A Control Method for Switching-Type Frequency Oscillations in

Hydropower System Based on Adjusting Dead Zone Structure

Hongtao Xiong

, Boxiong Zhang

, Boliang Lou

, Xiaobo Liu

Shan Hou

, Jiawei Wang

and Ancheng Xue

2,*

Zhejiang Electric Power Research Institute, Hangzhou, Zhejiang 310014, China

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (North China Electric Power

University), Changping District, Beijing 102206, China

Keywords: Enhanced Dead Zone, Hydropower System, Switching-Type Frequency Oscillation, Control Method.

Abstract: For the switching-type ultra-low frequency oscillation (ST-ULFO) that cannot be suppressed by increasing

damping, in the single-hydropower-generator power system (SHG-PS) with enhanced dead zone, a control

method based on adjusting the non-smooth structure of the dead zone is proposed. First, the model of the

SHG-PS with enhanced dead zone, corresponding to Filippov non-smooth system, and the ST-ULFO

phenomenon are introduced. Second, a case of oscillation cannot be suppressed by increasing system

damping, as the system equilibrium point disappeared under disturbance, is introduced. Next, a construction

method to make the enhanced dead zone continuous is given. Then, the Hopf-like non-smooth bifurcation

characteristics of the equilibrium point and the post fault dynamics of the SHG-PS before/after the enhanced

dead zone continuity are compared. Finally, the effectiveness of control method is verified in the SHG-PS

and the 2-area-4-generators system. The results show that, the oscillations which cannot be suppressed by

increasing system damping, will be effectively suppressed by adjusting the enhanced dead zone structure.

1 INTRODUCTION

Ultra-low frequency oscillations (ULFOs) have

occurred many times in the power grid with a high

proportion of hydropower at home and abroad (Li et

al., 2018). The ULFOs are quite different from the

traditional low-frequency oscillations and threaten the

safe operation of the power system Xue et al., 2021).

The ULFOs are mainly explained by the negative

damping oscillation (Liu et al., 2016), the smooth

forced oscillation (Ju et al., 2014), or the switching-

type oscillation (non-smooth oscillation)

Xue et al.,

2021). The negative damping oscillation and the

smooth forced oscillation correspond to the

oscillations in smooth dynamical systems. While the

switching-type oscillation correspond to the

oscillations associated with the switches, such as dead

zones, limits, or control switches in the non-smooth

dynamical system (Xue and Wang, 2020)(Xue et al.,

2021).

The negative damping oscillations and smooth

Corresponding author

forced oscillations can be suppressed or prevented by

increasing the system damping. Due to the influence

of water hammer effect, hydropower generators

exhibit negative damping characteristics in the ultra-

low frequency band. Thus, the control measures for

ULFOs also adopt the way of improving system

damping, such as optimizing the PID parameters of the

governor (Zhou et al., 2017).

It is worth noting that, although the control

measure of improving system damping has good

universality, the switching-type oscillations reflect the

large-scale dynamics of the system and have no clear

correspondence with the local properties of the

equilibrium point. For example, in a single-

hydropower-generator power system (SHG-PS) with

enhanced dead zone, the switching-type oscillations

occur when the system with no equilibrium point or

with a stable equilibrium point (SEP)

(Xue et al., 2021).

On the other hand, in addition to increasing system

damping, changing the size of dead zone can also

suppress the switching-type oscillations. For example,

Xiong, H., Zhang, B., Lou, B., Liu, X., Hou, S., Wang, J. and Xue, A.

A Control Method for Switching-Type Frequency Oscillations in Hydropower System Based on Adjusting Dead Zone Structure.

DOI: 10.5220/0012004400003612

In Proceedings of the 3rd International Symposium on Automation, Information and Computing (ISAIC 2022), pages 633-638

ISBN: 978-989-758-622-4; ISSN: 2975-9463

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

633

literature (Xue et al., 2021) points out that increasing

the size of dead zone of hydropower generators can

keep the system from non-smooth bifurcation and

avoid the switching-type oscillations. Although setting

a large dead zone of governor is beneficial to the

system stability, it also wastes part of frequency

regulation capability at the same time.

