International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1703
ISSN 2229-5518
Enhancement of SMIB Performance Using
Robust Controller
N.Siva Mallikarjuna Rao#1, B.V.Haritha*2
GITAM UNIVERSITY,HYDERABAD nsm.shiva@gmail.com harithabv91@gmail.com
—————————— ——————————
I. INTRODUCTION
The power system is a highly non linear system whose dynamic performance is influenced by a wide array of devices with different response rate and characteristics. The tendency of the system to develop restoring forces equal to or greater than the disturbing forces is known as stability. The stability problem is concerned with the behaviour of synchronous machines after a disturbance. Small disturbance stability is related to the ability of controlling signals such as voltage, rotor angle and speed etc., following small perturbations such as incremental changes in the system load. This form of
Under normal operating conditions, the relative positions of the rotor axis and the resultant magnetic field axis is fixed. The angle between the two is known as toque or power angle. When a disturbance occurs, the rotor will accelerate or decelerate with respect to rotating air gap MMF, and motion begins. The equation describing this motion is known as swing equation. After the oscillatory period, if the rotors locks back into synchronous speed, the generator will remain in stable state. If the disturbance is due to change in generation, load or any network conditions, the rotor goes to a new operating power angle relative to synchronously revolving field. The swing equation is represented as
stability is determined by the characteristics of load,
continuous controls and discrete controls at a given instant of time. Static analysis is done to determine the stability
𝐻
180 𝑓0
𝑑2 𝛿
𝑑𝑡2 = 𝑃𝑚 − 𝑃𝑒---- (1)
margins and to identify the factors influencing factors.
SINGLE MACHINE INFINITE BUS SYSTEM
Synchronous generators form the principal source of electrical energy in power systems. Many large loads are driven by synchronous motors. Power system stability deals with keeping of interconnected synchronous machines in synchronism. Therefore, the characteristics and accurate modelling of the system is very important for stability studies. A mathematical model is developed for the review of steady state and transient performance characteristics. Any disturbance, may be small or big leads to oscillation for which rotor angle stability analysis is significant and the behaviour of rotor dynamics is analysed using swing equation.
Where 𝛿 is expressed in electrical degrees and H as per unit
inertia constant, Pm and Pe are per unit mechanical and
electrical power respectively.
II. SMALL SIGNAL STABILITY OF SMIB
A general system configuration of SMIB connected to a large system through transmission lines is shown in Fig 1..
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𝑝𝜑𝑓𝑑 =
𝜔0𝑅𝑓𝑑
𝐿𝑎𝑑𝑢
𝐸𝑓𝑑 − 𝜔0𝑅𝑓𝑑𝑖𝑓𝑑 ---(3)
Where Efd is field voltage and Ladu is the mutual flux
linkage.
The state space form including the change in flux is given by
∆𝜔̇ 𝑟
� ∆𝛿̇ � =
̇
𝑓𝑑
𝑎11 𝑎12 𝑎13
∆𝜔𝑟
𝑏11 0
∆𝑇
�𝑎21 0 0 � � ∆𝛿
� + � 0 0 � �
𝑚 �—(4)
0 𝑎32 𝑎33
Where
−𝐾𝐷
∆𝜑𝑓𝑑
0 𝑏32
∆𝐸𝑓𝑑
𝑎11 =
--(5)
2𝐻
−𝐾1
Analysis of systems having such simple configurations is
useful to understand basic effects and concepts. In order to deal with large complex systems an equivalent transmission network is used which is known as single line representation.
𝑎12 = 2𝐻 --(6)
𝑎 = −𝐾2--(7)
2𝐻
𝑎21 = 𝜔0 = 2𝜋𝑓0 --(8)
−𝜔0𝑅𝑓𝑑
Developing the expressions for the elements of state matrix as
𝑎32 =
𝐿𝑓𝑑
𝑚1 𝐿𝑎𝑑𝑠 --(9)
explicit functions of system parameters is useful in
−𝜔0𝑅𝑓𝑑 1𝑎𝑑𝑠 1
developing the mathematical model of the system under
33 𝐿𝑓𝑑
+ 𝑚2 𝐿 𝑎𝑑𝑠 ]--(10)
𝐿𝑓𝑑
small signal disturbances.
𝑚1
𝐸𝐵(𝑋𝑇𝑞𝑠𝑖𝑛𝛿0−𝑅𝑇𝑐𝑜𝑠𝛿0 )
= ---(11)
𝐷
𝑚2 = 𝑎𝑑𝑠 ---(12)
The generator represented by the classical model is shown in
𝑋𝑇𝑞
𝐷
1
𝐿
𝐿𝑎𝑑𝑠+𝐿𝑓𝑑
Fig 2. Here E\1 is the voltage behind Xd 1 .Its magnitude is
𝑏11 =
2𝐻
--(13)
assumed to remain constant at pre-disturbance value .Let 𝛿 be
the angle by which E1 leads the infinite bus voltage EB . As the
rotor oscillates during a disturbance, 𝛿 changes.
