Standard passivity-based control for multi-hydro-turbine governing systems with surge tank

Applied Mathematical Modelling 79 (2020) 1–17

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Standard passivity-based control for multi-hydro-turbine governing systems with surge tank Walter Gil-González a,∗, Oscar Danilo Montoya b, Alejandro Garces a a b

Programa de Ingeniería Eléctrica, Universidad Tecnológica de Pereira. AA: 97, Pereira 660003, Colombia Programa de Ingeniería Eléctrica e Ingeniería Electrónica, Universidad Tecnológica de Bolívar, km 1 vía Turbaco, Cartagena, Colombia

a r t i c l e

i n f o

Article history: Received 1 February 2019 Revised 31 October 2019 Accepted 5 November 2019 Available online 20 November 2019 Keywords: Euler–Lagrange representation Decentralized control approach Lyapunov’S stability Hydro-turbine governing systems Standard passivity-based control

a b s t r a c t This paper addresses the problem of control design for hydro-turbine governing systems with surge tanks from the perspective of standard passivity-based control. The dynamic model of a synchronous machine is considered in conjunction with a model of the hydroturbine to generate an eleventh-order nonlinear set of differential equations. An Euler– Lagrange representati of the system and its open-loop dynamics is developed. Then, the standard passivity-based control is applied to design a global and asymptotically stable controller in closed-loop operation. The proposed control is decentralized to avoid challenges of communication between the hydro-turbine governing systems. The proposed standard passivity-based control approach is compared with two control approaches. First, a classical standard cascade proportional-integral-derivative controller is applied for the governing system, the automatic voltage regulator, and the excitation system. Second, a sliding mode control is also implemented in the governing system. Two test systems were used to validate the performance of the proposed controller. The first test system is a single machine connected to an infinite bus, and the second test system is the well-known Western System Coordinating Council’s multimachine system. Overall, simulation results show that the proposed controller exhibits a better dynamic response with shorter stabilization times and lower peaks during the transient periods. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Large-scale power systems provide electrical energy for vast regions that cover many countries [1], being the geographic location an influential factor in the structure of these systems. In Europe, the energy mix is mainly composed of thermal plants (based on carbon and nuclear fuels) with a high penetration of renewable generation, originated from large-scale wind and solar photovoltaic farms [2]. In Africa and Asia, the energy mix consists of renewable generation, thermal plants, and hydraulic power centrals. Electricity in North America is predominantly generated by thermal power plants, conversely, in Central and South America electricity comes mainly from hydraulic power plants [3]. Table 1 summarizes the composition of the energy mix in Colombia [4]. Table 1 shows that Colombian power generation comprises 63.39% from large-scale hydraulic power plants, 29.38% from thermal power plants, and the remaining 7.28% from small-generation sources. Thus, electricity in Colombia is generated

∗

Corresponding author. E-mail addresses: [email protected] (W. Gil-González), [email protected] (O.D. Montoya), [email protected] (A. Garces).

https://doi.org/10.1016/j.apm.2019.11.010 0307-904X/© 2019 Elsevier Inc. All rights reserved.

2

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Nomenclature Parameters TW1 starting time of water in the tunnel. TW2 starting time of water in the penstock. Tj time constant of the servomotor. Th storage constant of the surge tank. kf1 friction losses in the tunnel. kf2 friction losses in the penstock. L inductance matrix. Ldq dq−axis stator inductances. Lf field inductance. Lmdq dw−axis magnetization inductances. R resistance matrix. rs stator resistance. rf field resistance. rDQ DQ−axis rotor damping resistances. M inertia moment. k friction constant. τm mechanical torque. τe electrical torque. Ay proportionality constant of the hydraulic turbine. qnl no-load flow rate. Ay proportionality constant of the hydraulic turbine. i i-th generator connected to the power system. Variables q1 q2 h hs y

ω

i idq if iDQ

ψ ψ dq ψf ψ dq v vdq vf uy x xd

flow rate in the tunnel (x1 ). flow rate in the penstock (x2 ). head at the turbine. head at the surge tank (x3 ). servomotor position (x4 ). rotor speed (x5 ). current vector. dq−axis stator current (x6 , x7 ). Field current (x8 ). DQ−axis rotor damping windings currents (x9 , x10 ). flux linkages vector. dq−axis stator flux linkages. dq−axis stator field linkage. DQ−axis rotor damping windings flux linkages. voltage vector. dq−axis stator voltages. field voltage (u2 ). input control to the governor system (u1 ). state variables desired equilibrium point.

predominantly from hydraulic power plants, required to guarantee voltage and frequency regulation across the whole power system. This is a difficult task, especially when large disturbances occur in the power systems [5]. The main goal of the power system is to maintain continuous and reliable operation even when large disturbances are presented, such as the sudden disconnection of a generator, line, load, or dead short-circuits (also with low impedance) on buses and lines [6]. This level of operation involves sustaining the synchronization of all generators before, during, and after a fault event or large fluctuation in operating conditions [1]. Another essential issue arises when hydro-dominated power systems interact with low-frequency oscillation phenomena, which occurs in the Colombian power system [5]. These power system requirements have motivated studies on hydro-turbine governing systems (HTGSs) to improve their dynamic performance. The HTGS is a nonlinear non-minimum phase dynamic system with strong couplings among mechanical, hydraulic, and electrical subsystems. The control of such systems is a complex and challenging task [6]. Moreover, other important factors can complicate the design of controllers for HTGSs, including the occurrence of water hammer or large electrical disturbances in the power system [6,7]. Due to the importance of HTGSs to the reliable and secure operation of power systems,

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

3

Table 1 Power generation capacity in Colombia [4]. Central dispatched Installed power [MW]

Hydraulic plants 10974.00

Thermal plants 5087.00

Noncentral dispatch Installed power [MW]

Small hydraulic plants 862.57

Small thermal generation 171.90

TOTAL 16061 Cogeneration 149.00

Self-power generation 49.64

Wind plants 18.42

1251.53 17312.53

Fig. 1. Classification of the citations mentioned according to the dynamics considered.