In summary, for the control of switching-type

frequency oscillations, increasing system damping

may be ineffective and adjusting the dead zone size

has the disadvantage of wasting the frequency

regulation capability. In view of this, aiming at the

switching-type oscillation cannot be suppressed by

increasing system damping in the SHG-PS with

enhanced dead zone, this paper proposes a control

method based on adjusting the dead zone structure,

and verifies the control effect in the SHG-PS and 2-

area-4-generators systems.

2 SYSTEM MODEL AND

OSCILLATION PHENOMENA

This section presents the system model of the

simplified single-hydropower-generator power system

(SHG-PS) with enhanced dead zone, together with its

switching-type oscillation phenomena.

2.1 System Model

The system model studied in this paper is a simplified

SHG-PS with enhanced dead zone. Its corresponding

mathematical description is as follows (Xue et al.,

2021):

14prefP41I

2P412P1y

323P1WP412y W

43L4ref LJ

[( ) ( ( ) )]

(() ) /

[(())/]2/

[( )]/

xFx bY KFx xK

xKFxxxKT

xxKTKFx xxT T

xxKx PT

=+− −⋅





=+−⋅





=−− +− ⋅



=− − −





(1)

Where x

represents the angular frequency ω, and

the remaining parameters are given in the literature

(Xue and Wang, 2020). The values of the relevant

variables, if not otherwise specified, are the

standardized values. F(x

) is the description of the

enhanced dead zone, as shown in Figure 1.

−

area

area Barea C

()Fx

ref 4

−

Figure 1: The model of enhanced dead zone.

The mathematical description is:

(2)

Figure 1 and equation (2) show that the SHG-PS is

divided into three areas by the dead zone switching

manifolds, i.e., Σ

={x

=ω

ref

-ε} and

={x

=ω

ref

+ε}. The area A/C corresponds to the

regulation area and area B corresponds to the dead

zone area. Due to the step control logic at the

switching manifolds, the system vector field is

discontinuous, thus, the SHG-PS with enhanced dead

zone is a Filippov non-smooth system (Simpson,

2018).

2.2 Oscillation Phenomena

The SHG-PS with enhanced dead zone may occur

ULFOs under sudden load perturbations (Xue et al.,

2021).

For example, when the post-disturbance load P

=1.02, the system has no equilibrium point and an

ULFO of 0.084 Hz occurs, as shown in Figure 2.

0 100 200 300 400

t/s

-4

-2

Frequency

deviation/(×10

)

switching manifold b

area A

area B

area C

switching manifold b

Figure 2: The frequency oscillation when P

=1.02.

Figure 2 shows that the oscillation trajectory will

cross the switching manifold Σ

and change in the

area B and area C, corresponding to the switching-type

non-smooth oscillation.

ref 4

ref 4 ref 4

0||

()

ωε

ωωε

−<





−−>



ISAIC 2022 - International Symposium on Automation, Information and Computing

634

3 THE CASE OF OSCILLATION

CANNOT BE SUPPRESSED BY

INCREASING DAMPING

This section represents the case that increasing system

damping cannot suppress the switching-type

oscillation and analyzes the reasons.

Increasing system damping is generally considered

as a method to suppress oscillation. For the SHG-PS

of Section 2.1, the PI control parameters, i.e., K

and

, can be adjusted to increase the mechanical

damping coefficient D

(Liu et al., 2016).

In the case of Section 2.2, the mechanical damping

coefficient D

under different parameter groups are

shown in Table 1.

Table 1: The damping under different parameters.

initial parameters 5 1 -2.97

parameter group 1 3 1 -2.81

parameter group 2 1 0.25 -0.78

Table 1 shows that, after adjusting PI parameters,

the system damping is improved. Under the load

disturbance of Section 2.2, the frequency changes of

systems with different PI parameters are shown in

Figure 3.

0 50 100 150 200 250 300 350 400

t /s

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1.001

1.002

1.003

=5,K

=3,K

=1 K

=1,K

=0.25

Frequency(p.u.)

Figure 3: The frequency changes under different PI.

Figure 3 shows that after increasing the damping,

the system still occur the switching-type frequency

oscillation, that is, the switching-type oscillation of

SHG-PS with enhanced dead zone cannot be

effectively suppressed by increasing damping.

