Stability performance for primary frequency regulation of
hydro-turbine governing system with surge tank
Wencheng Guo
a , b , ∗, Jiandong Yang
a
a State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China b Maha Fluid Power Research Center, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette 47907, IN,
USA
a r t i c l e i n f o
Article history:
Received 11 April 2017
Revised 14 September 2017
Accepted 27 September 2017
Available online 12 October 2017
Keywords:
Hydro-turbine governing system
Surge tank
Primary frequency regulation
Stability
Critical stable sectional area
a b s t r a c t
This paper aims to study the stability for primary frequency regulation of hydro-turbine
governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-
turbine governing system considering the nonlinear characteristic of penstock head loss is
introduced. The nonlinear state equations under opening control mode and power control
mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine
governing system is investigated based on the stable domain for primary frequency regu-
lation. New feature of the nonlinear hydro-turbine governing system caused by the nonlin-
ear characteristic of penstock head loss is described by comparing with a linear model, and
the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally,
the concept of critical stable sectional area of surge tank for primary frequency regulation
is proposed and the analytical solution is derived. The combined tuning and optimiza-
tion method of governor parameters and sectional area of surge tank is proposed. The re-
sults indicate that for the primary frequency regulation under opening control mode and
power control mode, the nonlinear hydro-turbine governing system is absolutely stable
and conditionally stable, respectively. The stability of the nonlinear hydro-turbine govern-
ing system and linear hydro-turbine governing system is the same under opening control
model and different under power control model. The nonlinear characteristic of penstock
head loss mainly affects the initial stage of dynamic response process of power output,
and then changes the stability of the nonlinear system. The critical stable sectional area
of surge tank makes the system reach critical stable state. The governor parameters and
critical stable sectional area of surge tank jointly determine the distributions of stability
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 447
The successful operation of interconnected power systems requires matching the total generation with the total load
demand. With time, the operating point of a power system changes, and hence, systems may experience deviations in
nominal system frequency and scheduled power exchanges to other areas, which may yield undesirable effects [7] . Variation
in load frequency is an index for normal operation of power systems. To ensure good grid power quality when the grid load
changes, the grid frequency should be controlled in the allowable variation range of rated frequency [8,9] . Load frequency
control (LFC) is the primary measure to accomplish that task [10–12] . LFC is related to the short-term balance of energy and
frequency of the power systems and acquires a principal role to enable power exchanges and to provide better conditions
for electricity trading. The main goal of the LFC problem is to maintain zero steady-state errors for frequency deviation and
good tracking of load demands in a multi-area power system [13,14] .
Primary frequency regulation is one of the main control actions of LFC taken against frequency deviations in the grid as a
result of unbalances between demand and supply [15] . All the units contributing to primary frequency regulation give active
power support automatically by increasing/decreasing their active power output depending on the sign of the frequency
deviation [16,17] . For hydropower station, primary frequency regulation is actualized by the hydro-turbine governing system,
and the core component is governor. With the continuous development of hydroelectric energy, hydropower station with
long headrace/tailrace tunnel is becoming increasingly common. The long tunnel leads to large flow inertia [18] . To achieve
the security and stability of hydro-turbine unit operations, facilities of pressure reduction must be set on the tunnel. The
most commonly used facility of pressure reduction for hydropower station is the surge tank [19–21] . Since the influence of
water level oscillation in surge tank, the wave form of power response shows the characteristic of head wave and tail wave,
which is significantly different from the case without surge tank [22–24] . Under the action of surge tank, the dynamic
behaviors for primary frequency regulation are much more complicated than that without surge tank.
About the primary frequency regulation of governing system, previous researches mainly focus on the control strategies
of controller. Representative achievements are stated as follows. Wei [25] studied the numerical simulation of the primary
frequency operation for hydraulic turbine regulating systems without surge tank, and gave the tuning method of controller
parameters. Zhao et al. [26] presented a systematic method to design ubiquitous continuous fast-acting distributed load
control for primary frequency regulation in power networks by formulating an optimal load control problem. Miao et
al. [27] developed a coordination control strategy for wind farms with line commutated converter-based HVdc delivery.
Bao et al. [28] designed a hybrid hierarchical demand response control scheme to support primary frequency control
and discussed the parameters settings in detail. Pourmousavi and Nehrir [29] proposed a comprehensive central demand
response algorithm for frequency regulation, while minimizing the amount of manipulated load, in a smart microgrid.
Morel et al. [30] proposed a robust control approach to enhance the participation of variable speed wind turbines in the
primary frequency regulation during network disturbances. However, as a new topic, the stability for primary frequency
regulation of hydro-turbine governing system with surge tank has not been studied.
The hydro-turbine governing system is a nonlinear system. There are different nonlinear characteristics for different
subsystems. The nonlinear modeling and dynamic analysis of hydro-turbine governing system is always an important and
hot topic. Guo et al. [9] proposed a novel nonlinear mathematical model of hydro-turbine governing system with upstream
surge tank and sloping ceiling tailrace tunnel. Xu et al. [31] proposed a novel nonlinear fractional-order mathematical model
and studied the nonlinear dynamics of the fractional order hydro-turbine-generator unit system. Xu et al. firstly studied the
Hamiltonian mathematical modeling and dynamic characteristics of multi-hydro-turbine governing systems with sharing
common penstock under excitation of stochastic and shock load [32] , and then investigated the Hamiltonian mathematical
modeling for a hydro-turbine governing system including fractional item and time-lag [33] . Li et al. [34] addressed the
Hamiltonian mathematical modeling and dynamic analysis of a hydro-energy generation system in sudden load increasing
transient. Zhang et al. [35] studied the mathematical modeling of a hydro-turbine governing system in load rejection tran-
sient and illustrated the nonlinear dynamic behaviors by bifurcation diagrams, Poincare maps, time waveforms and phase
orbits. However, for the previous researches, attentions are paid on the hydro-turbine and generator. For the penstock, the
nonlinear characteristic of the head loss is not studied in further depth. The coupling effect of the nonlinear characteristic
of penstock head loss and surge tank is also not analyzed. For the hydro-turbine governing system with surge tank, the
penstock and hydro-turbine are directly coupled. Hence, a more precise mathematical model of penstock is important to
reflect the real effect of penstock on the hydro-turbine. Head loss is one of the main aspects that have obvious influence
on the dynamic performance of hydro-turbine governing system, and the nonlinear characteristic is the key characteristic
for head loss. Therefore, it is essential to consider the nonlinear characteristic of penstock head loss and reveal the effect
mechanism of nonlinear characteristic of penstock head loss on primary frequency regulation.
This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge
tank. The motivation and innovation are as follows. (1) Introduce a novel nonlinear mathematical model of hydro-turbine
governing system with surge tank considering the nonlinear characteristic of penstock head loss. Analyze the nonlinear
dynamic behaviors of the nonlinear hydro-turbine governing system and quantitatively evaluate the stability. (2) Investigate
new feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss
by comparing the novel nonlinear model with other models. Reveal the effect mechanism of the nonlinear characteristic of
penstock head loss, and propose a tuning and optimization method for determining the system parameters.
The paper is organized as follows. In Section 2 , for the hydropower station with surge tank, a novel nonlinear mathemat-
ical model of hydro-turbine governing system under primary frequency regulation considering the nonlinear characteristic
of penstock head loss is introduced. The nonlinear state equations for the nonlinear dynamic system under opening control
448 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Fig. 1. Hydropower station with surge tank under operating condition of primary frequency regulation.
mode and power control mode are derived. In Section 3 , the nonlinear dynamic performance of nonlinear hydro-turbine
governing system is investigated based on Hopf bifurcation theory. The stable domain for primary frequency regulation
is proposed and drawn under opening control mode and power control mode. In Section 4 , new feature of the nonlinear
hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing
with a linear model. The effect mechanism of the nonlinear characteristic of penstock head loss is revealed. In Section 5 , the
concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solu-
tion is derived. Based on the analytical solution, the combined tuning and optimization method of the governor parameters
and sectional area of surge tank is proposed. In Section 6 , the whole paper is summarized and the conclusions are given.
2. Nonlinear mathematical model
The pipeline and power generating system of hydropower station with surge tank is shown in Fig. 1 . For the operating
condition of primary frequency regulation, the change of the turbine unit frequency is the external disturbance. The
frequency disturbance acts as the input signal for the hydro-turbine governing system. Under the frequency disturbance,
the flow in the pipeline system and the components of the power generating system enter the transient processes, and
the dynamic responses of hydraulic parameters and mechanical parameters occur. Among all the dynamic responses of
parameters, the dynamic response of power output is the most important one because its response process is the key index
that evaluates the performances of the stability and regulation quality for primary frequency regulation of hydro-turbine
governing system.
In the following paragraphs of this section, a novel nonlinear mathematical model for hydro-turbine governing system
with surge tank considering the nonlinear characteristic of penstock head loss is introduced. For the governor, two control
modes, i.e. opening control mode and power control mode, are considered.
2.1. Basic equations
Hydro-turbine governing system contains the following subsystems: headrace tunnel, surge tank, penstock, hydro-
turbine, generator and governor. The basic equations for all subsystems are presented as follows. Note that the definition
and explanation for the notations and variables in Section 2.1 are presented in Appendix A .