operational controls must be studied and improved continuously to prevent undesired behaviors in the power system (such as blackouts or destruction of power system components). HTGSs typically incorporate classical control approaches based on linear cascade control strategies (specifically, proportional-integral-derivative (PID) strategies) for the governor system [8], and automatic voltage regulators (AVRs) in conjunction with power system stabilizers for the excitation system [9]. These control strategies are tested and tuned for the worst operating conditions in the power system (such as a three-phase symmetric dead short circuit [1]). Validations of these strategies are performed via multiple simulation scenarios under closed-loop operations [9]. These control schemes have been widely used for decades; however, they do not guarantee stability in accordance with Lyapunov’s theory, and they would be inadequate for the present-day composition of power system that include higher and more extensive penetration of renewable generation. Therefore, it is essential to investigate and develop new control schemes that guarantee the stability of the whole power system when subjected to strong disturbances [6]. Many recent studies have focused on proposing new control strategies for improving the dynamic response of HTGSs. Some approaches for the analysis of parameter uncertainties and unmodeled dynamics were proposed in [10]. Other studies were directed to reduce undesired oscillations by applying fuzzy and sliding control methods [11]. In [12], a power control mode for HTGS under primary frequency regulation taking into account the nonlinear characteristic of penstock head loss was developed. A nonlinear model of predictive control strategy for an HTGS was proposed in [13], and the robustness and efficiency of this approach was demonstrated using a performance index in conjunction with a penalty function. In [14], a control strategy was applied to the governing system for the frequency regulation of isolated AC networks, supporting voltage and frequency at the sending terminal of a high-voltage direct current system. In [15], a piece-wise function associated with the parameters of the dynamic model of a pumped storage hydroelectric power plant was studied. The effects of proportionalintegral gains on pumped storage hydroelectric power plants were analyzed in [16] by introducing random load behaviors. In [17], a novel perspective on the transient analysis and modeling of HTGSs was presented. In [18], the researchers used an improved gravitational search algorithm to design a fast fuzzy fractional order PID controller for a pumped storage hydro unit. Furthermore, strategies such as fuzzy approach [19], synergetic control [20], and intelligent method control [21] have been proposed in specialized literature. We note that these control techniques have performed well in their respective experiments on problems such as scalability to large-scale systems, modification of control rules, online optimization requirements, stability problems, or problems in adjusting control parameters. Besides, some of these studies did not consider all the dynamics of the HTGS (i.e., hydraulic, mechanical, and electrical dynamics), and they were not analyzed for large-scale power systems or under operational scenarios with strong disturbances. Other papers have reported studies on controllers based on port-Hamiltonian (pH) models that guarantee stability for HTGSs in accordance with Lyapunov’s theory [22–25]. These papers described the use of an orthogonal decomposition strategy known as generalized Hamiltonian theory for designing control strategies with a Hamiltonian structure in closed-loop systems [24,25]. Nevertheless, these methods required an exact model of the system, which complicated the designs and lacked physical interpretation. In [6,7], a mathematical formulation that represents the HTGS model in open-loop by a Hamiltonian form was proposed. Under this structure, the researchers obtained a passive control law that can guarantee stability under operating conditions based on Lyapunov’s theory via Hamiltonian equivalent formulations. In Fig. 1 is shown a compilation of the citations mentioned in this paper, which summarizes the dynamics considered by each one of them. Note that there are few works focused on considering the entire hydraulic, mechanical, and electrical dynamics. However these investigations have some limitations, such as: only the third-order model of the synchronous machine is considered, which only addresses mechanical and flux field dynamics; the

4

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Fig. 2. General scheme of a hydropower plant [6].

study did not consider a multimachine structure; and the proposed controller had a centralized structure, which made it difficult to extend its application to large-scale power systems because the long distances between HTGSs may require large communication systems with high costs and low reliability. Unlike the studies mentioned above, in this paper, we propose a standard passivity-based control (SPBC) approach, framed within an Euler-Lagrange representation, to design a decentralized control strategy for HTGSs in multimachine power systems. This approach addresses the stabilization problem of the rotor speed and attempts to regulate the output voltage (terminal voltage) for the HTGS. Additionally, the dynamic model (sixth-order model) of the synchronous machine is considered. An Euler-Lagrange representation and control strategy is employed to analyze the HTGS with surge tank, to control the rotor speed and voltage output of the multimachine power system. The proposed control scheme includes a decentralized design, which avoids the use of communication channels between geographically separated HTGSs. Additionally, the proposed control is designed under the SPBC paradigm by preserving the system open-loop dynamics to propose a closed-loop controller that guarantees global asymptotically stable in accordance with Lyapunov’s theory. The mathematical description of each HTGS is framed within an eleventh-order nonlinear set of differential equations that is reduced to a tenth-order model. This model reduction eliminates the rotor angular position as a dynamic variable because the stability of the angular rotor speed is guaranteed, and the rotor angular position is uncoupled from the system dynamics. To evaluate the robustness of the proposed Euler-Lagrange controller design, the standard controller and sliding mode control approaches are used for comparison purposes. The standard approach considers a voltage regulator, an exciter based on the IEEE type ST1A excitation system model, and power system stabilizers IEEE-PSS1A. Additionally, a PID controller with transient and static droops is employed for the governing system. This paper is organized into five sections. Section 2 presents a mathematical model of the power system, including a broad description of the hydro-turbine, synchronous machine, and electrical network models according to their EulerLagrange representations. In Section 3, the SPBC design is presented through the control inputs and calculations for the unknown controlled variables. The test system, simulation parameters, and results are presented and discussed in Section 4. Finally, Section 5 gives the concluding remarks of this study. 2. Power system model The power system dynamic model is composed of the hydraulic, electrical, and mechanical subsystems of each generator. This model also contains the electrical network. We assume that the power system meets the following assumptions: A1. The power system is represented by a balanced and symmetrical model. A2. The rotor angular position is known, thus it is possible to apply the Park transformation. 2.1. Hydro-turbine model Fig. 2 illustrates a nonlinear dynamic model of a hydraulic turbine scheme that includes a penstock and a surge tank. There are several models for Hydro-turbine system, which include a common penstock with several pipes [24,26]. Other models are approximated using Taylor decomposition equation for the moment and flow features curves provided by the manufacturer [20]. We employ a hydro-turbine model based on the water wave model from conditions of dynamic equilibrium and continuity, which it is used in frequency oscillation studies of hydro-dominant power systems [5,8]. This model considers the water as an incompressible fluid, as well as, the penstock, surge tank, servomotor dynamics, tunnel head and, penstock head and tank orifice losses [1,8]. The dynamical model for the hydraulic turbine is obtained by applying momentum laws on the tunnel and penstock where the rate of change of flow at the gate is modeled as the rate of change of the

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

5

water momentum in the penstock to the pressure head at the conduit [6], as described in (1).

