Optimal LQG controller to adjust the rudder supplying water to the turbine of small and medium hydro power plants

Research article. https://doi.org/10.16925/2357-6014.2021.03.07 1 Electric Power University in Vietnam Email: dangtientrung@gmail.com ORCID: https://orcid.org/0000-0001-7388-8037 2 AD-AF Academy of Viet Nam, Son Tay, Ha Noi, Viet Nam Email: lengocgianglinh@gamail.com ORCID: https://orcid.org/0000-0002-0198-2857 3 AD-AF Academy of Viet Nam, Son Tay, Ha Noi, Viet Nam Email: tinhpk79@gmail.com ORCID: https://orcid.org/0000-0003-3451-7721 4 Faculty of Electronics Engineering Technology, Hanoi University of Industry Email: dieulinh79@gmail.com ORCID: https://orcid.org/ 0000-0001-8596-392X Optimal LQG controller to adjust the rudder supplying water to the turbine of small and medium hydro power plants


INTRODUCTION
Article [1] presented a mathematical model describing the relationship between the rotation angle of the rudder to supply potential energy and kinetic energy of a water column for turbines of the combination "turbine + generator" in small and medium-sized hydroelectric power plants. However, the order value formation algorithm, to stabilize the frequency of transmission voltage at 50 Hz standard value, was not presented. In this paper, the authors present a solution to apply the optimal control theory to create a command to control the rudder angle to adjust the water flow into Proportional-integral structured optimal controllers are designed using a fullstate feedback control strategy employing performance index minimization criterion.
Some traditional single/multiarea and restructured multiarea power system models from the literature are explored deliberately in the present study. The dynamic performance of optimal controllers is seen to be superior in comparison to integral/proportional-integral controllers tuned using some recently published modern heuristic optimization techniques. It is observed that optimal controllers show better system results in terms of minimum value of settling time, peak overshoot/undershoot, various performance indices, and oscillations corresponding to change in area frequencies and tie-line powers along with maximum value of minimum damping ratio in comparison to other controllers. This paper's contribution lies in its employment of an effective optimal LQG control for varying operating conditions. Optimal LQG control is one of the most successful control algorithms and is widely used in handling multivariables and constraints. The designed control discipline is based on a mathematic model of a controlled object with the prescribed limit to acquire the optimal performance index [8][9][10].
According to previous research on PID control, the conventional control algorithm gives good results at infinite steady state; the only difficulty occurs when the reference trajectory is fluctuating [2,3,4]. This paper builds on the ideas presented in [5][6][7]. The performance of a controller using optimal LQG control and PID will be analyzed and discussed in the simulation section.

Building control algorithm
Article [1] described the relationship between the control signal to rotate the rudder and the rotation frequency of the turbine as follows: Ingeniería Solidaria Where: α is the rudder opening angle; ω is the rotation frequency of the turbine; Parameters T , T α , K , u K depend on the pressure and flow rate of the water column; Parameter 2 z depends on the pressure of the water column; Parameter 1 z depends on the pressure and flow rate of the water column, and load consumption. In that paper, we also presented an algorithm to identify these uncertainty parameters.
Because the difference between the generating frequency and the reference frequency 0 0 2 2 50 100 f ω π π π = = = (rad/sec) is the basic information for forming the control signal, the following should be set: 1 0 x ω ω = − 1 0 x ω ω = + (4) (4) can be substituted into (1) to get the equation: Equation (2) will have the following form: From the three linear differential equations (5), (7), (8) has the following linear dynamic system: Set state vector: From the three equations (9), (10), (11) there are dynamic equations in the form of state space as follows: (13) Where: The task of controlling generators in hydropower plants consists of two main sub tasks : control of generator excitation systems to stabilize the amplitude of the output voltage at the nominal value and control of the rudder supplying water to the turbine to stabilize the frequency of the output voltage at the nominal value.
The control of generator excitation systems has been published many times, so it is not considered in this paper. For all hydraulic generators currently available in Vietnam, the directional control algorithm often uses a PID control algorithm. However, this algorithm has different transient times when the load changes. In addition, the coefficient set for the PID controller is only reasonable when the parameters of matrices A, B, C in model (13) do not change.
During operation, due to the change of in power consumption load, the rotating frequency of the generator will change deviating from the standard frequency.
( 0 100 The control task must change the rudder angle so that the frequency returns to nominal frequency 0 ω , ie bringing the value of x 1 to zero ( 1 0 x → ).
So we can set up the optimal control problem as follows: Find the rule that changes the parameter value affecting the kinematic system (13) so that the function is: The function (21) can be written in the following standard form: T is the end of the control process 11 12 Apply the optimal control theory [3,4] to solve the above problem to determine the law of value change U. First, set the Hamilton functions as follows: Notation , is the scalar product of two vectors. Vector ( ) P t is determined by: (25) With boundary conditions: In the optimal orbit, satisfy the following equation: (24) and (27) that: Vector ( ) P t can be set as follows: To ensure the boundary condition (26), there are two conditions: Ingeniería Solidaria From (25) and (33) we have the following equation: From (13) and (34) we have the following equation: From (29) and (36) we have the following equation: From (30) and (37) we have the following equation: Equivalent to: For equation (39) to be true to all values of ( ) X t , then ( ) Combining equation (40) with boundary conditions (31) and combining equation (41) with boundary conditions (32) provides a system of two differential equations to determine the matrix ( ) x K t and determine the vector 1 ( ) K t : Where: (45) According to [2], in case the integral time f T is long and the vector ( ) V t does not change, the solution of equations (42) and (43) can be determined on the basis of solving the following algebraic equation system: Ingeniería Solidaria (47) [2] presents the solution of the Riccati quadratic nonlinear equation system (46).
After determining the coefficient matrix x K , the solution of (47) is: (48) After determining x K and 1 K , (29) and (30) provide the optimum control command to adjust the angle of the rudder supplying water to the turbine to rotate the generator as follows: (49) shows that to synthesize an optimal control law, one needs to determine the matrix x K through solving algebraic equations (46). To determine the vector Kt according to (48) it is necessary to determine the state vector X of the linear dynamic system (13), i.e. to measure or observe the deviation between the transmitter voltage frequency and the grid voltage frequency; the angle of the rudder and its opening speed. Thus, when designing and manufacturing the turbine and generator complex, it is necessary to arrange and install the corresponding measuring devices to measure or observe the information on the state of the dynamic system. In cases where it cannot be measured directly but must be observed, it is necessary to have the same algorithm and software to observe those parameters.

