Identification of the Conduit Water Starting Time Constant in Hydropower Plants Using LSTM and MLP Machine Learning Algorithms

In engineering and physics, system identification (SI) is a crucial technique for developing accurate models and control strategies [1]. Based on experimental data, SI involves finding out the mathematical model of a dynamics system. In this paper, the available dataset is for the hydro power plant control system, especially for parameters such as guide vane blade position, active mechanical power, and angular velocity. These data are further used for the identification of the water starting time constant of the conduit – $T_w$.

The crucial parameter when modeling the hydro power plant is $T_w$, especially in the Francis turbine and other reaction turbines. This parameter is related to the water inertia in the penstocks and the turbine system. It is important because it influences the plant’s dynamic response to load changes, meaning the position of the guide vane blade. Usually, large values for the $T_w$ mean slower response due to the delayed action of the water flow, but smaller values for the $T_w$ mean faster adjustment and stabilization in a steady-state position but increase the possibility of an easy control [2]. Sari et al. [3] examined various turbine technologies and their application on how the initiation of the water flow impacts the system efficiency of the penstock. If penstock dynamics of a hydropower plant are compared, Janevska et al. [4] represented the importance of the $T_w$. If the $T_w$ is large value, it especially means that the PID controller will require more aggressive tunning. Other control strategies, instead of PID, according to the study conducted by Zoby and Yanagihara [5], could be PI, or combination of PI-PD. Also, in the study conducted by Gangi [6], in a small hydropower plant, where the $T_w$=0.25 which is a small value makes turbines respond quickly but there are possibilities of overshoots and oscillation. As represented, the control type of the governor is PID. Other than the PID, another control strategies such presented in references [7], [8], involving ML based control that can adapt to different $T_w$ values based on past experiences and responses. As reported by Khaliel and Karrar [9], the importance of the optimal hydropower plant operation through PID tuning for the governor system controlling the guide vane mechanism is presented. Ensuring grit stability and efficiency frequency control, means accurate determination of the $T_w$ constant. It is also important coefficient when it comes to the prediction of the system’s behavior during the transient conditions and design of the control strategies. Some standard mandates that $T_w$ must be less or equal to 4 seconds to align with IEC 60308 and IEC 61362, ensuring compatibility with turbine-generator inertia, otherwise some advanced control strategies must be implying design modification, according to the technical guideline [10]. Pérez-Díaz et al. [11] presented another way of determining the $T_w$ constant. In this paper it is calculated by adding two additional elements, tail-race tunnel and penstock. The control type in that case was PI controller.

Saarinen et al. [12] presented standard methods, the prediction-error minimization and MATLAB system identification toolbox technique for system identification of the $T_w$ in various size hydropower plant. System identified are represented with a linear mathematical differential equation. Some hybrid deep learning models such as Temporal Convolutional Networks (TCN), residual LSTM, and Gated Recurrent Unit (GRU) as demonstrated by Ma et al. [13] are used for prediction the operation status of the hydropower plant units, also Wang et al. [14] presented adaptive learning model, recursive least squares (RLS) for modeling the hydropower turbine which is directly connected to $T_w$ identification by determining the state-space and the transfer function model. In this paper, LSTM as a ML method has been used because it successfully captures and processes the time-series data [15]. MLP as a ML method according to the research by Hagan et al. [16], is used for control systems analysis. As reported by Xiong et al. [17], LSTM is presented as a method for turbine operation using time series data. MLP and LSTM methods are selected to be used since both have completely different computational methods.

The hydropower plant represented in Figure 1 consists of a few subsystems such as hydraulic, electrical, and control subsystem. The control subsystem in this case is PID.

Based on the analysis presented by Babunski [18], hydro power plant model that includes the guide vane mechanism, the hydraulic turbine, and the electrical subsystem is mathematically represented in this paper according to the equation below.

Eq. (1) gives the water starting time constant of the conduit $T_w$, and its value depends on several parameters, the length of the pipeline $L$, the flow through the turbine at a fully open guide vane mechanism ${{q}_{base}}$, the pressure ${{h}_{base}}$, the cross-section of the pipeline $A$, and the acceleration due to gravity $g$. Eq. (2) represents the relationship between the generated mechanical power $\Delta P_m$ and the guide vane position $\Delta c$, which represents the linear model of a hydro turbine. Theoretically, the $T_w$ is defined as the time required for the flow in the pipeline to go from zero ${{q}_{base}}$ to when the pressure in front of the turbine will reach a value ${{h}_{base}}$.

The equation for the turbine mechanical power is represented by Eq. (3), where ${{h}_{t}}$ is the dimensionless unit pressure, ${{q}_{t}}$ is the dimensionless unit flow, ${{q}_{nl}}$ is the flow through a turbine when there is no load, ${{A}_{t}}$ is the amplifier factor, $D$ is the turbine damping coefficient, and $\Delta \omega$ is the difference between the reference and actual angular velocity of the hydraulic turbine generator.

The electrical subsystem consists of a power system, a generator, the power transferred between them and the changes in the load on the network. It is represented by Eqs. (4)-(6).