TW 1 q˙ 1 = 1 − hs − k f 1 q21 TW 2 q˙ 2 = hs − h − k f 2 q22 T j h˙ s = q1 − q2

(1)

Ty y˙ = uy − y where q1 and q2 denote the flow rates in the tunnel and penstock, respectively, y denotes the servomotor position; h and hs represent the heads at the turbine and surge tank, respectively. kf1 and kf2 denote the friction losses in the tunnel and penstock, respectively. TW1 and TW2 represent the starting times of water in the tunnel and penstock, respectively. Tj is the storage constant of the surge tank, Ty is the time constant of the servomotor, and uy is the input control to the governor system. The hydraulic head in the turbine relates the flow rate and gate position which can be expressed as

h=

 q 2 2

y

.

We make the following assumptions: A3. During normal operations, the water flow rate in the penstock is bounded according to qnl < q2 ≤ 1 [23]. A4. The gate position for each unit is bounded according to 0 < y ≤ 1 [23]. Note that these assumptions are given for the physical limitations of the system which are supported, as follows: Assumptions 3 and 4 are necessary for the HTGS to deliver active power. Additionally, the water flow rate on the conduit and the turbine always move towards the same direction. The gate position is limited by a maximum gate opening. Note that equilibrium points of (1) are given by:

q1 = q2 ,

q2 =



1 1 + (k f 1 + k f 2 ) y2

,

hs = 1 − k f 1 q21 .

(2)

By observing (2), it can be seen that y must be greater than zero to guarantee the feasibility of the system. Hence, the equilibrium point depends on the operation point of the power system. 2.2. Synchronous machine model Generally, a synchronous machine is composed of electrical and mechanical dynamics. Applying the Kirchhoff and Faraday laws to the stator, field, and damper windings of the synchronous machine, the electrical dynamics can be defined in dq reference frame as follows:

v = −Ri − ψ˙ + Cω ψ , where v = col(vd , vq , −v f , 0, 0 ), vd and vq denote the stator direct and quadrature axis voltages, respectively, vf represents the field voltage; i = col(id , iq , i f , iD , iQ ), id and iq denote the stator direct and quadrature axis currents, respectively, if represents the field current, iD and iQ are the rotor damping windings currents; ψ = col(ψd , ψq , ψ f , ψD , ψQ ), ψ d and ψ q denote the stator direct and quadrature axis flux linkages, respectively, ψ f represents the field flux, ψ D and ψ Q are the rotor   damping windings flux linkages; R = diag rs , rs , r f , rD , rQ , rs denotes the stator resistance, rf represents the field resistance, rD and rQ are the rotor damping windings resistances; and



Cω =

I (ω ) 03×2

02×3 03×3





I (ω ) =

0

ω



−ω . 0

(3)

Moreover, all the flux linkages of the generator are magnetically coupled in such a form that the flux in each winding depends on its current and the currents in all the other windings. This relation is expressed as

ψ = Li, with,



L L = T11 L12

(4)

L12 L22





L11

L = d 0

0 Lq





L12

L = md 0

Lmd 0

0 Lmq



 L22 =

Lf Lmd 0

Lmd LD 0



0 0 , LQ

where Ld and Lq denote the stator direct and quadrature inductances, respectively; Lf represents the field inductance; and Lmd and Lmd are the direct and quadrature magnetization inductances, respectively. Therefore, the electrical dynamics can be rewritten only in the flux linkages from (3) and (4), as follows:

 ψ˙ = −RL−1 + Cω ψ − v.

(5)

6

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

On the other hand, the mechanical dynamics can be posed as torque-acting balance on the rotor shaft, as follows:

Mω˙ = τm − τe − kω, θ˙ = ω,

(6)

where ω and θ denote the speed and angle of the rotor, respectively; M is the inertia moment of the rotor shaft; and k is the friction constant. Note that expression (6) models the unbalance of torque in the turbine between the mechanical and electrical torques that leads to its acceleration or deceleration. The mechanical torque τ m generated by the turbine is computed as the product of the flow rate and the head [6]. Since the turbine is not 100% efficient, it is necessary to include its losses, which are calculated by subtracting the no-load flow from the actual flow. Therefore, the turbine torque in per unit is given as

τm =

At q22 (q2 − qnl ) y2 ω

where At denotes the constant of proportionality of the hydro-turbine and qnl represents its no-load flow rate [1]. The electrical torque τ e is produced when the rotor magnetic field interacts with the stator magnetic field, as shown in (7).

τe = ψ q i d − ψ d i q ,

(7)

2.3. Euler-Lagrange model The dynamic systems described by (1), (5), and (6) can be represented as an Euler-Lagrange structure, which has the following form:

D x˙ + [C (x ) + R(x )]x = Gu + b(x ).

