Passivity-Based Control for Small Hydro-Power Generation With PMSG and VSC

This paper presents a passivity-based control method for a small hydro-power system that consists of a permanent magnet synchronous generator (PSMG) connected to the grid through a back-to-back converter. Nonlinear models of the hydraulic, mechanical, and electrical parts of the small hydro-power system are considered. Two approaches in the realm of passivity-based control are implemented, namely, standard passivity-based control and PI-passive. These controls consider the intrinsic characteristics of the model, which has a port-Hamiltonian (pH) structure with a small hydro-power system in open-loop. The purpose is to design a control law with passive output which ensures asymptotic stability for closed-loop operation in the sense of Lyapunov’s theory. The paper is practically oriented, and hence, the proposed controllers are tested and compared with a conventional approach, in a 13.2 kV distribution feeder. The proposed controllers have been assessed and compared with a classical PI controller considering steady state and transient behaviors in small hydro-power plants (SHPs). Simulation results show that the proposed methodology guarantees stability and offers better dynamical performance.


I. INTRODUCTION
Small hydro-power plants (SHPs) are well known renewable energy sources with foreseeable energy production, at least in the short term, compared to other types of renewable energy sources such as photovoltaic and wind power. An SHP is usually composed of a reservoir, water tunnel, penstock, hydraulic turbine, speed governor, and generator. Modern systems may include a power electronic converter, as shown in Fig. 1, which allows high efficiency due to the opportunity of variable speed operation. This system has a strong coupling between mechanical, hydraulic, and electrical dynamics which makes the system difficult to analyze [1], [2]. Moreover, it presents many different operating points, which also complicate the analysis [3]. These features make the design and control of this system challenging.
The associate editor coordinating the review of this manuscript and approving it for publication was Fengjiang Wu .
Several works have focused on proposing controllers for governor systems. Typically, two types of governor controller can be found in the literature, namely, which are proportionalintegral-derivative (PID) [4]- [6] and state feedback controller [7]- [10]. PID controllers' design is based on system outputs and does not use the internal information of the SHP. Therefore, it is challenging to guarantee the stability for all operating points since linear models oversimplify the system dynamics. In the case of state feedback controllers, several controllers have been proposed such as nonlinear control [9], [10], intelligent control [8], sliding mode control [2], [11], fuzzy control [12]- [14], fault-tolerant control [15], predictive control [7], [16], synergetic control [17] and finite-time control [18]. However, these controls are observed to be difficult to implement and require tuning of many parameters and perform an on-line optimization process [18]. Most of these investigations analyze only one part of the system, for example, the hydraulic and/or mechanical dynamics without taking into account the electrical part. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ FIGURE 1. Proposed small hydro-power system with PMSG and back-to-back converter with hydraulic, mechanical and electrical subsystems.
Recent investigations have demonstrated the advantages of including power electronic devices into the energy conversion system to improve efficiency by allowing variable speed [3]. In [19] and [20], a model SHP connected to the ac grid using a permanent magnet synchronous generator (PMSG), diode bridge rectifier, boost converter and voltage source converter (VSC) was studied. This configuration was also used in wind power applications in [21], [22]. The model proposed in [20] was improved in [3] and presented a detailed analysis where high efficiency was achieved by reducing the electrical power losses. In [23], a multilevel hierarchical structure based on a PI controller was proposed; however, in that work, hydraulic dynamics were not considered.
Unlike these previous works, we take advantage of the port-Hamiltonian (pH) structure that presents the SHP in open-loop for the purpose of designing a controller based on passivity theory that keeps its pH structure in closed-loop and guarantees asymptotic stability in the sense of Lyapunov. The main contributions of the paper are summarized below: • Presents a mathematical model for an SHP employing a PMSG which is connected to the grid through a back-to-back converter. This model contains hydraulic, mechanical, and electrical (PMSG and VSC) dynamics, all of which are shown as a pH structure.
• Designs a controller based on the passivity theory for SHP. Passivity-based control (PBC) is employed since the mathematical model for SHP in an open-loop exhibits a passive structure, making it suitable to apply the passivity theory. The proposed control considers the complete nonlinear model of the system and guarantees local asymptotic stability in the sense that Lyapunov maintains its passive structure.
• Makes an analysis of the performance of the proposed controller under conditions of stable transient responses to demonstrate its robustness.
• Compares the PBC with a conventional PI controller, where the proposed controller presents better performance in all cases considered.
The remainder of all the paper is organized as follows: In Section II, the dynamic model of the SHP and its pH representation are presented. Next, in Section III, the passivitybased control and controller design for the SHP are described. Following this, simulation results and main comments are presented. Finally, some conclusions are drawn in Section V.

