analysis of coupling dynamics for power systems with...

Analysis of Coupling Dynamics for Power Systems with Iterative DiscreteDecision Making Architectures

Zhixin Miao∗

∗Department of Electrical Engineering, University of South Florida, Tampa FL USA 33620.

Email: [email protected].

Abstract

Iterative “learning” by distributed control agents has been proposed for power system decision making.

Such decision making can achieve agreement among control agents while preserving privacy. The iterative

decision making process may interact with power system dynamics. In such cases, coupled dynamics are

expected. The objective of this paper is to propose a modeling approach that can conduct stability analysis

for these hybrid systems. In the proposed approach, the discrete decision making process is approximated

by continuous dynamics. As a result, the entire hybrid system can be represented by a continuous dynamic

system. Conventional stability analysis tools are then used to check system stability and identify key im-

pacting factors. An example power system with multiple control agents is used to demonstrate the proposed

modeling and analysis. The analysis results are then validated by nonlinear time-domain simulation.

The continuous dynamics models developed in this paper sheds insights into the control nature of each

distributed optimization algorithm. An important finding is documented in this paper: A consensus algo-

rithm based decision making may act as an integrator of frequency deviation. It can bring the frequency

back to nominal while the primal-dual based decision making cannot.

Keywords: Distributed optimization; frequency control; dynamic stability; hybrid system

1. Introduction

Introduction of numerous smart buildings, distributed energy sources and energy storages poses chal-

lenges in operation and control. A centralized control center may over burden its SCADA system and

computing machines to collect every piece of measurements and calibrate optimal operation schemes. On

the other hand, due to privacy concerns, communities are not willing to share all information. Therefore,

instead of one centralized control center, multiple control agents will handle the decision making process

while exchanging limited information. Agreement among agents is achieved through an iterative “learning”

Preprint submitted to Electric Power Systems Research March 22, 2016

process. These privacy-preserving decision making architectures for microgrids and power systems have been

proposed in the literature [1, 2, 3].

The mathematic foundation of the privacy-preserving decision making is distributed optimization [4].

For example, in dual decomposition, iterative updating of the Lagrangian variable translates in iterative

“learning” process among control agents. These iterative learning processes take place in a much faster

speed than the traditional hourly economic dispatch process. In addition, in many cases, feedback loop is

introduced in discrete decision making. For example, in [1, 5], a frequency deviation signal is fed into price

or power command update to reflect power imbanance.

Therefore, it is reasonable to suggest that the dynamics of the decision making may be coupled with the

power system dynamics.

To the authors’ best knowledge, there has been little research around that investigates dynamic stability

for such hybrid systems. It is the objective of this paper to provide a modeling approach to consider a

hybrid system as a whole and conduct dynamic stability analysis.

At least two approaches have been documented in the literature to tackle dynamic analysis of hybrid

systems. The first one is sampled data modeling approach. The discrete process is modeled in difference

equation to describe the relation between a discrete decision variable at (k + 1)-th step and that variable

at k-th step. Between the discrete sampling period T , the continuous system dynamics is integrated for a

period. That way, the continuous dynamics can be represented by difference equations as well. The entire

system is now represented by a discrete process and its stability can be judged based on the discrete system.

This approach has been seen in the literature, e.g., [6]. In power systems, discrete state-space models for

thyristor series controlled capacitor were developed using sampled data approach [7, 8].

The second approach is to approximate the discrete process as continuous dynamics. This approach has

been adopted in [5] to describe power market dynamics.

Power system dynamics is nonlinear and complex. Integrating power system dynamics over a period is

a daunting task. Therefore, we opt to the second approach to approximate the discrete decision making

process.

The rest of the paper is organized as follows. Section II describes the discrete decision making process and

the representation by continuous dynamic. Section III describes the test system, power system dynamics

and the continuous dynamic system representation of the entire hybrid system. Stability analysis will

be conducted in Section III. Section IV then presents validation results through time-domain simulation.

Section V concludes this paper.

2

The contributions of this paper include the following aspects.

• This paper proposes a straightforward modeling approach for hybrid system dynamic stability analysis.

Though straightforward, this modeling approach sheds insights into the hybrid system dynamics.

• Compared to the research work on coupled market and power system dynamics in [5], our research

work includes not only analysis but also validation through time-domain simulation. The validation

confirms the practical value of the proposed modeling approach.

2. Decision Making Process and Its Continuous Dynamic Model

In this section, we discuss two types of iterative-based distributed decision making procedures and their

corresponding dynamic models. The first type is based on primal dual decomposition [3]. The second type

is based on consensus algorithm and subgradient update [9].

2.1. Type 1 Primal-dual decomposition based decision making

Consider a two-area power system economic dispatch problem.