(8)

where x ∈ Rn is the vector of state variables, b(x ) ∈ Rn is a vector of stable and bounded inputs that depend on x, u ∈ Rm (m ≤ n ) is the input control vector, C (x ) = −C (x )T ∈ Rn×n and R = RT  0 ∈ Rn×n are the interconnection and damping matrices, respectively, D = D T  0 is the inertial matrix, and G ∈ Rn×m is the input matrix. Now, the vectors and matrices for HTGS are defined, as:

xi = col(q1i , q2i , hsi , yi , ωi , ii ), ui = col(uyi , v f i ), bi (xi ) = col(1, 0, 0, 0, τmi , −vdi , −vqi , 0, 0, 0 ),



Di = diag TW 1i , TW 2i , T ji , Tyi , Mi , L−1 i



Ri (xi ) = diag k f 1i x1i ,



C Ci ( xi ) = 1i 0







(9)



1 + k f 2i x2i , 0 , 1 , ki , Ri , x24i

0 , C 1i = C 2i



0 0 1

0 0 −1



−1 1 , C 2i = 0



0 0 0

0 C 3i 0



0 0 , C 3i = 03



0

ψqi

ψdi

0 0

−ψqi



−ψdi 0 , 0

where i denotes the generator connected to the power system. uyi is the input control to the governor system which controls servomotor position, and vfi is the field voltage which manages the field current. The variables to be controlled are the synchronous speed and the terminal voltage magnitude of each generator. It is important to note that the Euler-Lagrange system mentioned above does not account for the rotor angle dynamic, as the rotor angle converges to a constant value. The control scheme for the Euler-Lagrange system will be designed in the next section, whereby convergence is guaranteed for the rotor speed ω. 2.4. Electrical network model An electrical network model is composed of n generators and k loads that are connected via m transmission lines. Here, each generator is modeled in (9). The loads and transmission lines can be modeled using the phasor representation, thereby, omitting the load dynamics (which are much faster than the mechanical dynamics of the generators [9]). Obtaining the equilibrium point through a pre-fault flow analysis of all networks, the loads are computed as constant impedances after applying the Kron’s reduction; thus, the loads and transmission lines can be written as1

I = Y(δ1 , . . . , δn )V, Y(· ) ∈ R2n×2n , 1

See [27] for more details.

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

7

where Y(·) is the reduced admittance matrix in the terminal nodes of the synchronous machines, I = [id1 , iq1 , . . . , idn , iqn ]T and V = [vd1 , vq1 , . . . , vdn , vqn ]T are the vectors of the generator currents (id1 and iq1 ) and terminal voltage (vd1 and vq1 ), respectively; and δ i is the rotor power angle, which can be computed as

δ˙ i = ωi − ωb , where δ i is the rotor angle and ωb is the synchronous speed. 3. Power system control The control objectives of the power system are to regulate the rotor speed and the generator terminal voltage in order to improve stability under small and large disturbances. The control design is arranged under the following assumptions: A5. The nominal parameters of the hydro-turbines and synchronous generators in the power system are known. A6. All the states are available for measurement. 3.1. Standard passivity-based control The Standard passivity-based control (SPBC) was proposed in [28] as a variation of passivity-based control (PBC), suitable for systems described by Euler-Lagrange representations. There are several kinds of controllers based on passivity, and their applications depend on the open-loop structure in the system under study. Some PBC techniques are based on interconnection and need the model in open-loop to have port-Hamiltonian structure. These PBC approaches use another portHamiltonian system connected to the plant to add up their energy functions and thus, preserving the concept of energy in the interconnection, while the Standard PBC employs static state feedback based on the error dynamics [29]. The SPBC allows the design of a feedback law for the system (8) that generates the following closed-loop system [30]:

D x˜˙ + [C (x ) + Rd (x )]x˜ = 0.

(10)

where x˜ = x − xd is an error signal, xd is an equilibrium point, and Rd = be defined as

RTd

 0 is the desired damping matrix, which can

Rd = R + Ra , where Ra = RTa  0 is the damping injection matrix. To obtain the dynamic system (10), the following desired dynamic system is proposed:

D x˙ d + [C (x ) + R(x )]xd = Gu + b(x ) + Ra x˜.

(11)

The dynamic system (10) is yielded, by subtracting (8) and (10). The control law is obtained from (11), as follows:



u = GT G where

−1

(GT G )−1 GT

GT (D x˙ d + [C (x ) + R(x )]xd − b(x ) + Ra x˜),

(12)

is known as the Moore-Penrose pseudo-inverse, and D x˙ d = 0 since xd is an equilibrium point [30].

3.2. Design controller The design controller for the HTGS is based on the SPBC approach. To obtain the control laws from (12), it is necessary to define the damping injection matrix (Rai ). We can define the damping injection matrix as

Rai = Ra1i + Ra2i , with Ra1i = diag{r1i , . . . , r10i }, Ra2i4,5 = −k1i , and Ra2i5,4 = k1i , where r ji > 0 ∀ j = 1, . . . , 10 and k1i = 0. Applying (12), the control laws are

u1i = xd4i − k1i (x5i − xd5i ) − r4i (x4i − xd4i ), u2i = r f i xd8i − r8i (x8i − xd8i ). The speed governor usually controls the electrical power delivered by the synchronous machine and the rotor speed of the synchronous machine. We selected xd2 and xd5 to attain this objective. Analyzing (1) at equilibrium point, it is possible to obtain





k f 1i + k f 2i x3d2i − qnli (k f 1i + k f 2i )x2d2i − xd2i +

P

mdi

Ati



+ qnli = 0,

(13)

where Pmdi is the desired mechanical power delivered by turbine i, which can be written in terms of the electrical power delivered Pdi , as follows,



2

Pmdi = Pdi + idqi  rsi .

8

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Fig. 3. Error of xd2i in (13) and (14).

Note that xd2i has three solutions in (13) for a desired mechanical power Pmdi , but, only one of them satisfies the assumption A3. Therefore, to solve this polynomial online may be a complicated task. For this reason, we propose to compute xd2i with the following approximation:

xd2i ≈





 Pmdi 1 2 + qnli + qnli k f 1i + k f 2i 4Pmdi + P . Ati 2 mdi

(14)

Fig. 3 illustrates the error of xd2i in (13) and (14). The reference rotor speed xd5i is defined as

xd5i = ωnom . The field circuit of the exciter makes it possible to keep the terminal voltage magnitude vti at the desired value. The terminal voltage magnitude, defined in dq reference frame, is given by

vti2 = v2di + v2qi .

(15)

By analyzing (3) and (15) at an equilibrium point, we can obtain xd8i to control the terminal voltage magnitude as follows:

xd8i = where

vdqi =

vdqi + rsi xd7i + Ldi xd5i xd6i Lmdi xd5i



.

(16)

vti2 − v2di .