(1) Dynamic equation of headrace tunnel [18,20]
z − 2 h y 0
H 0
q y = T wy 0
d q y
d t (1)
(2) Continuity equation of surge tank [18,20]
q y = q t − T F d z
d t (2)
(3) Nonlinear dynamic equation of penstock
For the penstock, the nonlinear characteristic of the head loss is considered, and the nonlinear dynamic equation is
derived as follows. According to Newton’s second law of motion, the rate of change of momentum is equal to the resultant
force. Then we have
L t f t γ
g
d V t
d t = f t γ [ ( Z � − Z) − H − h t ] (3)
By simplifying Eq. (3) yields
L t
g
d V t
d t = ( Z � − Z) − H − h t (4)
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 449
Combining L t g
d V t d t
=
L t g f t
d( f t V t ) d t
and Q t = f t V t yields L t g
d V t d t
=
L t g f t
d Q t d t
. Then by combining L t g f t
d Q t d t
=
L t Q t0 g f t
d( Q t
Q t0 )
d t and Q t 0 = f t V t 0
yields L t g f t
d Q t d t
=
L t V t0 g
d( Q t
Q t0 )
d t . Therefore, we obtain
L t
g
d V t
d t =
L t V t0
g
d( Q t Q t0
)
d t (5)
Substitution of Eq. (5) into Eq. (4) gives
L t V t0
g
d( Q t Q t0
)
d t = ( Z � − Z) − H − h t (6)
At the initial time, i.e. t = 0 s, we have Z 0 = 0 m. Then the expression of Eq. (6) at t = 0 s is
L t V t0
g
d( Q t0
Q t0 )
d t = Z � − H 0 − h t0 (7)
By subtracting Eq. (7) from Eq. (6) yields
L t V t0
g
d( Q t −Q t0
Q t0 )
d t = −(H − H 0 ) − Z − ( h t − h t0 ) (8)
Dividing Eq. (8) by H 0 gives
L t V t0
g H 0
d( Q t −Q t0
Q t0 )
d t = −H − H 0
H 0
− Z
H 0
− h t − h t0
H 0
(9)
By using L t V t0 g H 0
= T wt0 , Q t −Q t0
Q t0 = q t ,
H−H 0 H 0
= h and
Z H 0
= z, Eq. (9) can be transferred to the following form:
T wt0 d q t
d t = −h − z − h t − h t0
H 0
(10)
For penstock head loss, we have h t = αt Q
2 t and h t0 = αt Q
2 t0
. Then we obtain h t = h t0 ( Q t Q t0
) 2 . By substituting Q t = Q t 0 + Q t 0 q t
into h t = h t0 ( Q t Q t0
) 2 yields h t = h t0 (1 + q t ) 2 = h t0 (1 + 2 q t + q 2 t ) . Therefore, we can obtain
h t − h t0 = h t0 (2 q t + q 2 t ) (11)
Substitution of Eq. (11) into Eq. (10) gives
T wt0 d q t
d t = −h − z − h t0
H 0
(2 q t + q 2 t ) (12)
Eq. (12) is the nonlinear dynamic equation of penstock. There is a nonlinear term
h t0 H 0
q 2 t , which is caused by the nonlinear
characteristic of the head loss of penstock.
(4) Moment equation and discharge equation of hydro-turbine [36,37]
m t = e h h + e x x + e y y (13)
q t = e qh h + e qx x + e qy y (14)
(5) Equation of generator [36,37]
p t = m t + x (15)
(6) Equation of governor
For the governor, the structure diagrams [36,37] for opening control mode and power control mode are shown in Fig. 2 .
From the structure diagrams of governor, we can obtain the equations of governor as follows:
Opening control mode : T y d
2 y
d t 2 + (1 + T y b p K i )
d y
d t + b p K i y = −K d
d
2 x
d t 2 − K p
d x
d t − K i x (16)
Power control mode : T y d
2 y
d t 2 +
d y
d t + e p K i p t = −K d
d
2 x
d t 2 − K p
d x
d t − K i x (17)
The expressions of K p , K i and K d are K p = 1/ b t , K i = 1/( b t T d ) and K d = T n / b t , respectively. In this paper, the condition of
T y = T n = 0 is considered. Then we have K d = 0. Therefore, Eqs. (16) and (17) can be transferred to the following forms:
Opening control mode : d y
d t + b p K i y = −K p
d x
d t − K i x (18)
Power control mode : d y + e K p = −K
d x − K x (19)
d t
p i t p d t
i
450 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Fig. 2. Structure diagrams of governor.
2.2. Nonlinear dynamic system
For the hydro-turbine governing system with surge tank, the nonlinear mathematical model is composed by Eqs. (1) , (2) ,
( 12 )–( 15 ) and (18) under opening control mode and Eqs. (1) , (2) , ( 12 )–( 15 ) and (19) under power control mode. Under pri-
mary frequency regulation, the input signal for the nonlinear hydro-turbine governing system is the external disturbance, i.e.
the change of turbine unit frequency. In this paper, step change of turbine unit frequency is considered and denoted as x S .
Then from the nonlinear mathematical model established in Section 2.1 , we can get that the nonlinear hydro-turbine govern-
ing system contains seven variables, i.e. q y , z, q t , h, m t , p t and y . While there are only four first-order differential equations,
i.e. Eqs. (1) , (2) , (12) and (18) under opening control mode and Eqs. (1) , (2) , (12) and (19) under power control mode. There-
fore, under primary frequency regulation, the hydro-turbine governing system is a fourth-order nonlinear dynamic system.
By integrating Eqs. (1) , (2) , ( 12 )–( 15 ) and (18) , we obtain the following fourth-order nonlinear state equation for the
nonlinear hydro-turbine governing system under opening control mode. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˙ q y =
1 T wy 0
(z − 2 h y 0 H 0
q y )
˙ z =
1 T F
( q t − q y )
˙ q t =
1 T wt0
[ −z −
(2 h t0
H 0 +
1 e qh
)q t − h t0
H 0 q 2 t +
e qx
e qh x S +
e qy
e qh y
] ˙ y = −K p
d x S d t
− K i x S − b p K i y
(20)
By solving Eq. (20) , we can obtain the dynamic response processes of q y , z, q t and y . Then by substituting q y , z, q t and
y into Eq. (21) yields the dynamic response processes of h, m t and p t . ⎧ ⎨
⎩
h =
1 e qh
( q t − e qx x S − e qy y )
m t =
e h e qh
q t + ( e x − e h e qh
e qx ) x S + ( e y − e h e qh
e qy ) y
p t =
e h e qh
q t + (1 + e x − e h e qh
e qx ) x S + ( e y − e h e qh
e qy ) y
(21)
By the same method and procedure used under opening control mode, the fourth-order nonlinear state equation for the
nonlinear hydro-turbine governing system under power control mode is obtained as Eq. (22) by integrating Eqs. (1) , (2) ,
( 12 )–( 15 ) and (19) . From Eqs. (22) and (21) , the dynamic response processes of q y , z, q t , y, h, m t and p t under power control
mode can be solved. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˙ q y =
1 T wy 0
(z − 2 h y 0 H 0
q y )
˙ z =
1 T F
( q t − q y )
˙ q t =
1 T wt0
[ −z − ( 2 h t0
H 0 +
1 e qh
) q t − h t0
H 0 q 2 t +
e qx
e qh x S +
e qy
e qh y
] ˙ y = −e p K i
e h e qh
q t − K p d x S d t
−[
e p K i (1 + e x − e h e qh
e qx ) + K i
] x S − e p K i ( e y − e h
e qh e qy ) y
(22)
For the nonlinear hydro-turbine governing system with surge tank under primary frequency regulation, the stability
performance is determined by the fourth-order nonlinear state equation, i.e. Eq. (20) or (22) . Because h, m t and p t are
composed by the linear combinations of q y , z, q t and y , the stability performances of h, m t and p t are consistent with those
of q y , z, q t and y . Hence, we can determine the stability performance of p t from Eq. (20) or (22) .
Up till now, a novel nonlinear mathematical model for hydro-turbine governing system with surge tank has been
established. The novelty of the nonlinear mathematical model embodies in three aspects:
(1) For the penstock, the nonlinear characteristic of the head loss is considered, and the nonlinear dynamic equation is
derived. For the hydro-turbine governing system with surge tank, the penstock and hydro-turbine are directly cou-
pled. Hence, a more precise mathematical model for penstock is important to reflect the real effect of penstock on
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 451
hydro-turbine. Head loss is one of the main aspects that have obvious influence on the dynamic performance of
hydro-turbine governing system, and the nonlinear characteristic is the key characteristic of head loss. Therefore, it is
essential to consider the nonlinear characteristic of the head loss of penstock.
(2) For the operating condition, primary frequency regulation is considered. Primary frequency regulation is one of the
main control actions of LFC taken against frequency deviations in the grid as a result of unbalances between demand
and supply. However, the stability for primary frequency regulation of hydro-turbine governing system with surge
tank has not been studied by previous researchers.
(3) For the governor, two control modes, i.e. opening control mode and power control mode, are considered. Therefore,
this study is comprehensive. Opening control mode and power control mode are the two basic control modes for
primary frequency regulation. Both of them are widely used in real applications.