It is worth noting that, although the system

damping is improved, the step characteristic at the

switching manifold of SHG-PS has not changed.

Under a certain load disturbance, the system still has

no equilibrium point, and the switching-type

oscillation will still occur. Therefore, the switching-

type oscillation of SHG-PS with enhanced dead zone

(Filippov non-smooth system) may not be suppressed

by increasing damping.

4 CONTINUOUS ENHANCED

DEAD ZONE

This section introduces a construction method to make

the enhanced dead zone continuous at the switching

manifold, so that the system has a SEP under certain

load disturbances.

To make the system always has an equilibrium

point, the dead zone characteristic should be

continuous. On the other hand, to make the governor

adjust quickly, the step characteristic shall be retained

at the switching manifold. The continuous function f(x)

simulating step characteristics, as shown in equation

(3), can meet these requirements.

()

xac

−

(3)

where a, b, c, and k are the constants, determining

the interval and degree of the step.

With the function f(x), the enhanced dead zone can

be continuous, as shown in Figure 4.

Figure 4: The continuous enhanced dead zone.

The mathematical description is：

A Control Method for Switching-Type Frequency Oscillations in Hydropower System Based on Adjusting Dead Zone Structure

635

()

ref

ref ref

ωωε

ωω ε

ωωε

εω ω

ωω ωωε

−−

−

−−

−



<−<



−−<−<





−−>



(4)

The SHG-PS with continuous enhanced dead zone

is no longer a Filippov non-smooth system with

discontinuous vector fields, but a piecewise-smooth

continuous system (a type of the non-smooth system)

(Simpson, 2018). That is, the type of non-smooth

systems is changed. The continuous system includes

area a, area b1, area b2 and area c.

5 SYSTEM BIFURCATION

CHARACTERISTICS BEFORE

AND AFTER CONTINUITY

This section presents the non-smooth bifurcation

characteristics of the equilibrium point and the post

fault dynamics of the SHG-PS before/after the

enhanced dead zone continued.

Considering typical system parameters, the pre-

fault system load is rated 1 and the system operates at

the SEP, i.e., (x

, x

) = (1, 1, 1, 1). Considering

the sudden load change (i.e., the load parameter P

takes different values after the perturbation), the non-

smooth bifurcation characteristics of the equilibrium

point and the post fault dynamics of the SHG-PS

before the enhanced dead zone continued can be

obtained, as shown in Figure 5 (Xue et al., 2021).

Figure 5:The non-smooth bifurcation characteristics before

continuity.

Figure 5 shows that with the change of the load

parameters, the system undergoes a Hopf-like non-

smooth bifurcation, manifesting that the equilibrium

point disappears and switching-type oscillation occurs

at the same time. Furthermore, P

and P

are the

bifurcation points (P

and P

correspond to another

type of bifurcation (Xue et al., 2021), which is not

considered in this paper).

Considering the continuous enhanced dead zone

parameters taken as: b=0.01, c=70, k=2, and ε=0.002,

respectively, and under the same simulation conditions,

the Hopf-like non-smooth bifurcation characteristics

of the SHG-PS after the enhanced dead zone continued

can be obtained, as shown in Figure 6.

Figure 6:The non-smooth bifurcation characteristics after

continuity.

Compared with Figure 5, Figure 6 shows that：

1) The range of disturbance load parameters that

may induce switching-type oscillation is decreased

after the enhanced dead zone is continuous, which is

beneficial to the system stability. For example, the

range 1.0015≤P

≤1.16 can be reduced to

1.007≤P

≤1.12, specifically.

2) After the enhanced dead zone is continuous, the

SHG-PS always has a SEP, and the switching-type

oscillation is no longer related to the system without

equilibrium point. That is, the Hopf-like non-smooth

bifurcation with the equilibrium point disappearing

does not occur, thus avoiding the switching-type

oscillations.

6 CONTROLLER

VERIFICATION

This section verifies the control effect of continuous

enhanced dead zone in SHG-PS and 2-area-4-

generators system, respectively.

6.1 Effect in SHG-PS

After the enhanced dead zone continuity, there is

always a SEP. In addition, optimizing PI control

parameters to increase damping can further increase

the global stability of system.