Prob1 minimize f1(P1) + f2(P2) (1a)

subject to λ1 : P1 − P12 = D1 (1b)

λ2 : P2 + P12 = D2 (1c)

−d ≤ P12 ≤ d (1d)

where Pi notates output active power from Area i, fi(Pi) is the cost related to power generation, P12 is

the tie-line flow from Area 1 to Area 2, λi notates the dual variable related to the power balance equality

constraint in Area i, Di notates the load power in Area i, and d is assumed to be the line limit.

The partial Lagrangian function of Prob1 with the two power balanced equality constraints relaxed is as

follows.

L(P1, P2, P12, λ1, λ2) = f1(P1) + λ1(D1 − P1 + P12)

+ f2(P2) + λ2(D2 − P2 − P12) (2)

Given the two price signals, can the each area determine its generation P1 and import P12? This question

will be examined by looking at Area 1. For a given λ1, for this objective function f1(P1)+λ1(D1−P1 +P12),

3

if P12 has no limit imposed, there is no solution unless λ1 = 0. The objective function can go −∞. Therefore,

this problem is considered not feasible.

We would like to treat P12 differently than P1 and P2 to help the problem solving. If we treat P12 as

given, similar as λ1 and λ2 are given, then the dual problem becomes:

d1(λ1, P12) = minP1

{f1(P1) + λ1(D1 − P1 + P12)} (3)

d2(λ2, P12) = minP2

{f2(P2) + λ2(D2 − P2 − P12)} (4)

The above two problems should have solutions.

We now consider the dual problem Prob2 with P12 given.

Prob2 maximize d1(λ1, P12) + d2(λ2, P12) (5a)

over λ1, λ2 (5b)

For this problem, we end up having a solution dependent on P12. Let us notate this solution of Prob2

as a function of P12: g(P12). g(P12) is a dual problem of the dual problem. Since the dual problem is a

concave function over λi, the dual’s dual problem should be a minimization problem over P12.

Therefore, the dual’s dual problem or the primal-dual problem Prob3 can be written as follows.

Prob3 minP12

maxλ1

(minP1

{f1(P1)− λ1P1}+ λ1

(D1 + P12

))+ max

λ2

(minP2

{f2(P2)− λ2P2}+ λ2(D2 − P12)

)(6a)

We can decompose a system that is connected by a tie-line by assuming a tie-line power flow. Each area

will consider the tie-line flow injection or exporting as a negative (or positive) load. Each area carries out

optimization and finds the locational marginal price (LMP) for the interfacing bus. The tie-line flow is then

updated based on the price difference.

The dual’s dual problem can be solved by subgradient updating of the primal variableP12. Hereinafter,

we will notate this virtual tie-line flow a different name: π. The subgradient of the line flow is (λ1 − λ2).

Since the primal problem is a minimization problem, therefore, in the update procedure, for a positive

4

gradient, π should be reduced. The updating procedure is presented as follows.

πk+1 = πk − α(λk1 − λk2) (7)

where α > 0. For a given virtual tie-line flow πk, the LMPs can be found by solving individual optimization

problem for each area.

The proposed decision making strategies have the assumption of lossless tie line. Therefore, the power

dispatched by the generators only takes care of loads. The total generation is less than the total consumption

including loads and tie-line power loss. To compensate the frequency deviation, the strategies are modified

to have the price calculation having an additional component that can reflect the power unbalance or energy

unbalance. Indicated in [5], the energy unbalance is proportional to the system’s average frequency deviation.

Therefore, at each step, the LMPs computation become as follows.

λk1 = 2a1(D1 + πk) + b1 −K∆f1 (8)

λk2 = 2a2(D2 − πk) + b2 −K∆f2 (9)

where ai, bi are coefficients of a generator quadratic cost function (fi(Pi) = aiP2i + biPi + ci), ∆fi is the

frequency deviation measurement at Area i, and K is a positive gain.

If the system’s frequency is below the nominal frequency, the prices will be increased. In turn, the

generators will increase their dispatch.

2.1.1. Modeling as continuous dynamics

We now proceed to give an approximate continuous model for the above mentioned iterative procedure.

Assuming that ∆f1 = ∆f2 (this assumption is valid as long as the iterative decision making dynamic is

much slower than the power system frequency dynamics), the iteration of the virtual tie-line flow is

πk+1 = (1− 2α(a1 + a2))πk + π0, (10)

where π0 = −α(2a1D1 − 2a2D2 + b1 − b2).