The equilibrium points for the non-controlled variables are obtained from (11), as follows:

ki xd5i − τmi + ψdi xd7i + ψqi xd6i + k1i x2i − r5i (x5i − xd5i , k1i −vdi + ψqi xd5i + r6i x6i = , rsi + r6i

xd4i = xd6i

xd7i =

−vqi + ψdi xd5i + r7i x7i . rsi + r7i

3.3. Stability analysis The dynamic system of each HTGS presented in (8) is stable at an equilibrium point in closed-loop if the control law in (12) is applied, by defining the following Lyapunov function:

W (x˜) =

1 T x˜ D x˜, 2

whose derivative with respect to time is

W˙ (x˜) = x˜T D x˜˙ = −x˜T [C (x ) + Rd (x )]x˜ = −x˜T Rd (x )x˜, This proves that the dynamic system in (8) is stable. If Rd 0, the dynamic system is asymptotically stable, which indicates that x converges asymptotically to xd [30].

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

9

Fig. 4. Schematic of single-machine system. Table 2 Governing system parameters. Description

Parameter

Value

Description

Parameter

Value

Proportional gain Integral gain Derivative gain Permanent Droop (pu) Temporary Droop (pu)

Kp Ki Kd

1.1072 0.0831 2.2144 0.0050 0.4500

Temporary Droop (pu) Power-Speed (pu) Maximum Gate Limit (pu) Minimum Gate Limit (pu) Maximum Gate Opening Rate (pu/s) Minimum Gate Opening Rate (pu/s)

σ

0.4500 0.0050 1.0000 0.000 0.1000 –0.1000

ρ σ

Rs – – – –

Table 3 SPBC parameters. Parameter

r1

r2

r3

r4

r5

r6

r7

r8

r9

r10

k1

k2

Value

10

5

5

15

80

100

4

4

1

1

40

80

4. Test system, simulation, and results The proposed SPBC controller was assessed in two test systems to verify its performance and robustness in the stability improvement in a power system. The first test system was a single machine connected to an infinite bus presented in [31]. The second test system was the well-known Western System Coordinating Council (WSCC) proposed in [27]. For comparison, each synchronous machine was implemented with a voltage regulator and exciter based on the IEEE type ST1A excitation system model, which presents a high initial and fast dynamic response [32,33]. The power system stabilizers IEEE-PSS1A were also considered. We also considered two controllers for the governing system. Firstly, we implemented the standard governing system with static and transient droop, to control the hydraulic turbine speed [1]. Secondly, a sliding mode control (SMC) for governing system proposed in [34] was also used. The power systems and control schemes were performed using the MATLAB/Simulink software. 4.1. Single-machine system The single-machine test system was connected to an infinite bus through a step-up transformer and two lines in parallel, as shown in Fig. 4. The parameters of the system, hydraulic turbine, and PSS were presented in [6], and the IEEE-ST1A excitation system parameters can be found in [33]. The control design parameters for the PID governing system and their limits are shown in Table 2. Here, the parameters were computed as recommended by [8], while static and transient droop were implemented as recommended by [1,35]. Table 3 also presents the SPBC parameters. To assess the performance of the proposed controller, two cases were analyzed. First, a change in the terminal voltage reference from 1.0 to 1.05, and second, a 120 ms short-circuit fault at the infinite bus. These events occur at 0.2 s. All figures shown in this paper are in per-unit and time in seconds. To quantify the performance of the SPBC controller, we used the integral of the time-weighted absolute error (ϕ ), which was determined for the generator terminal voltage and the rotor speed deviation, as follows:

ϕω = ϕvt =

 

tsim 0 tsim 0

t | ω|dt , t | vt |dt ,

where tsim is the simulation time range. The damping ratio (ζ ), settling time (ts ), and peak time (tp ) for the rotor speed were also used to quantify the performance of the controllers.

10

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Fig. 5. Dynamic responses of single-machine system for changes in reference voltage: (a) relative deviation of turbine speed, (b) generator terminal voltage, and (c) active power.

Fig. 6. Controller actions of single-machine system for changing reference voltage: (a) governor control, and (b) excitation controller.

Fig. 5 illustrates the relative deviation of the turbine speed ( ω), the generator terminal voltage (vt ), and the electric power (p) delivered by the synchronous machine when the reference voltage is changed. The control signals (uy and vf ) are also shown in Fig. 6. Table 4 shows the performance indexes for this case. Fig. 5 (a) indicates that the SPBC controller has a better regulation of the rotor speed deviation and needs less time to stabilize compared with the standard controller and SMC. This analysis is supported by comparing ϕ ω and ts between the proposed controller and other controllers, where these indexes are 60.5% and 47.2% lower, respectively, for the proposed controller. In addition, the damping ratio injected by the SPBC is greater than the standard controller and SMC. A similar comparison can also be done for the active power generated by the synchronous machine, as shown in Fig. 5(c). Fig. 5 (b) indicates that the SPBC controller presents a better dynamic performance in the control and regulation of the generator terminal voltage than the other controllers. This finding can be verified by comparing ϕvt between controllers, as shown in column for ϕvt in Table 4, where this index is 67.2% lower for the proposed controller.

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

11

Table 4 Performance indexes of single-machine system. Changing reference voltage

SPBC SMC Standard Control

Fault at infinite bus

ϕω

ϕvt

ts [s]

tp [s]

ζ [%]

ϕω

ϕvt

ts [s]

tp [s]

ζ [%]

0.0916 0.1378 0.1513

3.8366 8.4837 11.7153

4.0802 6.9406 8.6449

0.6700 0.9000 0.9700

0.5000 0.3300 0.2200

0.1906 0.3337 0.4708

10.8207 14.3475 20.2688

3.0903 6.6971 8.4942

0.3200 0.3200 0.3200

0.7906 0.4718 0.2876

Fig. 7. Dynamic responses of single-machine system for fault at infinite bus: (a) relative deviation of the turbine speed, (b) generator terminal voltage, and (c) active power.