II. SMALL HYDRO-POWER PLANT
The hydraulic and mechanical parts consist of a hydro-turbine, a governor system, and a penstock. The governor's main function is to control the speed and power output by adjusting the hydraulic turbine's water flow. Typical governor systems consist of PID controls [10], [24]. The electrical part is composed of a PMSG connected to the grid through a backto-back converter. Both parts exhibit a passivity structure which is used to design the control.

A. HYDRAULIC TURBINE
The model considers an incompressible fluid as recommended by the IEEE working group on prime movers and energy supply models [26]. This model considers the dynamics of the penstock, the tunnel and the turbine as well as the head losses (see Fig. 1), as follows: where h is the hydraulic head, k f defines the friction losses on the conduit; q is the normalized flow on the penstock and T w is the starting time of water on the penstock, which is defined as where q base is flow rate when the gates are fully open, h base defined as the static head of the water column above the turbine [10]; L and A are the length and the area of the penstock, and g denotes the acceleration of gravity. The hydraulic head and the mechanical power of the turbine in per-unit values are given by where y represents the gate position, A t is a proportional constant and q nl is the no-load flow rate of the hydro-turbine. Fig. 2 illustrates the nonlinear turbine model described from (1) to (3). The hydraulic servo model can be described by where T y is the time constant of the servomotor and u g denotes input control, which is generally built with PID controls. Fig. 1 shows a PMSG connected to the power grid with a back-to-back converter [25], [27]. The electrical and mechanical equations that describe the behavior of the dynamics in values per-unit of a PMSG are given by

B. PERMANENT MAGNET SYNCHRONOUS GENERATOR
with where v d , v q , i dg and i qg are the voltages and currents of PMSG in dq reference frame, respectively. R g and L g are the stator winding's resistance and inductance, respectively. ω m is the rotor speed (ω m = ω e g in per-unit). ψ denotes permanent magnetic flux produced by the rotor magnets, which is constant and depends on the material used for its construction. H g is hydro-turbine inertia time constant and ω o is the generator based angular speed; T m and T e are the mechanical and electrical torque. Mechanical torque T m is obtained readily from (3) as and the electrical torque is given by Typically a back-to-back configuration with voltage source converters is employed to integrate a PMSG to the ac grid since this requires a full-rated converter [27]. Therefore, the output voltages of a PMSG can be defined as a function of the modulation indices of the converter as follows: v dq = m dq g v dc (9) where m dq g ∈ [−1, 1] represents the modulation index and subscript g refers to the VSC g ; v dc is the voltage in the dclink. The PMSG output voltages are represented as modulation indexes by the voltage in the dc-link (for more details, see [28]).
The dynamical modeling of the second VSC in the dq reference frame is given by where i dq are the currents that flow through of the transformer and L and R are their inductance and resistance parameters, respectively. e dq is the ac voltage of the main grid; C is the dc-link capacitor of the VSC and its voltage is v dc and i s is the current delivered by the PMSG. ω e is the grid angular electrical frequency, which is found by employing a classical phase-locked loop (PLL) [29]. m dq ∈ [−1, 1] represents the modulation index of a VSC connected to the ac grid.

C. COMPLETE DYNAMICAL MODEL
The dynamical systems described from (1) to (5) are rewritten in a pH structure, 1 as follows where J (x) 34 = L g ω m , J (x) 45 = ψ, while and dynamical systems (10) can also be written as a pH structure as follows: where J o 13 = Lω e , J 1 13 = 1, J 2 23 = 1, while where x g , x v , d g , d v and u g , u v are state, external input and control signal vectors, respectively; D g , D v , R g (x), R v and J are the generalized inertia, damping and interconnection matrices, respectively. The term J i km denotes position km of matrix J i and fulfills that J i km = −J i mk . Observe that the matrices defined above satisfy that In order to design the control, we assume that: Assumption 1: The water flow rate in normal operation and in per-unit is limited between q nl < x 1 ≤ 1 (see [31] for a more physical justification of this assumption).
Assumption 3: Mechanical speed of PMSG is always greater than zero x 5 > 0 and the reference mechanical speed of PMSG ω ref m = 1. Assumption 4: The dc-link voltage is always greater than zero v dc > 0.
Assumption 5: The states and external inputs are measurable and all parameters of the system are known.
Assumptions 1 and 2 are necessary for a machine that delivers active power. The water flow rate is the conduit, and the turbine always moves in the same direction. Assumptions 3 is logical since the rotor speed should always be close to constant. Assumption 4 is necessary for a VSC to operate properly. Assumption 5 is reasonable since measuring rate flow, rotor speed, and currents are standard practice in an SHP.