𝑏32 =
𝜔0𝑅𝑓𝑑--(14)
𝐿𝑎𝑑𝑢
The input control signal to the excitation system is normally
the generator terminal voltage Er . In the classical generator model, Et is not a state variable. Here, Et is expressed in the terms of other state variables. The state space representation including the excitation system signal perturbations is given by
∆𝜔̇ 𝑟
⎛ ∆𝛿̇
𝑎11 𝑎12 𝑎13 0
⎞ = � 0 0 0 � �
∆𝜔𝑟
∆𝛿
� + �
𝑏1
� ∆𝑇
⎜∆𝜑𝑓𝑑 ⎟
⎝ ∆𝑣1̇ ⎠
Where ,
−𝜔0𝑅𝑓𝑑
𝑎21
0 𝑎32 𝑎33 𝑎34
0 𝑎42 𝑎43 𝑎44
− (15)
∆𝜑𝑓𝑑
∆𝑣1
0
0 𝑚
0
When the speed deviation ∆𝜔𝑟 and rotor angle deviation ∆𝛿
are considered the state space representation is
𝑎34 =
𝐿𝑎𝑑𝑢
𝐾5
𝐾𝐴 --(16)
𝑑 ∆𝜔̇ 𝑟
−𝐾𝐷
− 𝐾𝑆
∆𝜔𝑟
1
𝑎42 = --(17)
𝑅
𝐾
�
𝑑𝑡
∆𝛿̇
� = � 2𝐻
𝜔
2𝐻 � �
0
∆𝛿
� + �2𝐻� ∆𝑇𝑚 − (2)
0
𝑎43 =
6
𝑇𝑅
--(18)
0
Where KD is damping torque coefficient, KS is synchronizing
𝑎44 =
−1
𝑇𝑅
--(19)
torque coefficient and H is the inertia constant.
The field circuit dynamic equation is given by
Where KA is the exciter gain and TR is the terminal voltage
transducer time constant.
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ISSN 2229-5518
For the equations shown above , it can be induced that , with
∆𝜔̇ 𝑟
𝑎11
𝑎12
𝑎13
0 0 0
∆𝜔𝑟
positive K5 the effect of excitation will introduce a negative
⎛ ∆𝛿̇ ⎞
𝑎21
0 0 0 0 0
∆𝛿
synchronising torque and positive damping torque component.
⎜∆𝜑𝑓𝑑 ⎟
⎛ 0 𝑎 𝑎 𝑎
0 𝑎
⎞ ⎛ ⎞
∆𝜑𝑓𝑑
With negative K5 positive synchronising and negative ⎜
⎟ = ⎜
32 33 34
36 ⎟ ⎜ ⎟
damping torque component is introduced for which higher
⎜ ∆𝑣̇ 1 ⎟
⎜ 0 𝑎42 𝑎43 𝑎44 0 0 ⎟ ⎜ ∆𝑣1 ⎟
response exciter is beneficial in increasing synchronising
∆𝑣̇ 2
𝑎51 𝑎52 𝑎53 0 𝑎55 0
∆𝑣2
torque. But this results in negative damping. An effective way
to meet the required exciter performance with regard to system stability is by introducing power system stabilizer.
⎝ ∆𝑣̇ 𝑠 ⎠
⎝𝑎61 𝑎62 𝑎63 0 𝑎65 𝑎66 ⎠ ⎝ ∆𝑣𝑠 ⎠
− (20)
Where,
𝑎51 = 𝐾𝑠𝑡𝑎𝑏 𝑎11 --(21)
𝑎52 = 𝐾𝑠𝑡𝑎𝑏 𝑎12--(22)
𝑎53 = 𝐾𝑠𝑡𝑎𝑏 𝑎13 --(23)
−1
𝑎55 =
𝑊
--------(24)
IV.POWER SYSTEM STABILIZER
The basic function of a power system stabilizer is to add
damping to the generator rotor oscillations by controlling its
𝑎 = 𝑇1 𝑎 ----(25)
61 𝑇2 51
𝑎 = 𝑇1 𝑎 ----(26)
62 𝑇2 52
𝑇1
excitation using auxiliary stabilizing signal. To provide
damping, the stabilizer must produce a component of
𝑎63 =
𝑇2
𝑇1
𝑎53--(27)
1
electrical torque in phase with rotor speed deviations.
𝑎65 = 𝑇 𝑎55 + 𝑇 --(28)
The PSS consists of 3 blocks i) Phase compensation 2)a
signal washout signal 3)a gain block.