As shown in Fig. 6, the control signals stabilized by the SPBC controller are marked by larger changes at the beginning of the simulation than the standard controller and SMC. However, the stabilization time of the SPBC controller is faster. Fig. 7 illustrates the dynamic response of the relative deviation of the turbine speed ( ω), the generator terminal voltage (vt ), and the electric power delivered (p) by the synchronous machine when a short-circuit fault of 120 ms duration at infinite bus is presented. This situation indicates that, during the fault, the controllers exhibited similar behavior; furthermore, after the fault, the SPBC controller has a better dynamic behavior than other controllers because the proposed controller shows lower oscillations and stabilizes in a shorter time (the standard controller and SMC maintain small oscillations for more than 10 s). This analysis is supported by the performance indexes shown in Table 4, where ϕ ω , ϕ vt , and ts for the SPBC controller are, respectively, 59.5%, 46.6%, and 63.6%, less than the corresponding indexes for the standard controller. Fig. 8 depicts the control laws when a short-circuit fault of 120 ms duration at infinite bus is considered. The governor controller law for the SPBC controller shows some peaks due to the maximum gate opening rate used in the simulation (see Fig. 8(a)). 4.2. WSCC System A multimachine scenario is analyzed to assess the performance of the proposed controller. We used the WSCC 9-bus test system, as illustrated in Fig. 9, which is composed by three synchronous generators that are equipped with IEEE-ST1A excitation systems, PSSs, and turbine governors. This system also has three two-winding transformers, three loads, and six transmission lines. All the parameters of the test system are given in [27]. We considered that the generators #1 and #3 are hydro-turbines, while the generator #2 is a steam turbine. However, the control law for excitation presented in (16) can be used in a steam turbine without making any changes. The parameters used in this test system are the same presented in Tables 2 and 3.

12

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Fig. 8. Controller actions of single-machine system for fault at infinite bus: (a) governor controller, and (b) excitation controller.

Fig. 9. Schematic of WSCC system.

Two faults of 150 ms length are considered to assess the performance of the proposed controller. Fault 1 is a threephase short-circuit to ground located at bus 6, while Fault 2 is a short-circuit located at the middle of the transmission line connected between buses 7 and 8, as shown in Fig. 9). These faults occur at 0.2 s. For Fault 2, it is also considered that the protecting system works after 150 ms, i.e., the fault is isolated by opening the line interrupters.

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

13

Fig. 10. Dynamic response of relative deviation of turbine speed for fault at bus 6 during 150 ms (Fault 1).

Fig. 11. Dynamic response of the generator terminal voltage for fault at bus 6 during 150 ms (Fault 1).

Table 5 Performance indexes of WSCC system. Generator #1

Generator #2

Generator #3

ϕω

ϕvt

ts [s]

tp [s]

ζ [%]

ϕω

ϕvt

ts [s]

tp [s]

ζ [%]

ϕω

ϕvt

ts [s]

tp [s]

ζ [%]

Fault 1

SPBC SMC Standard Control

1.871 3.170 4.580

2.040 30.510 34.020

18.261 22.830 25.440

0.670 0.850 0.850

0.073 0.058 0.052

1.980 3.980 4.720

1.970 32.150 37.450

17.012 20.530 24.610

0.350 0.440 0.440

0.078 0.060 0.054

2.310 4.124 4.604

3.060 33.780 35.080

15.552 23.150 23.180

0.350 0.350 0.350

0.085 0.059 0.057

Fault 2

SPBC SMC Standard Control

1.090 5.970 6.450

1.000 3.740 3.930

17.980 21.480 24.690

0.850 0.850 0.850

0.070 0.050 0.050

1.160 5.970 6.710

0.590 26.850 27.010

16.850 23.290 23.290

0.350 0.350 0.350

0.080 0.060 0.060

0.950 4.420 5.510

1.860 167.280 170.130

11.980 21.940 21.990

0.610 0.590 0.590

0.110 0.060 0.060

4.2.1. Fault 1 This fault analyzes the ability of the SPBC to improve stability in a multimachine power system during and after a large disturbance. Figs. 10 and 11 illustrate the dynamic responses of the rotor speed deviations and the generator terminal voltages of the synchronous generators #1 and #2 for Fault 1. Table 5 summarizes the performance indexes for this test system. Fig. 10 shows that the rotor speed deviations of the generators provide a measure of the frequency behavior of the test system. The proposed controller produces lower peak values and fewer oscillations in the rotor speed deviations than the standard controller and SMC. Particularly, peak values and oscillations of the rotor speed deviations are reduced in 51.66%

14

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

Fig. 12. Dynamic response of relative deviation of turbine speed for fault located in the middle of transmission line L-7 during 150 ms (Fault 2).

Fig. 13. Dynamic response of generator terminal voltage for fault located in the middle of transmission line L-7 during 150 ms (Fault 2).

and 40.0% in the worst scenarios, respectively. Also, the proposed controller provides stability in a shorter time, which is 20.2% less according to other controllers. This analyzing is supported by the performance indexes shown in Table 5. In Fig. 11, it is noted that the voltage profiles have an enhanced response when the SPBC approach is implemented. It has fewer oscillations and a shorter settling time than other controllers. The oscillations and settling time are reduced in 93.27% and 59.65% compared to the worst case scenario. 4.2.2. Fault 2 This fault analyzes the ability of the SPBC to improve stability in a multimachine power system under topology changes represented by tripping the transmission line L-7. Figs. 12 and 13 depict the dynamic responses of the rotor speed deviations and the generator terminal voltages of the synchronous generators #3 and #2 for Fault 2. The performance indexes for this fault are also summarized in Table 5. In Fig. 12, it is observed that the Fault #2 generates longer oscillations of the rotor speed deviations in the test system, compared with Fault #1. However, this fault has lower peaks of the rotor speed deviations. On the other hand, the dynamic responses of the rotor speed deviations present fewer oscillations and faster settling times when the SPBCs are implemented. More precisely, the oscillations and settling times are reduced by 82.8% and 19.51% respectively, when the proposed controller is used. This indicates that the proposed controller can stabilize faster the power system even when there is a topology change. Therefore, the stability in the power system has been improved. Observe in Fig. 13 that the voltage profiles continuously presents a better performance when the SPBC approach is implemented. The oscillations and settling time are reduced by 78.51% and 41.56% respectively compared to the SMC and standard controller. This indicates that the power systems dynamic behavior is enhanced. This finding is supported by the performance indexes shown in Table 5.

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

15

Fig. 14. Dynamic response of relative deviation of turbine speed ω 1 when a random disturbance of ± 1% is considered: (a) Fault 1 and (b) Fault 2.