Moreover, the novelty of the nonlinear mathematical model proposed in this paper can also be revealed by comparing
with some recent and representative models, i.e. the models in Refs. [9,38] and [39] .
(1) In Ref. [9] , a nonlinear mathematical model of hydro-turbine governing system with upstream surge tank and sloping
ceiling tailrace tunnel is proposed. That model contains the nonlinear dynamic equation of sloping ceiling tailrace
tunnel, which can describe the motion characteristics of the interface of free surface-pressurized flow. However, the
nonlinear characteristic of the head loss of penstock is not considered, and the head loss term of penstock is linear.
Moreover, the operating condition in Ref. [9] is the load disturbance, not the primary frequency regulation. And only
one control mode of governor, i.e. frequency control mode, is studied.
(2) In Ref. [38] , a nonlinear mathematical model, which is composed of Francis turbine system, electrical generator sys-
tem, conduit system and governor system is established. But for the penstock, the authors of Ref. [38] assume that it
is an ideal model and then neglect the hydraulic friction losses. And only one control mode of governor, i.e. frequency
control mode, is studied.
(3) In Ref. [39] , the regular features of the state equations that describe small fluctuations of the system are investigated.
The equations for unsteady flow in pipeline system are given by graph theory. However, the unsteady flow equation
in pipeline system is linear, and the nonlinear characteristic of penstock head loss is not considered. Moreover, the
operating condition in Ref. [39] is the load disturbance, not the primary frequency regulation. And only one control
mode of governor, i.e. frequency control mode, is studied.
3. Nonlinear dynamic performance of hydro-turbine governing system
3.1. Methodology
In this section, the Hopf bifurcation theory [40–42] is adopted to study the nonlinear dynamic performance of the
nonlinear hydro-turbine governing system with surge tank under primary frequency regulation. The existence and direction
of Hopf bifurcation are discussed in the following paragraphs.
Having selected x = ( q y ,z, q t ,y ) T and μ as state vector and bifurcation parameter, respectively, the fourth-order nonlinear
dynamic systems Eqs. (20) and (22) can be expressed by ˙ x = f (x , μ) . The equilibrium point x E = ( q yE ,z E ,q tE ,y E ) T can be
obtained by solving ˙ x = 0 . At x E , we have d x S d t
= 0 .
For the opening control mode expressed by Eq. (20) , we can get z E =
2 h y 0 H 0
q yE from ˙ q y = 0 , q yE = q tE from ˙ z = 0 and
y E = − 1 b p
x S from ˙ y = 0 . Substitution of z E =
2 h y 0 H 0
q yE , q yE = q tE and y E = − 1 b p
x S into ˙ q t = 0 gives
h t0
H 0
q 2 tE +
(2 h y 0
H 0
+
2 h t0
H 0
+
1
e qh
)q tE −
(e qx
e qh
− e qy
e qh
1
b p
)x S = 0 (23)
Eq. (23) has two real roots: q tE−1 =
H 0 2 h t0
[ −( 2 h y 0 H 0
+
2 h t0 H 0
+
1 e qh
) +
√
( 2 h y 0 H 0
+
2 h t0 H 0
+
1 e qh
) 2
+ 4 h t0 H 0
( e qx
e qh − e qy
e qh
1 b p
) x S ] and
q tE−2 =
H 0 2 h t0
[ −( 2 h y 0 H 0
+
2 h t0 H 0
+
1 e qh
) −√
( 2 h y 0 H 0
+
2 h t0 H 0
+
1 e qh
) 2
+ 4 h t0 H 0
( e qx
e qh − e qy
e qh
1 b p
) x S ] . For the primary frequency regulation, q tE
should satisfy the following requirement: when x S > 0, q tE < 0; when x S < 0, q tE > 0. Only q tE − 1 meets that requirement. By the same method and procedure used under opening control mode, the equilibrium point for the power control
mode expressed by Eq. (22) is obtained as follows:
452 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Table 1
Judgement of the types and directions of emerged Hopf bifurcation.
Values of σ ′ (μc ) Types of Hopf bifurcation Directions of Hopf bifurcation
At x E , the Jacobian matrix J (μ) of ˙ x = f (x , μ) and characteristic equation det (J (μ) − χ I ) = 0 are:
J(μ) = D f x ( x E , μ) =
⎡
⎢ ⎢ ⎢ ⎣
∂ ̇ q y ∂ q y
∂ ̇ q y ∂z
∂ ̇ q y ∂ q t
∂ ̇ q y ∂y
∂ ̇ z ∂ q y
∂ ̇ z ∂z
∂ ̇ z ∂ q t
∂ ̇ z ∂y
∂ ̇ q t ∂ q y
∂ ̇ q t ∂z
∂ ̇ q t ∂ q t
∂ ̇ q t ∂y
∂ ̇ y ∂ q y
∂ ̇ y ∂z
∂ ̇ y ∂ q t
∂ ̇ y ∂y
⎤
⎥ ⎥ ⎥ ⎦
(25)
χ4 + c 1 χ3 + c 2 χ
2 + c 3 χ + c 4 = 0 (26)
For the opening control mode and power control mode, the expressions of coefficients in Eqs. (25) and (26) are
presented in Appendix B .
Assume that the following two conditions are satisfied when μ = μc :
(i) c i > 0 ( i = 1, 2, 3, 4), δ2 =
∣∣∣c 1 1 c 3 c 2
∣∣∣ > 0 and δ3 =
∣∣∣∣c 1 1 0 c 3 c 2 c 1 0 c 4 c 3
∣∣∣∣ = 0 .
(ii) The traversal coefficient σ ′ (μc ) crosses the imaginary axis at some nonzero speed, i.e. σ ′ ( μc ) = Re ( d χd μ
| μ= μc ) � = 0 .
Then, the system expressed by Eq. (20) or (22) undergoes a Hopf bifurcation at μ = μc . Moreover, at μ = μc , Eq. (26) has
a pair of purely imaginary eigenvalues, and the other eigenvalues all have negative real parts. The types and directions of
emerged Hopf bifurcation can be judged from Table 1 .
3.2. Stability analysis and numerical experiments
For the operating condition of primary frequency regulation, the nonlinear hydro-turbine governing system enters the
transient process after the turbine unit frequency disturbance. During the transient process, the dynamic responses of
system parameters are regulated by the governor. Under the different state parameters of system and governor parameters,
the performance of the transient process is different. Based on the automatic control theory [43] , the transient process of
the dynamic system can be divided into the following three categories: stable, critical stable and unstable. Under the three
transient processes, the dynamic responses of system variables are damped oscillation, persistent oscillation and divergent
oscillation, respectively.
In the practical applications, the tuning and optimization of governor parameters is the key issue for the stable operation
and control of hydro-turbine governing system. Selecting the governor parameters as coordinate axes, the domains that
present the performance of the transient process of the dynamic system can be drawn on the coordinate plane. The
obtained domains on the coordinate plane can provide an index for the evaluation of the performance of the transient
process and a guidance for the optimization of governor parameters. For primary frequency regulation, we select b p and
K i as coordinate axes under opening control mode, e p and K i as coordinate axes under power control mode. Based on the
Hopf bifurcation analysis in Section 3.1 , the domain that makes the system stable can be drawn on b p − K i or e p − K i plane
and is called the stable domain for primary frequency regulation in this paper.
In the following paragraphs of this section, we firstly conduct the stability performance analysis and then carry out the
numerical experiments to verify the stability analysis results. A hydropower station with surge tank is taken as example.
The basic data of this example are: H 0 = 45.45 m, Q 0 = 97.70 m
3 /s, T wy 0 = 42.80 s, T wt 0 = 2.66 s, h y 0 = 3.80 m, h t 0 = 0.79 m,
F = 100 m
2 , K p = 5 and g = 9.81 m/s 2 . The ideal turbine transfer coefficients are: e h = 1.5, e x = − 1, e y = 1, e qh = 0.5, e qx = 0 and
e qy = 1. The operating condition is as follows: the unit goes through a step frequency disturbance during normal operation
under 80% rated power output. The frequency disturbance is set at an anticlimax of rated frequency from 50 Hz to 49.8 Hz,
i.e. x = − 0.004.
S
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 453
Fig. 3. Nonlinear dynamic performance of nonlinear hydro-turbine governing system under opening control mode.
3.2.1. Opening control mode
By substituting the basic data of the hydro-turbine governing system into condition (i) in Section 3.1 , we can solve the
function curve composed by Hopf bifurcation points in the parameter plane. The Hopf bifurcation points are the critical
points of nonlinear system stability, and the function curve is referred as bifurcation line. In this paper, K i is selected as the
bifurcation parameter. Under opening control mode, the bifurcation line of the nonlinear hydro-turbine governing system is
determined and shown in Fig. 3 (a).
Fig. 3 (a) shows that there is no solution of condition (i) in the first quadrant of b p − K i plane, which indicates that the
bifurcation line under opening control mode does not locate in the first quadrant of b p − K i plane. Note that the above result
is always the same when the basic data of hydro-turbine governing system change. Therefore, the entire first quadrant of
b p − K i plane must locate on the same side of the bifurcation line, that is to say, the stability performances for all state
points in the first quadrant of b p − K i plane are the same.