The Kalman state observer [11,12]
To establish optimal control rules of the form (13) one must have status information x . This information is difficult to measure, so a Kalman filter must be applied.
Where ( ) w k is measurement error ( ) U k and parameter identification error 2 z . Using the Kalman filter algorithm: With two components of state vector x is measurable, as is the rudder rotation angle α : Where ( ) v t is the measuring error of the rudder rotation angle due to the measuring device. The base transfer matrix of the Kalman filter is as follows: From (52) the observation matrix of Kalman filter can be determined: From the system of equations (51), the noise intensity matrix is: The covariance matrix Q , R of the Kalman filter procedure is: Thus, it is possible to implement the Kalman filter algorithm to determine the composition of 3 x , ie the rudder rotation angle, serving the synthesis of optimal control law. The filtering algorithm will include the following steps: • Step 1: Establish initial matrix ( 1) P k + − as follows: Step 6: Build state vector: [ ] x are the components of ˆ( ) X k + determined by (59).

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Step 7: Provide X to the control device to synthesize the optimal control law.
• Step 8: Calculate the matrix The next filtering cycle will start from the second step (step 1 only performed once in the first cycle). The next filtering cycle is performed when the command T = 1.
The command T = 0 instruction is issued by the control device when ordered to stop the operation of the turbine.

RESULTS
For all hydraulic generators currently available in Vietnam, the directional control algorithm often uses a PID control algorithm. Therefore, in this paper, the authors conducted simulations with 2 algorithms used for the directional control of hydraulic generators: PID control algorithm and the LQG control algorithm, to compare and evaluate control efficiency. The simulated object is a combination of turbine and generator with a counter-type turbine model (using both the potential and kinetic energy of the water column). The specifications of the unit are as follows: • Rated power: 30 MW, • The rotor has 4 pairs of poles, corresponding to the standard rotation frequency:     The above graphs show that, with optimal control: the turbine rotation frequency acheives the standard value with small fluctuations, the setting time is small, the objective function J is also small (J = 0.0902). With PID control law, the objective function J also increases (J = 0.11157, J = 0,0947). This shows the superiority of the optimal control law.

DISCUSSION AND CONCLUSIONS
The use of renewable energies has been increasing in the recent years due to the current costs of oil and gas. Among them, hydroelectricity is the most developed renewable energy source throughout the world.
In [13,14] a classical technique, known as Direct Power Control was used, but this method is not robust when presented with grid instability. Various research projects have been conducted in this field ; for example [15] models a whole set of micro hydro power plants with electrical machines with control by fuzzy nonlinear systems.
Although, the fuzzy technique guarantees stability, it should be noted that non-linear constraints lead to complicated calculations.
In this paper, by identifiying the kinematic model parameters and estimating the consumption load, we are able to set up the optimal control problem for the water supply flow direction into small and medium hydropower turbines. An LQG optimal control theory was applied, building an algorithm to determine the status coefficient matrix of the kinematic system and the load vector in the control command structure.
The structure of the control command will determine the return request regarding hardware and software when designing and manufacturing the turbine -generator combination. The algorithm presented in the paper is the basis for setting up the software when designing and manufacturing the turbine -generator combination.
Applying this algorithm, the process of adjusting the transmitted power according to the required load will be performed with quality.