𝑎66 =
2
−1
𝑇2
2
---------(29)
V.ROBUST CONTROLLER
The phase compensation block provides the appropriate phase-lead characteristic to compensate for the phase lag between the exciter input and the generator electrical torque. The signal washout block serves as a high pass filter, with the time constant TW high enough to allow signals associated with oscillations in wr to pass unchanged. The stabilizer gain Kstab determines the amount of damping introduced by the stabilizer. The state space form including power system stabilizer is given by
Power system stabilizer is pertained to certain network configurations. If the configurations change, the system no longer operates in the stability range. In order to overcome this problem, advanced controlling techniques have been developed among which robust stabilisation is one of the modern control techniques.
Robust stability is the minimum requirement of any practical control system. Robust control theory was first developed by Zames and it addresses both the performance and the stability criterion of a control system.
H infinity controlling technique is used for robust stability of the system. Robust H∞ controllers are developed to provide high robust control environment to linear systems.
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Generally the H∞ norm of a transfer function F, is the maximum value over the complete spectrum and is represented as
‖𝐹(𝑗𝑤)‖∞ = 𝑠𝑢𝑝𝜎 (𝐹(𝑗𝑤))-(30). Here σ is the singular value
of the transfer function. In H∞ controller synthesis, two
transfer functions are used which split complex control
problem into separate sections one dealing with stability and the other with performance. The sensitivity function S, and complementary sensitivity function T are used in controller synthesis and is given by
𝑆 = 1
1+𝐺𝐾
--(31)
𝑇 = 𝐺𝐾 ---(32)
1+𝐺𝐾
Considerable advances have been made in the H∞ control synthesis since it’s inception. One can find number of theoretical advantages of this method such as high disturbance rejection , high stability and many more. Mixed weight H∞ controllers provide a closed loop response of the system according to the design specifications such as model uncertainty, disturbance attenuation at higher frequencies, required bandwidth of the closed loop plant etc. Practically, H∞ controllers are of high order which, may lead to large control effort requirement. Moreover, the design may also depend on specific system and can require its specific analysis. When H∞-optimal control approach is applied to a plant, additional frequency dependent weights are incorporated in the plant and are selected to show particular stability and performance specifications relevant to the design objective as defined.
Various techniques are available in literature for the design of H∞ controller and H∞ loop shaping is one of the widely accepted among them as the performance requirements can be embedded in the design stage as performance weights. H∞ based robust control deals with the characteristics such as amplifiers delay or sensors offset .Considering G (s) and K(s) as the open loop transfer function of the plant and controller transfer function respectively, robustness and good performance of closed loop system. Controller K(s) can be derived, provided it follows three criterions, which are:
1) Stability criterion: If the roots of characteristic equation 1+G(s)K(s)=0 are in left half side of s plane, then stability is ensured.
2)Performance criterion : It establishes that the sensitivity
Where G and K are transfer function and controller transfer functions respectively.
From Fig 4, shown above, w is the vector of disturbance signals ,z is the signal positioning of all errors. v is the vector of measurement variables and u is the vector of all control variables. Here the disturbance signal is considered as Vref . Controller is designed by minimising the norm
𝑚𝑖𝑛𝑘 = ‖𝑁(𝑘)‖----(33)
𝑁 = � 𝑤𝑠 𝑆 �-----(34)
𝑤𝑇 𝑇
Where Ws and Wt are the weight functions assigned to the
problem by the designer. The ultimate objective of the robust
control is to minimize the effect of disturbance on output. The sensitivity S and the complementary function T are to be reduced . To achieve this it is enough to minimize the magnitude of |S| and |T| which is done by making
|S(jw)|<1/Ws (jw) and |T(jw)|<1/Wt (jw) . Ws is the weighting function to limit magnitude of sensitivity function.
Wt is the robustness weighting function to limit magnitude of complementary sensitivity function .This technique is called
as loop shaping technique and is widely used for synthesis of the controller .The shaping objective is to make the output
𝑦 = ∆𝜔 (Generator frequency variation).
Thus, a stabilising controller k(s) is achieved by minimising
cost function ɣ. To obtain the desired frequency response for
the plant, loop shaping is employed with the weight functions. There are various methods for loop shaping. The parameters of the weight functions are to be varied so as to get the frequency response of the whole system within desired limits. The block diagram in Fig. 5 describes the mixed Sensitivity problem.
𝑠= 1
- is small for all frequencies where disturbances
1+𝐺(𝑆)𝐾(𝑠)
and set point changes are large.
3)Robustness criterion : It states that stability and performance
should be maintained not only for the nominal model but also for a set of neighbouring plant models that result from unavoidable presence of modelling errors. Robust controllers are designed to ensure high robustness of linear systems.
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The generalised plant P(s) is given by
𝑊𝑠 𝑆
𝑚𝑖𝑛‖𝑃‖ = 𝑚𝑖𝑛 �𝑊𝑘𝑠 𝐾𝑆� = 𝛾—(42)
𝑊𝑡 𝑇
Where P is the transfer function w to z. i.e., |Tzw|=ɣ.