When the coordinate values of the state point in b p − K i plane are given, the dynamic responses of the state vector
x = ( q y ,z, q t ,y ) T can be simulated from Eq. (20) by MATLAB. Based on the dynamic responses of state vector, we can get
the phase space trajectory of selected three variable responses (such as q y , q t and z ), which can present the stability
performance of the nonlinear hydro-turbine governing system. Now we optionally select a state point S Ex (0.01, 10 s −1 ) in
the first quadrant of b p − K i plane to determine its phase space trajectory. The numerical experiment result is shown in
Fig. 3 (b). Note that the similar result can be obtained for other state points in the first quadrant of b p − K i plane. The result
in Fig. 3 (b) indicates that: After several rounds of motion, the phase space trajectory of S Ex stabilizes at an equilibrium
point, indicating that the nonlinear hydro-turbine governing system under the state point S Ex is stable. Hence, under all the
state points in the first quadrant of b p − K i plane, the nonlinear hydro-turbine governing system is always stable.
Based on the above analysis, we can obtain the following conclusion: for the primary frequency regulation under
opening control mode, the nonlinear hydro-turbine governing system is absolutely stable.
3.2.2. Power control mode
By the same method and procedure used in Section 3.2.1 , the bifurcation line of the nonlinear hydro-turbine governing
system under power control mode is obtained and shown in Fig. 4 (a). Based on the coordinate values of the bifurcation
points in Fig. 4 (a), the values of σ ′ (μc ) corresponding to all bifurcation points can be calculated and the results are shown
in Fig. 4 (b).
From Fig. 4 (b) we have σ ′ (μc ) > 0, indicating that the Hopf bifurcation under power control mode is supercritical.
According to Table 1 , the locations of the stable domain and unstable domain can be determined and shown in Fig. 4 (a).
In Fig. 4 (a), S N -1 (0.01, 1 s −1 ), S N -2 (0.01, 1.96 s −1 ) and S N -3 (0.01, 2.47 s −1 ) are selected for numerical experiment, in
which S N -2 is bifurcation point, S N -1 and S N -3 locate in the stable domain and unstable domain, respectively. The numerical
experiment results are shown in Fig. 4 (c), (d) and 4 (e). The results indicate that: After several rounds of motion, the phase
space trajectory of S N -1 stabilizes at an equilibrium point, and the phase space trajectories of S N -2 and S N -3 enter stable
limit cycles. The stability performances shown in Fig. 4 (c), (d) and (e) are consistent with the theoretical analysis results in
Fig. 4 (a).
Based on the above analysis, we can get the following conclusion: for the primary frequency regulation under power
control mode, the nonlinear hydro-turbine governing system is conditionally stable.
454 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Fig. 4. Nonlinear dynamic performance of nonlinear hydro-turbine governing system under power control mode.
4. Effect mechanism of nonlinear characteristic of penstock head loss
This section aims to reveal the effect mechanism of nonlinear characteristic of penstock head loss. For the novel non-
linear mathematical model established in Section 2 , the nonlinear term is introduced from the nonlinear characteristic of
penstock head loss. The present research conducts the investigation of new feature of the hydro-turbine governing system
caused by the nonlinear characteristic of penstock head loss. The analysis is carried out by the comparisons of stability
between the novel nonlinear mathematical model and a linear mathematical model. In the following parts of this section,
the linear mathematical model for comparison is described firstly. Then, for opening control mode and power control mode,
the analysis for the new feature of the nonlinear hydro-turbine governing system and the effect mechanism of nonlinear
characteristic of penstock head loss is carried out.
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 455
4.1. Linear mathematical model for comparison
Among the basic equations of the novel nonlinear mathematical model in Section 2 , Eq. (12) is nonlinear and the other
equations are linear. For Eq. (12) , the nonlinear term is introduced from the nonlinear characteristic of penstock head loss. If
the nonlinear characteristic of penstock head loss is not considered, we can obtain the linear dynamic equation of penstock
as follows:
T wt0 d q t
d t = −h − z − 2 h t0
H 0
q t (27)
Then we get the linear mathematical model for hydro-turbine governing system with surge tank under primary fre-
quency regulation, i.e. Eqs. (1) , (2) , (27) , ( 13 )–( 15 ) and (18) under opening control mode and Eqs. (1) , (2) , (27) , ( 13 )–( 15 )
and (19) under power control mode.
Under primary frequency regulation of the above linear hydro-turbine governing system, the input signal is the step
disturbance of turbine unit frequency, i.e. x S , and the most important output signal is the dynamic response of p t . The
relationship between the input signal and output signal can be expressed by overall transfer function [44] , which reflects
the inherent characteristics of linear dynamic system. In this section, the overall transfer functions for primary frequency
regulation under opening control mode and power control mode are derived, and the stability criterion is presented.
4.1.1. Overall transfer function
According to the Laplace transform of Eqs. (1) , (2) , (27) , ( 13 )–( 15 ) and (18) , the overall transfer function for the primary
frequency regulation under opening control mode is obtained as follows:
G (s ) =
p tL (s )
x L (s ) =
{( e x + 1) A 1 +
[e qh ( e x + 1) − e h e qx
]A 2
}( s + b p K i ) +
[−e y A 1 + ( e h e qy − e qh e y ) A 2
]( K p s + K i )
a 0 s 4 + a 1 s 3 + a 2 s 2 + a 3 s + a 4 (28)
where p tL ( s ) and x L ( s ) are the frequency domain expressions for p t and x , respectively. The expressions of coefficients in
Eq. (28) are presented in Appendix C .
The purpose of the study on stability is to reveal the performance of dynamic system to return to the initial state after
the input disturbance disappears, i.e. the condition of x L ( s ) = 0. Based on Eq. (28) , x L ( s ) = 0 and the theory of overall transfer
function [44] , the stability for primary frequency regulation of the linear hydro-turbine governing system under opening
control mode is determined by the following characteristic equation:
a 0 λ4 + a 1 λ
3 + a 2 λ2 + a 3 λ + a 4 = 0 (29)
where λ is the root of the characteristic equation.
Proceeding similarly as opening control mode, the overall transfer function and characteristic equation for the primary
frequency regulation of the linear hydro-turbine governing system under power control mode are
G (s ) =
p tL (s )
x L (s ) =
{( e x + 1) A 1 +
[e qh ( e x + 1) − e h e qx
]A 2
}s +
[−e y A 1 + ( e h e qy − e qh e y ) A 2
]( K p s + K i )
e 0 s 4 + e 1 s 3 + e 2 s 2 + e 3 s + e 4 (30)
e 0 λ4 + e 1 λ
3 + e 2 λ2 + e 3 λ + e 4 = 0 (31)
The expressions of coefficients in Eqs. (30) and (31) are presented in Appendix C .
4.1.2. Stability criterion
For the linear dynamic system expressed by Eqs. (28) and (29) , the stability can be judged by the Routh–Hurwitz
criterion [43,44] . Under the opening control mode, the stability criterion is
a i > 0 ( i = 0 , 1 , 2 , 3 , 4 ) (32)
�3 =
∣∣∣∣∣a 1 a 0 0
a 3 a 2 a 1 0 a 4 a 3
∣∣∣∣∣ = a 1 a 2 a 3 − a 2 1 a 4 − a 0 a 2 3 > 0 (33)
When a i ( i = 0, 1, 2, 3, 4) satisfy the discriminants Eqs. (32) and (33) simultaneously, the linear hydro-turbine governing
system under opening control mode is stable.
Under the power control mode, the similar stability criterion can be obtained as follows:
e i > 0 ( i = 0 , 1 , 2 , 3 , 4 ) (34)
�3 =
∣∣∣∣∣e 1 e 0 0
e 3 e 2 e 1 0 e 4 e 3
∣∣∣∣∣ = e 1 e 2 e 3 − e 2 1 e 4 − e 0 e 2 3 > 0 (35)
Based on the above stability criterion, the stable domain for the linear hydro-turbine governing system can be drawn on
b p − K i or e p − K i plane.
456 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Table 2
Values of e 1 , e 3 and �3 under state points S 1 , S 2 and S 3 when F = 100 m
2 .