In these equations, $\omega $ is the angular velocity at which the generator rotates, ${{P}_{l}}$ is the load on the network, i.e., the disturbance, ${{P}_{e}}$ is the generated electrical energy, ${{T}_{m}}$ is the mechanical time constant whose value depends on the number of generator revolutions per minute, but is calculated according to the equations defined in reference [19]. Also ${{T}_{m}}$ (mechanical time constant) is represented in Eq. (7), where $G\cdot {{D}^{2}}$ is the moment of inertia, ${{n}_{r}}$ is the number of revolutions per minute, and ${{P}_{r}}$ is the generated power in megawatts (MW).

The model for the servo mechanism is represented in formula (8) where ${{T}_{A}}$ is the time constant of the servo mechanism.

Because in this paper, two real hydropower plants are going to be of research interest, in Table 1 some details about them are represented.

The optimization function is also important because it quantifies the accuracy of the turbine model, particularly when identifying the $T_w$. The more minimal the value for $F_\text{opt}$, the more the $T_w$ is optimal, which means that the system’s dynamics is better represented. $F_\text{opt}$ is represented with Eq. (9). In this equation, $\omega $ is the angular velocity at which the generator rotates and the ${{\omega}_{ref}}$ is the referent angular velocity, $c$ guide vane position and ${{c}_{d}}$ is requested guide vane position. The weighting coefficients k1 and k2 determine the influence of the corresponding terms in $F_\text{opt}$. The influence of the speed error is greater because the main objective of the speed controller is speed regulation, i.e., ensuring a stable output frequency of the generated energy, which is why the weighting coefficient k1=0.7, i.e., the speed error has an influence of 70%, while the weighting coefficient k2=0.3, which means that the influence of the second term is 30%.

In this paper, LSTM and MLP have been used as ML methods for $T_w$ identification. The general structure of the LSTM method is represented in Figure 2 and its equations of the signal flow are represented by Deva Sarma et al. [20].

Figure 3 from the study of Salim et al. [21] shows the MLP network architecture and its mathematical representation.

Because Python has been used as a programing language, the algorithm for $T_w$ identification is represented through the following order:

1. Definition of the required libraries to perform the calculation.

2. Definition of the inputs (input data set).

3. Definition of the target outputs (target output data set).

4. Definition of the type of ANN with which the data will be processed.

5. Definition of the number of hidden layers and the number of neurons in each layer.

6. Definition of the activation function (AF) for each layer separately.

7. Definition of the type of optimizer.

8. Making a percentage distribution of the data for validation, testing and training.

9. Training the model.

10. Defining the upper and lower bounds of the value of $T_w$ that needs to be obtained.

11. The generated values for the time constant of water are extracted.

12. The obtained values are placed in a simulation model.

13. The obtained outputs are compared with the outputs of the model before applying ML by applying MLP and LSTM.

Because there are going to be represented results from two power plants, the input data set for Plant 1 is the position of the guide vane blades and the output is generated power from Power Plant 1, and the dataset for the Plant 2 is active mechanical power from the second power plant (data taken from supervisory control and data acquisition (SCADA) system) and active mechanical power of the first, referent power plant. The structure of the network for both LSTM and MLP networks used for $T_w$ identification are represented in Figure 4 and Figure 5.

According to Figure 4, the network architecture has one input layer; next is the LSTM architecture, and then is the MLP with three hidden layers. The first one is with 258 nodes, the second one is with 128 nodes and the third one is with 24 nodes and an output layer. The AF are sigmoid, sigmoid and tanh; the learning rate is 0.01, and ADAM is used as an optimizer.

According to Figure 5, the network architecture has one input layer, 2 hidden layers (the first one has 10 nodes and the second one has 4 nodes) and an output layer. The AF is tanh for both hidden layers, and ADAM is used as an optimizer.

According to Table 1, $T_w$=1.434 (s) for Power Plant 1, while for Power Plant 2, $T_w$=0.479 (s). The obtained values for $T_w$ by using MLP and LSTM are defined in Table 2 if the available dataset is a guide vane blade position and active mechanical power.

Another $T_w$ identification is done by using a different dataset. It is a combination of a dataset of mechanical power inputs whose values (static parameters) are taken from the SCADA system of the hydro power plant itself, while the output dataset is experimental measurements made by the referent plant, Power Plant 1 (dynamic parameters). The reason for such a dataset is that when the identification for the $T_w$ coefficient is in process, the dynamics of Power Plant 2 are to be approximted to the dynamics of Power Plant 1 as a fast response and stable system. The network structure remains the same as in Figure 5 and the results are represented in Table 3.

If the values obtained are analyzed for root-mean squared error (RMSE) and for the $F_\text{opt}$, the results for points 1 to 4 of Table 2, by applying the MLP method, the datasets for the guide vane blade position and the active mechanical power are combined; it could be noted that the closer the predicted value of $T_w$ is to the real one, the lower the RMSE is. As the value of $T_w$ increases, the RMSE increases, which is expected because it moves away from the value of the already existing $T_w$. At the same time, if the $F_\text{opt}$ is analyzed, it could be noted that the difference appears in the fourth or fifth decimal place if the $F_\text{opt}$ value is compared for the existing $T_w$ (first value) and the ML predicted one (second value), which means that the predicted $T_w$ values are close to the real ones and identify the new system with the requested dynamics.