III. PASSIVITY-BASED CONTROL
The PBC theory is a well-founded technique that has proven to be very useful and powerful in designing robust controllers when applied to dynamical systems with a pH structure [32]. The PBC preserves the pH structure and guarantees stability in closed-loop in the sense of Lyapunov [32], [33]. The standard PBC (S-PBC), IDA-PBC, and PI passive techniques have been widely used in a number of applications, including mechanical, electro-mechanical, and power electronic systems [34]. In this section, the controller designs for SHP and VSC are shown.
A. S-PBC Given that the system (11) is an input-affine system, it is appropriate to use an S-PBC. 2 Therefore, the system (11) can be described in closed-loop as follows where e g = x g − x g , x g is a desired equilibrium point, v is an additional damping injection, which is selected as with the damping injection matrix given by 2 See [35] for further details of the input-affine system and S-PBC.
A replication of the system dynamics of SHP (11) is proposed, as follows Now, subtracting (11) from (28) and employing (26) yields the system described in (29): where R = R g (x) + R a > 0.
The system (25) is asymptotically stable at any equilibrium point with the control law u g . In order to see this, let us define a Lyapunov function candidate and, taking its time derivative yieldṡ V g (e g ) = e g D gėg = e g (J −R ) e g = −e g R e g < 0, if R is a positive definite matrix,V g (e g ) < 0 is asymptotically stable, which implies that x g converges asymptotically to x g [30].

1) CONTROLLER DESIGN FOR THE SHP
This section presents the controller design for the SHP based on the previously presented S-PBC theory. The control laws are found from (28), as follows Now, the reference values for the SHP needs to be defined. Typically, the PMSG's the control variables are rotor speed and direct axis current. x 5 allows control of the rotor speed, i.e., x 5 = ω nom m and to guarantee the maximum torque x 3 = 0. To control the electrical power delivered by PMSG, we use the following expression with, where P e is the reference electrical power, and the admissible trajectories for the non-controlled variables are achieved from (28), which generate: B. PASSIVITY-BASED PI CONTROL Given that the system (18) is an input-non-affine system, it is sufficient to employ PI-PBC. 3 Let x v define an admissible trajectory of the system (18). I.e., x v is an equilibrium trajectory on (18)

if it exists, is differentiable, is bounded if it satisfies
and it is produced by some bounded control signals u v . An admissible trajectory x v of a dynamical system is just a solution which belongs to the flow of the differential equation for a bounded input u v . An admissible trajectory x v is not reached if there is no bounded u v that generates it. The admissible trajectory concept is indispensable for designing controllers based on passivity-based theory.

1) PASSIVE CONTROL FOR pH INCREMENTAL MODELS
This section presents an incremental model of the dynamical system shown in (18) and it is demonstrated that the incremental model is passive.
The dynamical system presented in (38) is defined based on the incremental model, i.e., (18).
and for an output function defined as is passive, if it satisfies the dissipation inequalityṠ ≤ỹ vũ v , whereỹ and the storage function is given by Applying the temporal derivative of the storage function (41) along the trajectories of (38), the following equation is obtained.
This proves that the system (38) is passive from u v −→ y v . 3 See [36] for further details of the input-non-affine system and PI-PBC. 4 This matrix was already established in [33].

2) PI GLOBAL TRACKING CONTROLLER
The VSC described by (18) can be controlled in closed-loop by the controllerż and output y v by (40). Consider gain matrices in which K p = K p > 0, K i = K i > 0, and fulfilling some admissible trajectory x v given by (37). To achieve global tracking for any initial conditions (x v (0), z v (0)) of the system (18), it is necessary to satisfy the rank condition rank(M ) = n, with This property was already reported in [33]. Let us define a Lyapunov function candidate as Calculating the time derivative (47) along (38) giveṡ

3) CONTROLLER DESIGN OF THE VSC
This section presents the controller design of the VSC based on PI-passive theory shown above. Applying (40), the passive outputs areỹ and, the asymptotic convergence condition is checked by means of the following matrix while x 6 or x 7 are bounded away from zero, the full-rank condition is fulfilled and x v will converge to x v asymptotically. This situation occurs in normal operation of the VSC since PMSG is always generating the active power, i.e., x 6 > 0. Defining x 8 and x 7 as the reference values for the VSC. x 8 is selected to control dc-link voltage on the capacitor C, i.e., x 8 = V nom dc . The x 7 permits us to control the reactive power delivered (or absorbed) by VSC to the ac grid, which is calculated as where Q is the reference reactive power and P e is the active power delivered by PMSG. VOLUME 8, 2020  The controller design is fulfilled when x 6 , m d2 and m q2 are determined. Therefore, the admissible trajectory for VSC is achieved from (37), and given by m d2 = Lẋ 6 + Lωx 7 + e d + Rx 6 x 8 , m q2 = Lẋ 7 − Lωx 6 + e q + Rx 7 x 8 , Fig. 3 illustrates the control scheme of the proposed controllers. Fig. 4 depicts the test system employed to assess the proposed controller. This test system is a modification of the one introduced in [28], which has a 13.2 kV distribution feeder with a 2-MW SHP and four loads. The parameters of the system and SHP are listed in Table 1.