Points/ Values e 1 e 3 �3
S 1 : e p = 0.01, K i = 1 s −1 1983.0 0.722 38,283
S 2 : e p = 0.03, K i = 5 s −1 1241.6 − 4.592 − 2,040,800
S 3 : e p = 0.07, K i = 10 s −1 − 1671.3 − 25.465 54,666,0 0 0
4.2. Opening control mode
For the linear hydro-turbine governing system under opening control mode, the stability criterion is Eqs. (32) and
(33) . It is easy to judge that a i > 0 ( i = 0, 1, 2, 3, 4) always holds. Therefore, the stability depends on the value of
�3 = a 1 a 2 a 3 − a 2 1 a 4 − a 0 a 2 3 . Based on the hydro-turbine governing theory [36,37] , the stability of the system is the worst
when b p → + ∞ and K i → + ∞ . At the state point S + ∞
( + ∞ , + ∞ ) on the b p − K i plane (as shown in Fig. 3 (a)), we have
b p K i → ( + ∞ ) 2 . The stability of the system at S + ∞
is judged as follows. �3 can be rewritten as �3 = a 1 ( a 2 a 3 − a 1 a 4 ) − a 0 a 2 3 ,
where a 2 a 3 − a 1 a 4 = ( b p K i ) 2 ( b 1 b 2 − b 0 b 3 ) + b p K i b
2 2 + b 2 b 3 . b 0 , b 1 , b 2 and b 3 are always greater than 0, and it is easy to prove
that b 1 b 2 − b 0 b 3 is also always greater than 0. Hence, we have a 2 a 3 − a 1 a 4 → ( + ∞ ) 4 when b p K i → ( + ∞ ) 2 . Moreover, we have
a 1 ( a 2 a 3 − a 1 a 4 ) → ( + ∞ ) 6 based on a 1 → ( + ∞ ) 2 and a 2 a 3 − a 1 a 4 → ( + ∞ ) 4 . Because of a 1 ( a 2 a 3 − a 1 a 4 ) → ( + ∞ ) 6 and a 0 a 2 3
→(+ ∞ ) 4 , we can obtain that �3 = a 1 ( a 2 a 3 − a 1 a 4 ) − a 0 a
2 3 > 0 always holds when b p → + ∞ and K i → + ∞ , which indicates that
the system at S + ∞
is stable. Because the stability of the system at S + ∞
is the worst, we can determine that the system is
always stable at other points on the b p − K i plane. Therefore, the stable domain is the same with that in Fig. 3 (a). Taking the
hydropower station in Section 3.2 as example, by computations it is found that a i > 0 ( i = 0, 1, 2, 3, 4) and �3 = a 1 a 2 a 3 −a 2 1 a 4 − a 0 a
2 3 > 0 always hold for all the points on the b p − K i plane, which verify the results of the above theoretical analysis.
Based on the above analysis, we can obtain the following conclusion: for the primary frequency regulation under
opening control mode, the linear hydro-turbine governing system is absolutely stable. From the above conclusion, we can
get that the stability of the nonlinear hydro-turbine governing system in Section 2 and the linear hydro-turbine governing
system in Section 4.1 is the same, indicating that the nonlinear characteristic of penstock head loss cannot change the
stability for primary frequency regulation of hydro-turbine governing system under opening control mode.
4.3. Power control mode
For the linear hydro-turbine governing system under power control mode, we cannot judge the stability of the system
from Eqs. (34) and (35) directly because the expressions of e i and �3 are extremely complicated. Taking the hydropower
station in Section 3.2 as example, we can draw the curves of e i = 0 ( i = 0, 1, 2, 3, 4) and �3 = e 1 e 2 e 3 − e 2 1 e 4 − e 0 e 2 3 = 0 , which
are called the discriminant curves of stability. The results are shown in Fig. 5 (b). From Fig. 5 (b) we can get the following
conclusion: for the primary frequency regulation under power control mode, the linear hydro-turbine governing system is
conditionally stable. In the following parts of this section, Fig. 5 (b) is adopted to study the stable domain of the system.
Meanwhile, the discriminant curves when F are valued as 50 m
2 , 200 m
2 and 400 m
2 are also given to illustrate the changes
of the positions and relative position relationships of the discriminant curves. The results are shown in Fig. 5 (a), (c) and (d).
Fig. 5 shows that:
(1) On the e p − K i plane, there are four smooth discriminant curves of stability. To be specific, e i = 0 includes e 1 = 0 and
e 3 = 0 because e i > 0 ( i = 0, 2, 4) always holds; �3 = 0 is composed of two curves, which are denoted as ( �3 = 0) 1 and ( �3 = 0) 2 . e 1 = 0 and e 3 = 0 locate in the region between ( �3 = 0) 1 and ( �3 = 0) 2 . e 1 = 0 is close to ( �3 = 0) 1 ,
and e 3 = 0 is close to ( �3 = 0) 2 . With the increase of the sectional area of surge tank, the interval between e 1 = 0
and ( �3 = 0) 1 becomes smaller, while the interval between e 3 = 0 and ( �3 = 0) 2 becomes larger. During the above
change processes, the relative position relationships between e 1 = 0 and ( �3 = 0) 1 as well as e 3 = 0 and ( �3 = 0) 2 keep unchanged.
(2) Taking F = 100 m
2 as an example, three state points, i.e. S 1 , S 2 and S 3 in Fig. 5 (b), are selected to calculate the values
of e 1 , e 3 and �3 . The results are shown in Table 2 . For e 1 , we have e 1 > 0 under S 1 and S 2 while e 1 < 0 under S 3 ,
indicating that the state points in the region of bottom left corner of e 1 = 0 satisfy the condition of e 1 > 0. For e 3 , we
have e 1 > 0 under S 1 while e 1 < 0 under S 2 and S 3 , indicating that the state points in the region of bottom left corner
of e 3 = 0 satisfy the condition of e 3 > 0. For �3 , we have �3 > 0 under S 1 and S 3 while �3 < 0 under S 2 , indicating
that the state points in the region of top right corner of ( �3 = 0) 1 and bottom left corner of ( �3 = 0) 2 satisfy the
condition of �3 > 0.
(3) Based on the analysis results of the relative position relationships among e 1 = 0, e 3 = 0, ( �3 = 0) 1 and ( �3 = 0) 2 and
the calculated values of e 1 , e 3 and �3 under S 1 , S 2 and S 3 , we can obtain that the region of bottom left corner of
( �3 = 0) 2 can satisfy the discriminants Eqs. (34) and (35) simultaneously. That region is the stable domain under
power control mode and shown in Fig. 6 . Correspondingly, ( �3 = 0) 2 is called the critical curve, and the region of top
right corner of ( �3 = 0) 2 is the unstable domain.
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 457
Fig. 5. Discriminant curves of stability for linear hydro-turbine governing system under power control mode.
Table 3
Performance indexes of dynamic response processes of p t .
Indexes/Models/ h t 0 (m) Peak value Damping ratio Period (s)
Nonlinear model Linear model Nonlinear model Linear model Nonlinear model Linear model
From the above analysis we get that the nonlinear hydro-turbine governing system and linear hydro-turbine governing
system are both conditionally stable. Now, we conduct the investigation of new feature of the nonlinear hydro-turbine
governing system and reveal the effect mechanism of nonlinear characteristic of penstock head loss by the comparisons of
stability between the nonlinear mathematical model and linear mathematical model.
Taking the hydropower station in Section 3.2 and F = 100 m
2 as example, Fig. 7 shows the comparisons of stable domain
between the nonlinear mathematical model and linear mathematical model under x S = − 0.004 (negative disturbance) and
x S = 0.004 (positive disturbance), respectively. Meanwhile, the penstock head loss h t 0 is valued as 0.79 m and 1.29 m to illus-
trate its effect on the stability. In Fig. 7 , the dynamic response processes of p t under the state point S N − 1 (0.01, 1s − 1 ) are also
given for assistant analysis. The dynamic response processes of p t under the nonlinear mathematical model are solved from
Eqs. (22) and (21) . And the dynamic response processes of p t under the linear mathematical model are determined based on
Eq. (30) and x L ( s ) = x S / s , in which the function residue() in MATLAB [45] is used for equation solving. Table 3 gives the perfor-
mance indexes of the dynamic response processes of p t that quantitatively describe the dynamic performance of the system.
Note that:
458 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Fig. 6. Stable domain, critical curve and unstable domain for linear hydro-turbine governing system under power control mode.
(1) For the negative disturbance, i.e. x S = − 0.004, the peak value represents the value of the first wave crest for the
dynamic response process of p t . For the positive disturbance, i.e. x S = 0.004, the peak value represents the value of
the first wave trough for the dynamic response process of p t .
(2) The damping ratio and period represent the dynamic response process of p t after the first wave crest under negative
disturbance or the first wave trough under positive disturbance, which is the periodic sine attenuation stage.
Fig. 7 and Table 3 show that:
(1) There is an obvious difference of stable domain between the nonlinear mathematical model and linear mathematical
model. To be specific,
(2) Under the negative disturbance, i.e. x S = − 0.004, the stable domain of the nonlinear mathematical model is larger than
that of the linear mathematical model, indicating that the stability of the nonlinear system is better than that of the
linear system and the nonlinear characteristic of penstock head loss is favorable for stability. For both the nonlinear
system and linear system, the stable domain becomes larger and the stability becomes better with the increase of h t 0 .
(3) Under the positive disturbance, i.e. x S = 0.004, the stable domain of the nonlinear mathematical model is smaller than
that of the linear mathematical model, indicating that the stability of the nonlinear system is worse than that of
the linear system and the nonlinear characteristic of penstock head loss is unfavorable for stability. For the nonlin-
ear system, the stable domain becomes smaller and the stability becomes worse with the increase of h t 0 . For the
linear system, the stable domain is the same with that under negative disturbance, indicating that the type of the
disturbance has no effect on the linear system stability.
(4) There is an obvious difference of dynamic response processes of p t between the nonlinear mathematical model and
linear mathematical model. And the difference among the response processes mainly embodies in the initial stage,
not the periodic sine attenuation stage. That phenomenon indicates that the nonlinear characteristic of penstock head
loss mainly affects the initial stage of dynamic response process of p t , and then changes the stability of the nonlinear
system. And the stability of the system mainly determined by the initial stage of dynamic response process of p t . The
nonlinear characteristic of penstock head loss and h t 0 almost have no effect on the periodic sine attenuation stage.