If the values of $T_w$ obtained by ML are analyzed (values for no. 5 do 13), it could be noted that when applying the MLP method, obtained results are also closer to the real one, and are with a lower RMSE, compared to the results obtained when applying the LSTM method. MLP as a method is far simpler compared to the LSTM method, therefore, from Table 2 and Table 3, it could be noted that the RMSE value when applying the MLP method as a simpler method is smaller compared to the LSTM method. That is expected because data processing does not require large computational power where the information moves in one direction without considering the influence of previous information. It could be concluded that it is not always necessary to use the strongest ML method or a complicated NN (neural network) consisting of more than one hidden layer or a huge number of neurons because it does not always guarantee the calculation of the best solution to the given problem.

In Figure 6, a summary diagram is presented for different values of $T_w$ obtained by applying the MLP method for the first real hydropower facility where the real value of $T_w$=1.1434 and the green line represents the grid load. According to the simulation results for the generated mechanical power, angular velocity, RMSE and $F_\text{opt}$, it can be noted that if the value of $T_w$ as part of the hydraulic subsystem is 0.175, the lowest value of RMSE=0.3% is obtained, but the value of $F_\text{opt}$ is also in the middle between the highest and lowest. If the responses for the turbine power are analyzed, it can be said that for $T_w$=0.157, a response without oscillations is provided, and the speed of reaching a steady state is identical with the remaining values of $T_w$. According to the results, in the design process, hydro turbines with $T_w$=0.8791 should be taken into consideration.

If responses from Figure 7 are analyzed, the hydraulic subsystem with $T_w$=0.5822 is considered, the generated mechanical power at a steady state approaches the requested value (in this case, the step function that represents the network load – blue line) will be slower, but with smaller dumping compared to the real value, $T_w$=1.434, and also with the smallest $F_\text{opt}$.

The responses shown in Figure 8 refer to Power Plant 2, where the same ML methods, MLP and LSTM, have been applied. For Power Plant 2, the real $T_w$=0.479. Figure 8 shows the responses from the MLP method. When analyzing, it could be noted that for $T_w$=0.195, a response with the smallest error is obtained, without oscillations when approaching, and at the same time for stabilizing the system in a steady state compared to the responses with other values of $T_w$. For $T_w$=0.60458, where the RMSE is as much as 8% and is the largest of the given values, it gives a response with the largest oscillations and the same time for achieving the steady state as for all other values of $T_w$.

If the results in Figure 9 are analyzed, for all new $T_w$ values, similar responses are obtained, from which $T_w$=0.8602 or $T_w$=0.87875963 can be chosen as the most appropriate because they generate a similar response, but slightly faster in approaching the steady state, and the time for settling the oscillations is similar to that of the existing value for $T_w$. From Table 3, it could be noted that for $T_w$=0.829, RMSE=76%, which means that the response constantly oscillates around a steady state with a tendency for the oscillations to decrease, but after a very long period of time.

If all identified values for $T_w$ are analyzed, it could be noted that the values obtained by applying MLP and LSTM are in the range of the real value, which is 0.479. This is also a good example from which we can say that not always the most powerful method and network will give the best results, sometimes a simpler method such as MLP is quite enough to obtain satisfactory results that will introduce improved dynamic characteristics of the control system.

According to the results in Figure 6, Figure 7, Figure 8 and Figure 9, could be noted that reverse power of the active power signal is still present after $T_w$ coefficient identification which means that using ML algorithm does not idealize the signal or neglect some physical conditions, because in some cases, ML can try to eliminate that part of the signal making it to look like it does not exist, which is not correct. According to references [8], [18], the reverse power should exist when analyzing the mechanical power and is experimentally verified.

This paper represented the importance of two ML algorithms, MLP and LSTM, in the SI purpose. These two were chosen because they are processing the signal/dataset in a completely different way. The model that has been worked on was two different hydro power plants. Because the $T_w$ – water starting time constant of the conduit is an important parameter for the hydro turbine definition, instead of its calculation, based on the available dataset, learning rate value, and optimizer type, the identification of the $T_w$ has been done by using MLP and LSTM.

According to the results represented in the figures above, it could be concluded that ML can do SI using different algorithms, meaning that its algorithms can be used in the systems design process. It should be noted that not every time the complicated and big network will give better results compared to a simpler network. In this paper, MLP has appeared as an algorithm whose capacities are enough for obtaining results that introduce improved dynamics of the controlled system. Another important part is that the results obtained by using the ML algorithm represent the signal with all its characteristics, such as reverse power, which means that the represented signals from the simulation could be compared with the measured and experimentally verified results.

Further work is needed for the laboratory measurement results to be represented. Also, to be presented is how ML methods are affecting the system control and how the problem is solved.