B. SIMULATION CASES
We propose three simulation cases in order to investigate the effectiveness of the proposed controller of the SHP, and the simulations were implemented using MATLAB/Simulink. For validation, the proposed controller is compared with a conventional PI controller. It is important to note that when we refer to the PI controller for the SHP, it is divided as follows: the governor system is controlled with the PID controller, while the PMSG and the VSC are controlled with PI controllers.
The parameters of the proposed controllers and PI controller are listed in Table 2.

1) CASE I
In this case, it presents the ability of the PBC to control the active and reactive output power of the SHP on the point of common coupling (PCC).
The simulation results illustrated in Fig. 5 show the active power delivered by the hydro-power plant on the dc-link and the rotor speed deviation. Note in Fig. 5(a) that the response of both controllers presents the intrinsic non-minimum phase characteristic of the hydro-power plant. However, the PBC has a better response than the PI controller with an average error of  around 5%, while the PI controller presents an average error of around 8.4%. For rotor speed regulation, the proposed controller continues to show a better performance than the PI controller since this maintains an error of around 1 · 10 −4 (see Fig. 5(b)). Fig. 6 shows the active and reactive power delivered on PCC by the SHP and the dc-link voltage. By comparing Fig. 6(a) with Fig. 5(a), it is observed that the active power behavior is maintained where the PBC presents an average error of around 6.4%, while the PI controller has an average error of around 9.2%.
Observe in Fig. 6(b) that both controllers follow the desired reference. However, the proposed controller shows a better dynamic response than the PI controller with a standard deviation of 0.6%, while the PI controller has a standard deviation of 0.9%. For the voltage regulation, both controllers present similar behavior.

2) CASE II
This case investigates the performance of the proposed controller when the SHP system delivers from low to high power. The active power reference starts at 0.1 pu and in t = 0.4 s the reference changes to 0.93 pu. The results are illustrated in Figs. 7 and 8. The active power provided by the PMSG for both controllers is depicted in Fig. 7(a). Note that the proposed controller continues to show a better response than the PI controller. The proposed controller has an average error of around 22.5%, while the PI controller has an average error of around 25.4%. It is important to mention that average errors are high since the settling time is approximately 0.7 s.
Observe in Fig. 7(b) that rotor speed deviation has the same behavior as shown in case I (see Fig. 6(b)). Hence, the PBC continues to demonstrate better performance than the PI controller since the conventional controller maintains an error of around 1.16 · 10 −4 .
Analyzing Figs. 7(a) and 8(a), it can be observed that the active power behavior is maintained when the PBC and the PI controllers are implemented and the average errors are around 24.4% and 28.6%, respectively.
Note that in Fig. 8(b), both controllers can follow the desired reference. However, the PBC has a better dynamic response than the PI controller with a standard deviation of 1.1%, while the PI controller has a standard deviation

3) CASE III
The transient behavior of the SHP is analyzed in this case. The active power generated by the SHP is maintained constant at 0.9 pu. A three-phase short-circuit to ground at node-3 is considered in t = 0.4 s. The results are presented in Figs. 9 and 10. The active power generated for both controllers on the dc-link voltage decreases due to the grid voltage drop at Node-3. However, both controllers returned to initial values (see Fig. 9(a)). The rotor speed maintains its speed during the fault (see Fig. 9(b)).
The active power delivered on PCC decreases to almost zero for both controllers. Next, when the fault is removed, the active power provided by the PBC has a higher peak than the PI controller. However, the proposed controller stabilizes the system in less time compared with the PI controller (see Fig. 10(a)). Note in Fig. 10(b) and 10(c) that both controllers show the same dynamic response for the reactive power and dc-link voltage, respectively.

4) COMPLEMENTARY ANALYSIS
To quantify the performance of the proposed controller, the integral of the time-weighted absolute error ( ) is used. The settling time t p for the active power delivered by the PMSG is also considered. -values are computed for the rotor speed deviation, the dc-link voltage, the active power delivered by the PMSG, and the reactive power, as follows where t sim is the simulation time. Table 3 shows the performance indices for each case considered. Note that these indices validate the enhanced performance for the proposed controller from the perspective  of less steady-state error and reduced settling time of each case proposed.

V. CONCLUSION
A new approach for the control of SHPs based on passivity theory was described in this paper. The SHP consisted of a PMSG connected to a three-phase grid through a backto-back converter. The S-PBC and PI-PBC approaches for an SHP connected to a test feeder system were described. These controllers guarantee asymptotic stability through Lyapunov's theory. The controllers were assessed and compared with the classic PI controller considering state and transients behaviors in SHPs. It was observed that the controllers based on PBC showed better performance in all cases considered compared to the PI controllers. This has been confirmed by comparing with the integral of time-weighted absolute error and settling time.