(5) Tuning and optimization method for determining the system parameters.
Stable domain shows the states of stability of hydro-turbine governing system on the b p − K i or e p − K i plane. The tuning
and optimization of the system parameters can be easily achieved by using the stable domain. For the opening control
mode, the hydro-turbine governing system is absolutely stable on the b p − K i plane. So from the perspective of stability, the
system parameters can take any values. For the power control mode, we can firstly draw the stable domain on the e p − K i
plane based on the analysis in Sections 3 and 4 . Then, ( e p ,K i ) should fall in the stable domain during the parameter tuning
of governor. As a result, the hydro-turbine governing system under primary frequency regulation is stable. Moreover, ( e p ,K i )
should keep away from the bifurcation line/critical curve as much as possible to obtain a better stability.
For the nonlinear hydro-turbine governing system, the effect of penstock head loss is opposite under the negative distur-
bance and positive disturbance. For the design of hydropower station, the occurrence frequency for the negative disturbance
and positive disturbance should be considered comprehensively. The penstock head loss also affects the generation benefit
and construction investment, which should be coordinated together with the stability of hydro-turbine governing system.
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 459
Fig. 7. Comparisons of stability between nonlinear mathematical model and linear mathematical model.
5. Critical stable sectional area of surge tank for primary frequency regulation
It is well known that F is the most important hydraulic parameter for the design of surge tank, and the value of F has
a significant influence on the operation of the hydropower station. Hence, the issue of the critical stable sectional area of
surge tank is studied in this section.
For primary frequency regulation under opening control mode, the hydro-turbine governing system is absolutely stable.
There is not a critical stable state for the system. The sectional area of surge tank can take any value. For primary frequency
regulation under power control mode, the hydro-turbine governing system is conditionally stable. The system may be stable
or unstable for different sectional areas of surge tank.
Based on the results in Table 3 , the period of the periodic sine attenuation stage for dynamic response process of p tis in the range from 421.69 s to 424.54 s and denoted as T PS . It is well known that the period of water level oscillation in
surge tank can be determined as T ST = 2 π√
L y F /g f y = 419 . 23s . Then we have T PS ≈ T ST , indicating that the periodic sine
attenuation stage is caused by the water level oscillation in surge tank. The results in Section 4.3 show that the nonlinear
characteristic of penstock head loss almost has no effect on the periodic sine attenuation stage. Therefore, there is almost
no difference for the stability of the water level oscillation in surge tank between the nonlinear mathematical model and
linear mathematical model. Hence, the linear mathematical model can be adopted to derive the critical stable sectional area
of surge tank, which is reasonable in theory and much simpler than nonlinear mathematical model.
According to the results obtained in Section 4.3 , the decisive discriminant for the stability of the linear system is �3
> 0. Correspondingly, the critical condition for the stability is �3 = 0, which is called the critical stable discriminant. The
expression of �3 contains F . Therefore, the expression of F that solved from �3 = 0 is the sectional area of surge tank that
makes the system reach the critical stable state. That sectional area is called the critical stable sectional area of surge tank
for primary frequency regulation in this paper and is denoted as F . The critical stable sectional area is an important basis
PFR
460 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Fig. 8. F PFR -1 and F PFR -2 on e p - K i plane.
for the design of the surge tank. To achieve the stable oscillations of water level in surge tank and the dynamic response of
power output, the practical value of the sectional area of surge tank should be selected based on the critical stable sectional
area. In the following parts of this section, the analytical solution for F PFR is studied.
For �3 = e 1 e 2 e 3 − e 2 1 e 4 − e 0 e
2 3
= 0 , F is set as the independent variable and then e i ( i = 0, 1, 2, 3, 4) can be converted into
the functions of F as follows: ⎧ ⎪ ⎪ ⎨
⎪ ⎪ ⎩
e 0 = g 0 F e 1 = g 1 F e 2 = g 2 + g 3 F e 3 = g 4 + g 5 F e 4 = g 6
(36)
By substituting Eq. (36) into �3 = e 1 e 2 e 3 − e 2 1 e 4 − e 0 e 2 3 = 0 yields
l 0 F 2 + l 1 F + l 2 = 0 (37)
The expressions of coefficients in Eqs. (36) and (37) are presented in Appendix D .
By solving Eq. (37) , we obtain the analytical formulas for the critical stable sectional area of surge tank for primary
frequency regulation
F PF R −1 =
−l 1 +
√
l 2 1
− 4 l 0 l 2
2 l 0 (38)
F PF R −2 =
−l 1 −√
l 2 1
− 4 l 0 l 2
2 l 0 (39)
Taking the hydropower station in Section 3.2 as example, the values of F PFR − 1 and F PFR − 2 can be calculated from
Eqs. (38) and (39) , and the results are shown in Fig. 8 . Note that, in the practical applications, only the positive real so-
lutions of F PFR − 1 and F PFR − 2 are meaningful. Hence, only the positive real solutions of F PFR − 1 and F PFR − 2 are given in Fig. 8 .
Fig. 8 shows that, for the value, distribution and change law, there is a great difference between F PFR − 1 and F PFR − 2 . To
be specific,
(1) F PFR − 1 : In the left side of the e p − K i plane, i.e. the region where e p is small, the value of F PFR − 1 is small and about
100 m
2 . With the increase of e p and K i , F PFR − 1 increases rapidly and reaches 50 0–60 0 m
2 . Then with the further
increase of e p and K i , F PFR − 1 keeps almost unchanged first and decreases rapidly afterward. Finally, in the top right
corner of the e p − K i plane, i.e. the region where e p and K i are both large, F PFR − 1 reduces to less than 10 m
2 and keeps
almost unchanged.
(2) F PFR − 2 : The value of F PFR − 2 is distributed in two isolated regions. The first isolated region is the left side of the e p − K i
plane, i.e. the region where e p is small. In the first isolated region, F PFR − 2 is less than 100 m
2 and is very sensitive to
the change of e p . The second isolated region is the top right corner of the e p − K i plane, i.e. the region where e p and
K i are both large. In the second isolated region, F PFR − 2 is between 500 m
2 and 600 m
2 and is not sensitive to the
change of e p or K i .
To further analyze the relationship between F PFR − 1 and F PFR − 2 , we take the cross sections of Fig. 8 under different K i ,
i.e. 4 s −1 , 8 s −1 , 12 s −1 and 16 s −1 , and then draw the transversals of F PFR − 1 and F PFR − 2 under the same K i in one figure.
The results are shown in Fig. 9 .
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 461
Fig. 9. Variations of F PFR -1 and F PFR -2 with respect to e p under different K i .
Fig. 9 shows that:
(1) Both F PFR − 1 and F PFR − 2 consist of two smooth curves. Taking Fig. 9 (b) as an example, the two smooth curves of
F PFR − 1 are denoted as L 1 − 1 and L 1 − 2 , respectively; the two smooth curves of F PFR − 2 are denoted as L 2 − 1 and L 2 − 2 ,
respectively. L 2 − 1 , L 1 − 1 and L 2 − 2 are joined end to end smoothly. The intersection point of L 2 − 1 and L 1 − 1 is denoted
as S C − 1 . S C − 1 is the left endpoint for both L 2 − 1 and L 1 − 1 . When e p is in the right side of S C − 1 , there are one F PFR − 1
and one F PFR − 2 with respect to one e p . The intersection point of L 1 − 1 , L 1 − 2 and L 2 − 2 is denoted as S C − 2 . S C − 2 is the
right endpoint for L 1 − 1 and the left endpoint for both L 1 − 2 and L 2 − 2 . When e p is in the right side of S C − 2 , there
are one F PFR − 1 and one F PFR − 2 with respect to one e p . With the increase of K i , both S C − 1 and S C − 2 move to the side
where e p is smaller.
(2) Actually, S C − 1 and S C − 2 are the intersection points of F PFR − 1 and F PFR − 2 . In other words, for a K i , the two values of e pthat satisfy F PFR − 1 = F PFR − 2 are the abscissa values of S C − 1 and S C − 2 . Correspondingly, the two values of F PFR − 1 and
F PFR − 2 that satisfy F PFR − 1 = F PFR − 2 are the ordinate values of S C − 1 and S C − 2 . Based on Eqs. (38) and (39) , l 2 1 − 4 l 0 l 2 = 0
yields F PFR − 1 = F PFR − 2 . For a given K i , e p can be solved from l 2 1
− 4 l 0 l 2 = 0 and the two roots corresponding to the
abscissa values of S C − 1 and S C − 2 are denoted as e p − C − 1 and e p − C − 2 , respectively.
e p−C−1 = min
{ e p ( K i ) | l 2
1 −4 l 0 l 2 =0
} (40)
e p−C−2 = max
{ e p ( K i ) | l 2
1 −4 l 0 l 2 =0
} (41)
The ordinate values of S C − 1 and S C − 2 are
F C−1 =
−l 1 2 l 0
( e p−C−1 , K i ) (42)
F C−2 =
−l 1 2 l 0
( e p−C−2 , K i ) (43)
(3) Taking K i = 8 s −1 , i.e. Fig. 9 (b), as an example, four points S 1 − 1 , S 1 − 2 , S 2 − 1 and S 2 − 2 are selected on L 1 − 1 , L 1 − 2 ,
L 2 − 1 and L 2 − 2 , respectively. S 1 − 1 and S 2 − 1 have the same abscissa values; S 1 − 2 and S 2 − 2 have the same abscissa
values. Then, under the same abscissa value, S A − 1 which locates in the middle of S 1 − 1 and S 2 − 1 as well as S B − 1 and
462 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Table 4
Values of discriminants corresponding to the ten selected points.