Fig. 15. Dynamic response of generator terminal voltage vt2 when a random disturbance of ± 1% is considered: (a) Fault 1 and (b) Fault 2.

Fig. 16. Dynamic response of the rotor speed deviation ω1 and the flow rates on penstock q2 when the surge tank is and is not considered: (a) Fault 1 and (b) Fault 2.

4.2.3. Complementary analysis In this part, it is analyzed the robustness of the proposed controller under a noise in the control variables ( ω i and vti ). We considered that there is a random disturbance of ±1% in the control variables. Figs. 14 and 15 illustrate the dynamic responses of the rotor speed deviation ω1 and the generator terminal voltage vt2 for Faults 1 and 2. Note in Figs. 14 and 15 that the rotor speed deviation and the generator terminal voltage are affected when a random disturbance of ±1% in their measurements is considered. Furthermore, the proposed controller is able to maintain the regulation of the rotor speed and voltage with better performance and a lower ripple than other controllers. It is important to highlight that in all the scenarios analyzed; the proposed controller shows an improved response for the regulation of rotor speeds and generator terminal voltages, in comparison to the standard control and SMC approaches. In addition, the damping ratios are greater for the SPBC, which demonstrates an improvement on the rotor oscillations damping This benefits the HTGSs by reducing the stress on the shaft, however, the proposed controller has some disadvantages compared with the standard controller, i.e., the state variables of the system must be known at each time as well as their parameters.

16

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

4.2.4. Surge tank analysis In this part, the response of the rotor speeds is analyzed when the surge tank is and is not considered, using the SPBC approach. Fig. 16 depicts the dynamic responses of the rotor speed deviation ω1 and its flow rates on tunnel q1 for Faults 1 and 2. As can be seen in Fig. 16(a), the dynamic behavior of the rotor speed deviation is very similar whether the surge tank is considered. This demonstrates that all the dynamics of the power system work together, and compared with the surge tank, the damping injected by the proposed control has a more meaningful participation in the response. The flow rates on tunnel indicate a reduction on the pressure on tunnel since the influence of water-level fluctuation is lower. Lastly, comparing the results in Fig. 16 it can be concluded that the effect of the surge tank in the system affects the performance of the controller since it does not depend on any measure of the surge tank. 5. Conclusions This paper proposes a standard passivity-based controller to regulate voltage and frequency in multimachine power systems composed by HTGS that guarantees stability properties in the sense of Lyapunov by exploiting the Euler-Lagrange representation of this system. The complete eleventh order dynamical model of the system that comprehends the HTGS with surge tanks in conjunction with the synchronous machine model is analyzed to propose an asymptotic stable controller that allows injecting damp to reduce voltage and frequency oscillation in the power systems under the effect of large disturbances. The major conclusions are summarized as follows: (1) The frequency response of the system depends on the nature of the controller applied to the HTGS, especially when large disturbances occur. Comparing the nonlinear SPBC approach against a sliding mode controller, as well as the PID controller, it was evidenced that the proposed method injects enough damping to reduce frequency oscillations with faster convergence than other controllers. Therefore, the stress suffered by the shafts of turbines are less when proposed controller is implemented, being reduced the frequency oscillations and their settling time about 41.1% and 41.2% for the WSCC system in the worst scenarios, respectively. This shows that the proposed controller injects greater damping than the other controllers in around 20% more. (2) The control of the generator terminal voltages allows maintaining the voltage profiles within their established limits in stable state operation and to maintain stability under large disturbances. The proposed controller shows that it can track and regulate the terminal voltages at their reference values under normal and transient regime in a multimachine power system with better performance than other controllers. The proposed controller presents fewer oscillations and stabilizes in shorter times than other controllers, reducing these indicators by 77.58% and 40.42% in the worst case. (3) Nonlinear controllers based on Euler-Lagrange representation such as the SPBC, take advantage of the open-loop dynamic of the system (i.e., passive system), to propose a set of control laws that makes power system stable in the sense of Lyapunov; in addition, the most important fact on the implementation of SPBC is related to its scalability as demonstrated for single-machine and multimachine power systems. Additionally, The developed control is decentralized avoiding all communication challenges between the hydro-turbine governing systems. (4) The effect of the surge tank in the electrical dynamics of the power system is minimum since the SPBC controllers, that inject damping are able to support all the frequency and voltage oscillations, regardless the inclusion of the surge tank within the dynamic model. However, when it is included, and the whole dynamic of the HTGS is observed, the pressure on the tunnel is affected. Simulation results show that the proposed controller does not react when the surge tank is/is not presented, since its structure does not depend on any measure coming from itself. This situation shows that the proposed controller can be implemented for multimachine systems, even if some of the HTGS have/do not have surge tanks. In future studies, the proposed controller will be evaluated in combination with renewable energy sources and energy storage systems to improve transient stability in multimachine power systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

M. Jan, W. Janusz, R. James, Power System Dynamics: Stability and Control, 2 ed., John Wily & Sons, 2008. V. Smil, Chapter 1: Energy System: Their Basic Properties, Energy Transitions: Global and National Perspectives, 2nd ed., Praeger, Santa Barbara, 2016. I.E. Agency, Key World Energy Statistics 2018, International Energy Agency Paris, 2018. XM, Effective capacity by type of generation, 2018, (Online). https://www.xm.com.co/Paginas/Generacion/tipos.aspx. H.V. Pico, J.D. McCalley, A. Angel, R. Leon, N.J. Castrillon, Analysis of very low frequency oscillations in hydro-dominant power systems using multi-unit modeling, IEEE Trans. Power Syst. 27 (4) (2012) 1906–1915. W. Gil-González, A. Garces, A. Escobar, Passivity-based control and stability analysis for hydro-turbine governing systems, Appl. Math. Modell. 68 (2019) 471–486. W. Gil-González, A. Garces, A. Escobar-Mejía, O.D. Montoya, Passivity-based control for hydro–turbine governing systems, in: Proceedings of the IEEE PES Transmission Distribution Conference and Exhibition - Latin America (T D-LA), 2018, pp. 1–5. IEEE working group report, Hydraulic turbine and turbine control models for system dynamic studies, IEEE Trans. Power Syst. 7 (1) (1992) 167–179. H. Huerta, A. Loukianov, J. Cañedo, Passivity sliding mode control of large-scale power systems, IEEE Trans. Control Syst. Technol. (99) (2018) 1–9. O. Cerman, P. Huse˘ k, Adaptive fuzzy sliding mode control for electro-hydraulic servo mechanism, Expert Syst. Appl. 39 (11) (2012) 10269–10277. J. Liang, X. Yuan, Y. Yuan, Z. Chen, Y. Li, Nonlinear dynamic analysis and robust controller design for francis hydraulic turbine regulating system with a straight-tube surge tank, Mech. Syst. Sig. Process. 85 (Supplement C) (2017) 927–946.