Taking the load perturbation parameter P

=1.02 as

an example, the dynamic characteristics of the SHG-

PS with enhanced dead zone and with continuous

enhanced dead zone under different PI parameters can

be obtained, as shown in Figure 7.

ISAIC 2022 - International Symposium on Automation, Information and Computing

636

Figure 7: Control effect in SHG-PS.

Figure 7 shows that:

1) With the initial parameters, i.e., K

=5 and K

=1,

the oscillation amplitude will be somewhat reduced

after the enhanced dead zone continuity. The possible

reason is that after the enhanced dead zone continuity,

the governor has begun to participate in the regulation

when the frequency deviation gradually approaches

the set value of dead zone.

2) With the optimized parameters, i.e., K

=1 and

=0.25, the frequency of the SHG-PS with

continuous enhanced dead zone, will return to stability.

While the SHG-PS with enhanced dead zone still

occur frequency oscillation, as shown in Figure 3. The

switching-type frequency oscillation of the SHG-PS is

effectively suppressed by adjusting dead zone

structure and optimizing PI control parameters.

6.2 Effect in 2-area-4-generators System

In this subsection, a 2-area-4-generators system, as

shown in Figure 8, is used to verify the control effect.

Figure 8: The 2-area-4-generators system.

The 2-area-4-generators system is set as follows:

four generators are all hydropower units, where the

dead zone of G1 governor is enhanced or continuous

enhanced dead zone, and the dead zones of remaining

three generators are traditional dead zone.

The main system parameters are set as follows:

dead zone

0.05Hz

; water hammer effect

coefficient T

=0.3; PI parameters K

=5 and K

=2.5.

Two fault conditions are set as follows：

1) Small disturbance: 5MW load is reduced at bus

2) Large disturbance: 100MW output is reduced in

generator G1.

Under large/small disturbances, the frequency

characteristics of the system with enhanced dead zone

or continuous enhanced dead zone, are shown in

Figure 9 and Figure 10, respectively.

Figure 9: The system frequency under small disturbance.

Figure 10: The system frequency under large disturbance.

Figure 9 and Figure 10 show that:

1) When the dead zone of G1 governor is enhanced

dead zone, the system will occur the oscillation with

oscillation frequency of 0.09Hz under small

disturbance. While the dead zone is replaced with a

continuous enhanced dead zone, the system frequency

gradually stabilizes and the oscillation disappears.

2) Under the large disturbance, the 2-area-4-

generators system will occur ULFO, no matter the

enhanced dead zone is discontinuous or continuous.

However, the system with continuous enhanced dead

zone has smaller oscillation amplitude, which may

reduce the degree of harm to the system.

Therefore, under large/small disturbances, the

A Control Method for Switching-Type Frequency Oscillations in Hydropower System Based on Adjusting Dead Zone Structure

637

governor with continuous enhanced dead zone can

effectively suppress the oscillation, and the system

stability is better.

ACKNOWLEDGMENTS

This work is supported by the project of "Research on

the optimization technology of coordinated control of

differentiated grid sources based on transient response

under the background of large power reception" by the

State Grid Zhejiang Province Electric Power

Company Science and Technology

(No.5211DS200086).

7 CONCLUSIONS

Aiming at the ST-ULFO in hydropower systems with

enhanced dead zone, this paper provides a control

method based on adjusting the non-smooth structure

of the enhanced dead zone and verifies its

effectiveness. The conclusions are as follows:

a) SHG-PS with enhanced dead zone is a Filippov

non-smooth system. It may have no equilibrium point

and occur the switching-type oscillation under certain

load disturbance. Only increasing the system damping

cannot suppress the switching-type oscillation.

b) The continuous function simulating step

characteristics can make the enhanced dead zone

continuous. The corresponding continuous system

maintains the rapidity of governor action, and there is

no longer a Hopf-like non-smooth bifurcation with the

equilibrium point disappearing.

c) The continuous enhanced dead zone can

effectively suppress frequency oscillation, that is,

adjusting the dead zone non-smooth structure can

suppress the switching-type oscillation.

It is worth noting that, the continuous enhanced

dead zone is an ideal model that has not yet been put

into use in real systems. To be practical, its structural

parameters still need to be optimized and analyzed.

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