)=\mathit{aM}1x\left(\mathit{kT}\right)+\mathit{bM}1u\left(\mathit{kT}\right)$$E9

$$y\left(\mathit{kT}\right)=\mathit{cM}1x\left(\mathit{kT}\right)$$E10

For all identification processes using a sampling interval of 0.1 s with a discrete equation mode of order 3. The program listing on the system identification toolbox is as follows

S%Identification ModelSMIB

dat = idddata(y11 y12 y13],u11 u12],0.1);

SMIB = pem(dat,nx,…

  ‘DisturbanceModel’,‘none’,…

  ‘InitialState’,‘zero’);

[aM1d, bM1d, cM1d, dM1d, ke] = ssdata(SMIBDskrt);

aM1d

bM1d

cM1d

3.3 Validation power generation system

Power generation system validation model represents the mathematical model of the actual model and the 3rd order approach model. The representation of the simulation model approach with Matlab/Simulink is in the following block diagram.

The sub-system model of the generation system validation is shown in the following Matlab/Simulink block diagram (Figure 4).

Figure 4.

Sub-system model of generation system validation.

The results of the identification and validation of the generating system are expressed in the form of the equation state space order 3 matrices A, B, and C below.

Modeling the identification of the power generation system parameters through the simulation of loading of electrical power in each cluster obtained a linear model of the input state matrix (A matrix), the output state matrix (C matrix) and the control signal state matrix (B matrix).

3.4 Power generation system parameter simulation results

The identification result is a state space matrix equation as shown in Eqs. 9 and 10 in Table 1 above. The identification process is outlined in a flowchart as shown in Figure 5 below:

Cluster	Matrix A	Matrix B	Matrix C
1	$\begin{array}{ccc}1,0006& 0,4699& -0,0567\\ -0,0562& 0.8521& 0,0317\\ -0,0208& 0,1863& 0,9662\end{array}$	$\begin{array}{cc}-1,1014& -0,4644\\ 0,0906& -0,8310\\ -0,5709& -0,7381\end{array}$	$\begin{array}{ccc}-0,0333& -0,0208& 0,0594\\ -0,0179& -0,1015& -0,0531\\ 0,2325& 0,5239& -0,4321\end{array}$
2	$\begin{array}{ccc}0,9499& 0,6371& -0,0459\\ -0,0397& 0,8488& -0,0070\\ 0,0234& -0,3796& 0,9921\end{array}$	$\begin{array}{cc}-0,6493& -0,4438\\ 0,0823& -0,9872\\ 0,4765& 0,6902\end{array}$	$\begin{array}{ccc}-0,0056& 0,0043& -0,0084\\ -0,0246& -0,1114& 0,0185\\ 0,1572& 0,3463& 0,2393\end{array}$
3	$\begin{array}{ccc}1,0513& 0,5022& 0,2361\\ -0,0662& 0,8710& -0,0124\\ 0,0251& -0,1192& 0,8908\end{array}$	$\begin{array}{cc}0,1099& -0,9611\\ -0,2356& -0,6486\\ 0,5362& 0,5751\end{array}$	$\begin{array}{ccc}0,0022& 0,0125& 0,0049\\ -0,0735& -0,1159& -0,0107\\ 0,0960& 0,4697& 0,2131\end{array}$
4	$\begin{array}{ccc}0,8189& -1,1845& -0,5249\\ 0,0910& 1,3565& 0,2761\\ -0,1205& -0,4570& 0,5857\end{array}$	$\begin{array}{cc}0,1141& 2,4466\\ -0,1583& 0,7748\\ 0,2317& -1,7243\end{array}$	$\begin{array}{ccc}-0,0188& -0,1593& -0,1006\\ 0,0151& 0,0293& -\mathrm{0,068}\\ 0,0310& -0,2124& -0,1050\end{array}$

Table 1.

Cluster based state space matrix.

Figure 5.

Flowchart research.

The identified state space equation model is then validated as a reference equation for the mathematical model of the power generation system load cluster parameters compared to the actual model, with the following results:

1. Validate the following cluster 1 power generation system parameters (Figure 6):

2. Validate the following cluster 2 power generation system parameters (Figure 7):

3. Validate the following cluster 3 power generation system parameters (Figure 8):

4. Validate the following cluster 4 power generation system parameters (Figure 9):

Figure 6.

Simulation results of cluster 1 load, (a) frequency, (b) bus voltage, (c) electrical power.

Figure 7.

Simulation results of cluster 2 load, (a) frequency, (b) bus voltage, (c) electrical power.

Figure 8.

Simulation results of cluster 3 load, (a) frequency, (b) bus voltage, (c) electrical power.

Figure 9.

Simulation results of cluster 4 load, (a) frequency, (b) bus voltage, (c) electrical power.

Comparative representation between the actual model and the validation model is presented in Table 2 form as follows:

Kluster	IAE actual output			IAE identification model output
	F	V	P	F	V	P
1	2.046	51.32	51.49	0.7489	51.27	50.54
2	1.856	51.32	35.81	0.3652	51.55	30.8
3	1.909	51.31	31.17	0.4764	51.46	17.48
4	2.184	51.3	37.7	0.4927	51.45	20.72
Average	1.999	51.313	39.043	0.5208	51.433	29.885

Table 2.

Comparison of integral absolute error (IAE) identification model output and actual output.

Based on the results obtained in Table 2, it can be seen that the IAE calculation is very significant between the actual output and the identification results, especially for changes in frequency and electrical power.

To calculate the performance comparison between the actual results and identification, the Mean Absolute Percentage Error (MAPE) formula is used as follows:

$$\mathit{\text{MAPE}}=\frac{{\sum }_{t=1}^n\left|\frac{Z_t-{\widehat{Z}}_t}{Z_t}\right|}{n}\times 100\%$$E11

where $Z_t$ and ${\widehat{Z}}_t$ are the actual value and the identification value, while $n$ is the number of calculation data sett.

Based on the MAPE calculation, the average frequency value is 73.95 percent, nominal voltage is 0.23 percent, and electric power is 23.46 percent.

4. Conclusion

The identification process of the power generation system in this paper is very supportive for further research, especially in the field of controlling the power generation system.

Prediction Error Minimization is very helpful in the identification process that is able to adjust the response to the desired signal model. Response time delay is calculated so that the generation system equipment on the mechanical and electrical side can work optimally. The delay time in Figure 4 is included so that the power generation system equipment on the mechanical and electrical side can work optimally. The simulation results are obtained with better performance between the actual output and the identification model, namely the calculation of Integral Absolute Error (IAE), with MAPE for the average frequency value of 73.95 percent, nominal voltage of 0.23 percent, and electric power of 23,46 percent.

For future research work related to the identification of parameters of multi-engine generating systems as well as interconnection systems, it is necessary to carry out experiments. Even though it is in the form of a simulation, this work will help researchers to get closer to solving problems in real conditions.

Finally, this paper can be used as a reference for further research on the identification of power generation system parameters. In maintaining the stability of the frequency value on the mechanical side, the nominal voltage of the generator terminal and electric power when the electric load fluctuates.