Applying minimum gain theorem, we can make the H∞ norm
of |Tzw| | less than unity .
𝑊𝑠 𝑆
𝑚𝑖𝑛‖𝑇𝑧𝑤 ‖ = 𝑚𝑖𝑛 �𝑊𝑘𝑠 𝐾𝑆� < 1—(43)
𝑊𝑡 𝑇
The weights Ws, Wks and Wt are the tuning parameters and it
requires some iterations to obtain weights that yields a good
controller . The starting point of weights is given as
𝑠 +𝜔0
𝑊𝑠 = 𝑀 --(44)
𝑠+𝜔0𝐴
Wks =constant
𝑠 𝜔0
𝑊𝑡 = 𝑀 ---(45)
𝐴𝑠+𝜔0
𝑍1
𝑊𝑠 −𝑊𝑆 𝐺
𝑤
�𝑍2 � = � 0 𝑊𝑘𝑠 � �
�--(35)
Where A is the allowed steady state offset,ω0 is the desired
𝑍3
𝑒
0 𝑊𝑡 𝐺 𝑢
𝐼 −𝐺
bandwidth and M is the sensitivity peak. Typically, A=0.01
and M=2.
Considering the following state space realizations
𝐺 𝑠 = �𝐴 𝐵�--(36)
𝐶 𝐷
𝐴𝑠 𝐵𝑠
A. Algorithm to synthesize Robust controller:
𝑊𝑠 𝑠 = �
𝐶𝑠 𝐷𝑠
�--(37)
1 .Defining the subsystems: In this the open loop transfer
function (G) of the system is defined .
𝑊𝑘𝑠 𝑠 = �
𝐴𝑘𝑠 𝐵𝑘𝑠
𝐶𝑘𝑠 𝐷𝑘𝑠
�--(38)
2 .Weights are to be selected.
3. Creating the generalised plant.
𝑊𝑡 𝑠 = �
𝐴𝑡 𝐵𝑡
𝐶𝑡 𝐷𝑡
�--(39)
4. Synthesizing the controller.
5.Reducing the order of controller
6.Obataining the controller transfer function of desired phase margin.
The possible state space realization for P(s) is given as
VI. RESULTS
For the SMIB system represented in this paper, P=0.9,Q=0.3
Et =1.0 at 360 and EB =0.995
𝑊𝑠 −𝑊𝑆 𝐺
𝐴 𝐵1 𝐵2
Thyristor exciter K
A=200, T
R =0.02S
𝑃 = � 0 𝑊𝑘𝑠
� = �𝐶1 𝐷11 𝐷12 �--(40)
PSS added is K
=9.4,T
=1.4s,T =0.09s,T =0.3s
stab W 1 2
0 𝑊𝑡 𝐺
𝐼 −𝐺
𝐶2 𝐷21 𝐷22
The constants calculated from the equations are given by
From the equations 35 &40, a mixed sensitivity problem is written as
𝑊𝑠 𝑆
𝑃 = �𝑊𝑘𝑠 𝐾𝑆�--(41)
𝑊𝑡 𝑇
In case of mixed sensitivity problem our objective is to find a
rational function controller K(s) and to make the closed loop system stable satisfying the following expression
K1 =0.7643 K2 =0.86 K3 =0.32 K4 =1.41 K5 = -0.14
K6 =0.416
T3 =2.365 TR =0.02 KA=200. The transfer function of the
lower order robust controller obtained is given by
𝑠8 + 1884𝑠7 + 8.9𝑒05𝑠6 + 2.578𝑒06𝑠5 + 2.494𝑒06𝑠4 + 1.064𝑒06𝑠3
𝐾 = +2.05𝑒05𝑠2 + 1.522𝑒04𝑠 + 132.7
𝑠9 + 1884𝑠8 + 8.9𝑒05𝑠7 + 2.554𝑒06𝑠6 + 2.503𝑒06𝑠5 + 1.069𝑒06𝑠4
+2.054𝑒05𝑠3 + 1.457𝑒04𝑠2 − 2.666𝑠 − 1.2
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And the reduced form of controller by increasing the desired phase margin is given by
66.3𝑆+148
--(46)
𝑆+148
Fig 6:Characteristic of Rotor angle using power system stabilizer
Fig 7: characteristic of Rotor angle using H∞ controller
VII CONCLUSION
The output characteristic using power system stabilizer has settling time of 5.5sec and peak overshoot of 1.5. For the output characteristic using H∞ controller settling time is 4sec and the peak overshoot is 1.2 .
From the results, it is observed that both the settling time and the peak overshoot has reduced for the controller designed using H∞ loop shaping technique compared with conventional power system stabilizer characteristics.
.
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