The power references are determined by the prices. Therefore,

P k1 = (πk + D1)− K

2a1∆fk (11)

5

Considering that the frequency measurement of the previous step is taken in the price calculation, the

k + 1 step power reference is modified as

P k+11 = (πk+1 + D1)− K

2a1∆fk (12)

Substituting πk+1 and πk by P k+11 and P k1 , we find

P k+11 = [1− 2α(a1 + a2)]P k −Kα(a1 + a2)

a1∆fk + P10 (13)

where P0 = 2α(a1 + a2)1.

Using forward Euler method, we can express the derivative at k-step is

P1k ≈ P k+1

1 − P k1τ

. (14)

where τ is the step size.

Therefore, the discrete equation can now be approximated by a continuous dynamic equation.

τP1 = −2α(a1 + a2)P −Kα(a1 + a2)

a1∆f + P10 (15)

To this end, we have derived the continuous dynamic model for the discrete decision making process. In

Laplace domain, the power command can be expressed as

∆P refi =

K

2ai

1

1 + τ ′s∆fi (16)

where τ ′ = τ2α(a1+a2) .

2.2. Type 2 Consensus algorithm and subgradient update based decision making

The second type of iterative based decision making is based on consensus algorithm and subgradient

update. A consensus problem will be identified from the original economic dispatch problem. For the

following two-area system, the original economic dispatch problem is as follows.

min C1(P1) + C2(P2) (17a)

subject to: P1 + P2 = D1 + D2 (17b)

6

where Ci(Pi) is the cost of generation, Pi is the power generation at Area i and Di is the load consumption

at Area i.

The dual problem is described in (18).

maxλ

minP1,P2

C1(P1) + C2(P2) + λ(D1 − P1 + D2 − P2) (18)

The above problem can be converted to a consensus problem by introducing λ1 and λ2 for each area. λ1

should be equal to λ2. Therefore, the optimization problem is converted to a maximization problem with a

consensus constraint.

maxλ1,λ2

minP1,P2

C1(P1) + λ1(D1 − P1) + C2(P2) + λ2(D2 − P2)

s.t. λ1 = λ2 (19)

The consensus algorithm that utilizes a stochastic matrix to conduct weighted averaging only guarantees

consensus of multiple λi. It cannot guarantee that the λ can maximize the dual problem’s objective function.

To guarantee maximization, subgradient update has to be used.

The subgradient of λ is the total power unbalance. This information requires again global information.

Fortunately in power systems, frequency deviation is a measure of power unbalance. Frequency is a

local measurement. Therefore, distributed control can be realized by substituting the subgraident of by the

frequency deviation.

The iterative procedure can be described by the following equations.

λ1

λ2

k+1

= A

λ1

λ2

k

−K

∆f1

∆f2

(20)

where A is a stochastic matrix. For the test two-area system, we select A =

12

12

12

12

.

The decision making again introduces feedback signals of frequencies. For each area, the power command

is related to the Lagrangian multiplier. Ignore the limits of each generator, we can find

λ1 = 2a1P1 + b1 (21)

λ2 = 2a2P2 + b2 (22)

7

The iteration procedure (20) is now expressed in terms of the power commands:

P1

P2

k+1

+

b12a1

b22a2

= A

P1

P2

k

+

b12a1

b22a2

−K

∆f12a1

∆f22a2

(23)

The difference equation is now converted to a continuous dynamic equation.

∆P1

∆P2

= − (τs−A+ I)−1

K2a1

0

0 K2a2

︸︷︷︸

G1(s)

∆f1

∆f2

(24)

The gain matrix G2(s) defines the transfer function matrix from the frequency deviation to the power

commands.

G1(s) =K

2τs(τs+ 1)

2τs+12a1

12a2

12a1

2τs+12a2

(25)

Remarks: Converting discrete decision making process to continuous dynamics sheds light into each

algorithm. Through this study, we have the following important findings.

• Compared to the two dynamics of the decision making algorithms, the consensus one has an integrator

unit. We expect that consensus algorithm based Type-2 decision making can bring the frequency

deviation to zero. The primal-dual algorithm is similar as a first-order filter. Therefore, we do not

expect Type-1 decision making can bring the frequency back to nominal.

3. Test system and power system dynamic model

In Section III, the power system dynamics model and the integrated system model will be described and

analyzed. The test power system is a two-area four-machine system shown in Fig. 1. This system comes

from the classical two-area four-machine power system [10] with the following modification: the tie-line has

been shortened; the inertia constants of the machines are reduced to 2.5 pu to have faster electromechanical

dynamics; the damping coefficients are set to be 1 pu. Generators are modeled as classical generators with

turbine-governor blocks. Primary frequency droops with the regulation constant at 4% are all included.