W. Gil-González, O.D. Montoya and A. Garces / Applied Mathematical Modelling 79 (2020) 1–17

17

[12] W. Guo, J. Yang, Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank, Appl. Math. Modell. 54 (2018) 446–466. [13] R. Zhang, D. Chen, X. Ma, Nonlinear predictive control of a hydropower system model, Entropy 17 (9) (2015) 6129–6149. [14] G. Zhang, Y. Cheng, N. Lu, Q. Guo, Research of hydro-turbine governor supplementary control strategy for islanding AC grid at sending terminal of HVDC system, IEEE Trans. Energy Convers. 31 (4) (2016) 1229–1238. [15] H. Zhang, D. Chen, C. Wu, X. Wang, J.-M. Lee, K.-H. Jung, Dynamic modeling and dynamical analysis of pump-turbines in s-shaped regions during runaway operation, Energy Convers. Manag. 138 (2017) 375–382. [16] H. Zhang, D. Chen, B. Xu, E. Patelli, S. Tolo, Dynamic analysis of a pumped-storage hydropower plant with random power load, Mech. Syst. Sig. Process. 100 (2018) 524–533. [17] H. Zhang, D. Chen, P. Guo, X. Luo, A. George, A novel surface-cluster approach towards transient modeling of hydro-turbine governing systems in the start-up process, Energy Convers. Manag. 165 (2018) 861–868. [18] Y. Xu, J. Zhou, X. Xue, W. Fu, W. Zhu, C. Li, An adaptively fast fuzzy fractional order pid control for pumped storage hydro unit using improved gravitational search algorithm, Energy Convers. Manag. 111 (2016) 67–78. [19] C. Li, Y. Mao, J. Zhou, N. Zhang, X. An, Design of a fuzzy-PID controller for a nonlinear hydraulic turbine governing system by using a novel gravitational search algorithm based on cauchy mutation and mass weighting, Appl. Soft Comput. 52 (2017) 290–305. [20] W. Zhu, Y. Zheng, J. Dai, J. Zhou, Design of integrated synergetic controller for the excitation and governing system of hydraulic generator unit, Eng. Appl. Artif. Intell. 58 (2017) 79–87. [21] N. Kishor, S. Singh, Simulated response of NN based identification and predictive control of hydro plant, Expert Syst. Appl. 32 (1) (2007) 233–244. [22] Y. Zeng, L. Zhang, T. Xu, H. Dong, Improvement rotor angle oscillation of hydro turbine generating sets based on hamiltonian damping injecting method, in: Proceedings of the Power and Energy Engineering Conference (APPEEC), 2010, pp. 1–5. [23] T. Xu, L. Zhang, Y. Zeng, J. Qian, Hamiltonian model of hydro turbine with sharing sommon conduit, in: Proceedings of the Asia-Pacific Power and Energy Engineering Conference, 2012, pp. 1–5. [24] B. Xu, F. Wang, D. Chen, H. Zhang, Hamiltonian modeling of multi-hydro-turbine governing systems with sharing common penstock and dynamic analyses under shock load, Energy Convers. Manage. 108 (2016) 478–487. [25] H. Li, D. Chen, H. Zhang, C. Wu, X. Wang, Hamiltonian analysis of a hydro-energy generation system in the transient of sudden load increasing, Appl. Energy 185, Part 1 (2017) 244–253. [26] Y. Xu, L. Ren, Z. Zhang, Y. Tang, J. Shi, C. Xu, J. Li, D. Pu, Z. Wang, H. Liu, et al., Analysis of the loss and thermal characteristics of a SMES (superconducting magnetic energy storage) magnet with three practical operating conditions, Energy 143 (2018) 372–384. [27] P.M. Anderson, A.A. Fouad, Power System Control and Stability, Wiley-IEEE Press, 2003. [28] R. Ortega, M.W. Spong, Adaptive motion control of rigid robots: a tutorial, Automatica 25 (6) (1989) 877–888. [29] R. Ortega, A. Van Der Schaft, F. Castanos, A. Astolfi, Control by interconnection and standard passivity-based control of port-hamiltonian systems, IEEE Trans. Autom. control 53 (11) (2008) 2527–2542. [30] R. Ortega, J.A.L. Perez, P.J. Nicklasson, H.J. Sira-Ramirez, Passivity-based Control of Euler–Lagrange Systems: Mechanical, Electrical and Electromechanical Applications, Springer Science & Business Media, 2013. [31] A. Leon, J. Solsona, M. Valla, Comparison among nonlinear excitation control strategies used for damping power system oscillations, Energy Convers. Manage. 53 (1) (2012) 55–67. [32] IEEE Standard definitions for excitation systems for synchronous machines, IEEE Std 421.1–2007 (Revision of IEEE Std 421.1–1986) (2007) 1–33. [33] K. Máslo, A. Kasembe, M. Kolcun, Simplification and unification of IEEE standard models for excitation systems, Electr. Power Syst. Res. 140 (2016) 132–138. [34] J. Liang, X. Yuan, Y. Yuan, Z. Chen, Y. Li, Nonlinear dynamic analysis and robust controller design for francis hydraulic turbine regulating system with a straight-tube surge tank, Mech. Syst. Signal Process. 85 (2017) 927–946. [35] W. Gil González, A. Garces, O. Fosso, A. Escobar, Passivity-based control of power systems considering hydro-turbine with surge tank, IEEE Trans. Power Syst. (2019), doi:10.1109/TPWRS.2019.2948360. 1–1