Fig. 10. Distributions of stability states of system and critical stable sectional areas of surge tank.
S C − 1 which locate on both sides of S 1 − 1 and S 2 − 1 are selected. Similarly, under the same abscissa value, S A − 2 which
locates in the middle of S 1 − 2 and S 2 − 2 as well as S B − 2 and S C − 2 which locate on both sides of S 1 − 2 and S 2 − 2 are
selected. The coordinate values for the above ten selected points are shown in Table 4 . The values of the discriminants
corresponding to the ten points are also shown in Table 4 .
Table 4 shows that:
(1) For S 1 − 1 and S 2 − 1 , the coordinate values can satisfy �3 = 0 and e i > 0 ( i = 0, 1, 2, 3, 4) simultaneously. Moreover, for
S A − 1 , S B − 1 and S C − 1 , e i > 0 ( i = 0, 1, 2, 3, 4) always holds. We also have �3 < 0 under S A − 1 and �3 > 0 under S B − 1
and S C − 1 . The results indicate that S 1 − 1 and S 2 − 1 are the real critical stable points, and the corresponding F PFR − 1
and F PFR − 2 are the real critical stable sectional areas of surge tank.
(2) For S 1 − 2 and S 2 − 2 , the coordinate values can satisfy �3 = 0 while e 1 < 0 and e 3 < 0 hold. Moreover, for S A − 2 , S B − 2
and S C − 2 , e 1 < 0 always holds. The results indicate that the system is always unstable. Hence, S 1 − 2 and S 2 − 2 are not
the real critical stable points, and the corresponding F PFR − 1 and F PFR − 2 are not the real critical stable sectional areas
of surge tank.
(3) Based on the above analysis, we can obtain the following conclusions on the critical stable sectional area of surge
tank. For any K i ,
(a) If e p > e p − C − 2 , the system is absolutely unstable. There does not exist a critical stable sectional area of surge tank.
(b) If e p − C − 1 ≤ e p ≤ e p − C − 2 , the system is conditionally stable. The critical stable sectional areas of surge tank are
F PFR − 1 and F PFR − 2 that determined by Eqs. (38) and (39) . When F PFR − 2 < F < F PFR − 1 , the system is unstable; when F
> F PFR − 1 or F < F PFR − 2 , the system is stable.
(c) If 0 < e p < e p − C − 1 , the system is absolutely stable. There does not exist a critical stable sectional area of surge tank,
and the sectional area of surge tank can take any value.
The above conclusions are illustrated in Fig. 10 .
Application guidances: In the practical applications for the primary frequency regulation under power control mode, the
combined tuning and optimization of the governor parameters, i.e. e p and K i , and the sectional area of surge tank F can be
achieved based on the above conclusions.
(1) The smaller K i , the larger the absolutely stable region and the smaller the absolutely unstable region. For the same e p ,
the critical stable sectional area of surge tank decreases with respect to small K i . Hence, K i is recommended as small
value.
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 463
(2) For a preliminarily selected K i , e p − C − 1 and e p − C − 2 are calculated from Eqs. (40) and (41) . Then we determine the
distributions of absolutely stable region, conditionally stable region and absolutely unstable region. Based on practical
conditions, e p can be selected in absolutely stable region or conditionally stable region.
(3) If e p is selected in the absolutely stable region, the sectional area of surge tank F can take any value to achieve the
stable operation for the primary frequency regulation under power control mode. If e p is selected in the conditionally
stable region, we can firstly determine F PFR − 1 and F PFR − 2 from Eqs. (38) and (39) . Then the sectional area of surge
tank F should satisfy F > F PFR − 1 or F < F PFR − 2 .
6. Summary and conclusions
For the hydropower station with surge tank, a novel nonlinear mathematical model of the hydro-turbine governing
system under primary frequency regulation considering the nonlinear characteristic of penstock head loss is introduced.
The nonlinear state equations under opening control mode and power control mode are derived. The nonlinear dynamic
performance of nonlinear hydro-turbine governing system is investigated. The stable domain for primary frequency regula-
tion is proposed and drawn under both opening control mode and power control mode. The new feature of the nonlinear
hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing
with a linear model. The effect mechanism of nonlinear characteristic of penstock head loss is revealed. The concept of
critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is
derived. Based on the analytical solution, the combined tuning and optimization method of the governor parameters and
sectional area of surge tank is proposed. Several conclusions can be drawn from this study:
(1) For the primary frequency regulation under opening control mode, the nonlinear hydro-turbine governing system is
absolutely stable. The stable domain is the whole b p − K i plane. For the primary frequency regulation under power
control mode, the nonlinear hydro-turbine governing system is conditionally stable. The stable domain is the region
of the bottom left corner of bifurcation line on the e p − K i plane.
(2) The nonlinear characteristic of penstock head loss cannot change the stability of hydro-turbine governing system
under opening control mode. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine
governing system is the same under opening control model.
(3) There is an obvious difference of stable domain between the nonlinear mathematical model and linear mathematical
model under power control model. Under negative disturbance, the stability of nonlinear system is better than that
of linear system and the nonlinear characteristic of penstock head loss is favorable for the stability. Under positive
disturbance, the stability of nonlinear system is worse than that of linear system and the nonlinear characteristic of
penstock head loss is unfavorable for the stability.
(4) The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of p t ,
and then changes the stability of the nonlinear system. The stability of the system mainly determined by the initial
stage of dynamic response process of p t . The nonlinear characteristic of penstock head loss and h t 0 almost have no
effect on the periodic sine attenuation stage of dynamic response process of p t .
(5) The critical stable sectional area of surge tank for primary frequency regulation makes the system reach the critical
stable state, and is an important basis for the design of surge tank. The linear mathematical model can be adopted to
derive the critical stable sectional area of surge tank for primary frequency regulation. The analytical formulas for the
critical stable sectional area are expressed by Eqs. (38) and (39) . For any K i ,
(a) If e p > e p − C − 2 , the system is absolutely unstable. There does not exist a critical stable sectional area of surge tank.
(b) If e p − C − 1 ≤ e p ≤ e p − C − 2 , the system is conditionally stable. When F PFR − 2 < F < F PFR − 1 , the system is unstable; when
F > F PFR − 1 or F < F PFR − 2 , the system is stable.
(c) If 0 < e p < e p − C − 1 , the system is absolutely stable. There does not exist a critical stable sectional area of surge tank,
and the sectional area of surge tank can take any value.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Project No. 51379158 ) and the China
Scholarship Council (Project No. 201506270057 ).
Appendix A
(1) Basic notations and variables
464 W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466
Q y discharge in headrace tunnel, (m
3 /s) Q t discharge in penstock (i.e. turbine unit discharge),
(m
3 /s)
V y velocity in headrace tunnel, (m/s) V t velocity in penstock, (m/s)
L y length of headrace tunnel, (m) L t length of penstock, (m)
f y sectional area of headrace tunnel, (m
2 ) f t sectional area of penstock, (m
2 )
h y head loss of headrace tunnel, (m) h t head loss of penstock, (m)
T wy flow inertia time constant of headrace tunnel, (s) T wt flow inertia time constant of penstock, (s)
H turbine net head, (m) F sectional area of surge tank, (m
2 )
g acceleration of gravity, (m/s 2 ) γ specific weight of water, N/m
3
Z change of water level in surge tank (relative to the
initial level, positive direction is downward), (m)
Z � elevation difference between initial water level in
surge tank and tailwater level, (m)
M t kinetic moment, (N · m) P t power output, (kW)
N turbine unit frequency, (Hz) Y guide vane opening, (mm)
e h , e x , e y moment transfer coefficients of turbine e qh , e qx , e qy discharge transfer coefficients of turbine
K p proportional gain K i integral gain, (s −1 )
K d differential gain, (s) b t temporary droop
b p permanent droop e p power permanent droop
T y servomotor response time constant, (s) T d damping device time constant, (s)
T n acceleration time constant, (s) T F time constant of surge tank, (s)
αt coefficient of penstock head loss, (s 2 /m
5 ) t time, (s)
s complex variable
(2) Relative variables (The subscript ‘0’ refers to the initial value.) h =
H−H 0 H 0
, z =
Z H 0
, q y =
Q y −Q y 0 Q y 0
, q t =
Q t −Q t0 Q t0
, x =
N−N 0 N 0
,
y =
Y −Y 0 Y 0
, m t =
M t −M t0 M t0
and p t =
P t −P t0 P t0
are the relative deviations of corresponding variables.
(3) Other variables (The subscript ‘0’ refers to the initial value.)
T wy 0 =
L y Q y 0
g H 0 f y , T wt0 =
L t Q t0
g H 0 f t , T F =
F H 0
Q y 0
, e h =
∂ m t
∂h
, e x =
∂ m t
∂x , e y =
∂ m t
∂y , e qh =
∂ q t ∂h
, e qx =
∂ q t ∂x
, e qy =
∂ q t ∂y
.
Appendix B
Opening control mode:
∂ ˙ q y ∂ q y
= − 1
T wy 0
2 h y 0
H 0
, ∂ ˙ q y ∂z
=
1
T wy 0
, ∂ ˙ q y ∂ q t
= 0 , ∂ ˙ q y ∂y
= 0 , ∂ ̇ z
∂ q y = − 1
T F ,
∂ ̇ z
∂z = 0 ,
∂ ̇ z
∂ q t =
1
T F ,
∂ ̇ z
∂y = 0 ,
∂ ˙ q t ∂ q y
= 0 ,
∂ ˙ q t ∂z
= − 1
T wt0
, ∂ ˙ q t ∂ q t
= − 1
T wt0
(2 h t0
H 0
+
1
e qh
)− 1
T wt0
2 h t0
H 0
q tE , ∂ ˙ q t ∂y
=
1
T wt0
e qy
e qh
, ∂ ˙ y
∂ q y = 0 ,
∂ ˙ y
∂z = 0 ,
∂ ˙ y
∂ q t = 0 ,
∂ ˙ y
∂y = −b p K i ,
c 1 = −(
∂ ˙ q y ∂ q y
+
∂ ˙ q t ∂ q t
+
∂ ˙ y
∂y
), c 2 =
∂ ˙ q y ∂ q y
∂ ˙ y
∂y +
∂ ˙ q t ∂ q t
∂ ˙ y
∂y +
∂ ˙ q y ∂ q y
∂ ˙ q t ∂ q t
− ∂ ˙ q y ∂z
∂ ̇ z
∂ q y − ∂ ̇ z
∂ q t
∂ ˙ q t ∂z
,
c 3 =
∂ ˙ q y ∂z
∂ ̇ z
∂ q y
∂ ˙ y
∂y +
∂ ̇ z
∂ q t
∂ ˙ q t ∂z
∂ ˙ y
∂y +
∂ ˙ q y ∂z
∂ ̇ z
∂ q y
∂ ˙ q t ∂ q t
+
∂ ̇ z
∂ q t
∂ ˙ q t ∂z
∂ ˙ q y ∂ q y
− ∂ ˙ q y ∂ q y
∂ ˙ q t ∂ q t
∂ ˙ y
∂y ,
c 4 = −(
∂ ˙ q y ∂z
∂ ̇ z
∂ q y
∂ ˙ q t ∂ q t
∂ ˙ y
∂y +
∂ ̇ z
∂ q t
∂ ˙ q t ∂z
∂ ˙ q y ∂ q y
∂ ˙ y
∂y
).
Power control mode:
∂ ˙ q y ∂ q y
= − 1
T wy 0
2 h y 0
H 0
, ∂ ˙ q y ∂z
=
1
T wy 0
, ∂ ˙ q y ∂ q t
= 0 , ∂ ˙ q y ∂y
= 0 , ∂ ̇ z
∂ q y = − 1
T F ,
∂ ̇ z
∂z = 0 ,
∂ ̇ z
∂ q t =
1
T F ,
∂ ̇ z
∂y = 0 ,
∂ ˙ q t ∂ q y
= 0 ,
∂ ˙ q t ∂z
= − 1
T wt0
, ∂ ˙ q t ∂ q t
= − 1
T wt0
(2 h t0
H 0
+
1
e qh
)− 1
T wt0
2 h t0
H 0
q tE , ∂ ˙ q t ∂y
=
1
T wt0
e qy
e qh
, ∂ ˙ y
∂ q y = 0 ,
∂ ˙ y
∂z = 0 ,
∂ ˙ y
∂ q t = −e p K i
e h e qh
,
∂ ˙ y
∂y = −e p K i
(e y − e h
e qh
e qy
), c 1 = −
(∂ ˙ q y ∂ q y
+
∂ ˙ q t ∂ q t
+
∂ ˙ y
∂y
),
c 2 =
∂ ˙ q y ∂ q y
∂ ˙ y
∂y +
∂ ˙ q t ∂ q t
∂ ˙ y
∂y +
∂ ˙ q y ∂ q y
∂ ˙ q t ∂ q t
− ∂ ˙ q y ∂z
∂ ̇ z
∂ q y − ∂ ̇ z
∂ q t
∂ ˙ q t ∂z
− ∂ ˙ q t ∂y
∂ ˙ y
∂ q t ,
c 3 =
∂ ˙ q y ∂z
∂ ̇ z
∂ q y
∂ ˙ y
∂y +
∂ ̇ z
∂ q t
∂ ˙ q t ∂z
∂ ˙ y
∂y +
∂ ˙ q y ∂z
∂ ̇ z
∂ q y
∂ ˙ q t ∂ q t
+
∂ ̇ z
∂ q t
∂ ˙ q t ∂z
∂ ˙ q y ∂ q y
+
∂ ˙ q t ∂y
∂ ˙ y
∂ q t
∂ ˙ q y ∂ q y
− ∂ ˙ q y ∂ q y
∂ ˙ q t ∂ q t
∂ ˙ y
∂y ,
c 4 =
∂ ˙ q y ∂ ̇ z ∂ ˙ q t ∂ ˙ y −(
∂ ˙ q y ∂ ̇ z ∂ ˙ q t ∂ ˙ y +
∂ ̇ z ∂ ˙ q t ∂ ˙ q y ∂ ˙ y )
.
∂z ∂ q y ∂y ∂ q t ∂z ∂ q y ∂ q t ∂y ∂ q t ∂z ∂ q y ∂y
W. Guo, J. Yang / Applied Mathematical Modelling 54 (2018) 446–466 465
Appendix C
The expressions of coefficients in Eq. (28) are presented as follows.
a 0 = b 0 , a 1 = b p K i b 0 + b 1 , a 2 = b p K i b 1 + b 2 , a 3 = b p K i b 2 + b 3 , a 4 = b p K i b 3 ,
b 0 = e qh T wy 0 T wt0 T F , b 1 =
[T wy 0 + e qh
(T wy 0
2 h t0
H 0
+ T wt0
2 h y 0
H 0
)]T F , b 2 = e qh ( T wy 0 + T wt0 ) +
(1 + e qh
2 h t0
H 0
)2 h y 0
H 0
T F ,
b 3 = 1 + e qh
2( h y 0 + h t0 )
H 0
, A 1 = T wy 0 T F s 2 +
2 h y 0
H 0
T F s + 1 ,
A 2 = T wy 0 T wt0 T F s 3 +
(T wy 0
2 h t0
H 0
+ T wt0
2 h y 0
H 0
)T F s
2 +
(T wy 0 + T wt0 +
2 h y 0
H 0
2 h t0
H 0
T F
)s +
2( h y 0 + h t0 )
H 0
.
The expressions of coefficients in Eqs. (30) and (31) are presented as follows.
e 0 = b 0 , e 1 = e y e p K i b 0 + b 1 + f 0 , e 2 = e y e p K i b 1 + b 2 + f 1 , e 3 = e y e p K i b 2 + b 3 + f 2 , e 4 = e y e p K i b 3 + f 3 ,
f 0 = −e h e qy e p K i T wy 0 T wt0 T F , f 1 = −e h e qy e p K i
(T wy 0
2 h t0
H 0
+ T wt0
2 h y 0
H 0
)T F , f 2 = −e h e qy e p K i
(T wy 0 + T wt0 +
2 h y 0
H 0
2 h t0
H 0
T F
),
f 3 = −e h e qy e p K i
2( h y 0 + h t0 )
H 0
.
Appendix D
g 0 = e qh T wy 0 T wt0 H 0
Q y 0
, g 1 =
{[1 + e y e p K i
(e qh −
e h e qy
e y
)T wt0
]T wy 0 + e qh
(T wy 0
2 h t0
H 0
+ T wt0
2 h y 0
H 0
)}H 0
Q y 0
,
g 2 = e qh ( T wy 0 + T wt0 ) , g 3 =
{e y e p K i
[T wy 0 +
(e qh −
e h e qy
e y
)(T wy 0
2 h t0
H 0
+ T wt0
2 h y 0
H 0
)]+
(1 + e qh
2 h t0
H 0
)2 h y 0
H 0
}H 0
Q y 0
,
g 4 = 1 + e qh
2( h y 0 + h t0 )
H 0
+ e y e p K i
(e qh −
e h e qy
e y
)( T wy 0 + T wt0 ) , g 5 = e y e p K i
[1 +
(e qh −
e h e qy
e y
)2 h t0
H 0
]2 h y 0
H 0
H 0
Q y 0
,
g 6 = e y e p K i
[
1 +
(e qh −
e h e qy
e y
)2
(h y 0 + h t0
)H 0
]
, l 0 = g 1 g 3 g 5 − g 0 g 2 5 , l 1 = g 1 g 2 g 5 + g 1 g 3 g 4 − g 2 1 g 6 − 2 g 0 g 4 g 5 ,
l 2 = g 1 g 2 g 4 − g 0 g 2 4 .
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