The underlying power system dynamic model ∆f∆P ref

1is to be found. The two generators in each area

are coherent and therefore will be considered as one generator. The two-area four-machine system is now

8

represented by a two-generator system.The two rotor angles are expressed as:

∆δ1 =1

M1s2 +D1s+ T1(∆Pm1 + T1∆δ2) (26)

∆δ2 =1

M2s2 +D2s+ T2(∆Pm2 + T2∆δ1) (27)

where M,D, T are inertia constants, damping and synchronizing coefficients. T1 = T2. Rearranging the

equations, we have

∆δ1 =(M1s

2 +D1s+ T1)∆Pm1 + T1∆Pm2

(M1s2 +D1s+ T1)(M1s2 +D1s+ T1)− T1T2. (28)

The transfer function matrix G2(s) that defines the relationship from the power command to the speed

deviations due to the power system dynamics is expressed in (30).

∆f1

∆f2

= G2(s)

∆P ref1

∆P ref2

(29)

where

G2(s) =Gtg(s)s

ω0[(M1s2+D1s+T1)(M2s2+D2s+T2)−T1T2]

M1s2 +D1s+ T1 T1

T2 M2s2 +D2s+ T2

(30)

where Gtg is the turbine-governor transfer function representing the relationship from the power order P ref

to the mechanical power Pm.

In addition, the droop control has to be included. Therefore, the diagonal components of G2(s) have to

be modified to include the droop control.

The entire system block diagram is obtained and shown in Fig. 2.

3.0.1. Root loci

To examine the stability of the closed-loop system, the open-loop gain matrix G2G1 will be examined.

G2G1 is a two by two matrix. The root loci of the first row first column element are shown in the following

figures.

Figs. 3 and 4 are the root loci for Type 1 system. It can be shown that droop related poles will go to

the right-half-plane (RHP) when the gain K is increasing. Increasing the step size τ will make the system

9

more stable.

Figs. 5 and 6 are the root loci for Type 2 system. It can be shown that droop related poles will go to

the right-half-plane (RHP) when the gain K is increasing. Increasing the step size τ will make the system

more stable.

Remarks: The analysis conducted in the section shows that the hybrid system could suffer low frequency

oscillation of approximately 0.1-0.2 Hz. Increasing the gain of frequency deviation in the discrete decision

making steps will make the system go unstable.

4. Dynamic simulation results

This section gives dynamic simulation results to validate the claims made in the previous section. The two

types of discrete decision making procedures are implemented in the two-area four-machine power systems

as shown in Fig. 1. Power System Toolbox [11] is selected as the dynamic simulation platform.

The power system and Type 1 decision making architecture are shown in Fig. 1. The discrete decision

making will take place every 2 seconds or every 5 seconds. The power commands from Agent 1 and Agent 2

will be sent to change the turbine-governors’ power reference inputs. Among the two agents, the information

exchanged includes the virtual tie-line power flow and the price signal. Area 1 consists of Gen 1 and Gen 2

and Load 1. Area 2 consists of Gen 3, Gen 4 and Load 2. The two areas are connected through tie-lines.

Initially, the four generators are dispatched at 7.0207 pu, 7.00 pu, 7.16 pu and 7.00 pu. Assume that in

Area 1 the two generators are having the same quadratic cost functions: 1.5P 21 , 1.5P 2

2 and in Area 2 the

two generators are also having the same quadratic cost functions P 23 and P 2

4 . The the total load is 27.41 pu.

Initially the four generators’ dispatch levels are similar. After the decision making procedures, Area 2’s

generators will have higher dispatch levels as Gen 3 and Gen 4 are much cheaper than Gen 1 and Gen 2.

4.1. Type 1 primal-dual based decision making

Three scenarios are compared to show the effect of the step size τ of discrete decision making and the

gain K in frequency deviation feedback.

• τ = 2, K = 300, Figs. 7-8.

• τ = 2, K = 500, Figs. 9-10.

• τ = 5, K = 500

10

Oscillations at 0.2 Hz are observed in Scenario 2 when the gain increases. In Scenario 3, the step size τ

is increased to 5 seconds for a 500 gain. Oscillations are then damped. The machine speeds for Gen 1 for

the three scenarios are compared in Fig. 11.

The simulation results corroborate with the findings made in Section III root locus analysis. The slower

the discrete decision making process, the system is more stable.

4.2. Type 2 consensus based decision making

Two scenarios are compared to show the effect of the step size τ of discrete decision making and the gain

K in frequency deviation feedback.

• τ = 2, K = 500, Figs. 12-13.

• τ = 5, K = 500, Figs. 14-15.

The comparison of the two scenarios is presented in Fig. 16. It is observed that when τ = 2 seconds,

0.15 Hz oscillations are observed. When the step size increases to 5 seconds, the oscillations have better

damping.

Remarks: The dynamic simulation results corroborate with the finding made through linear system

analysis in Section III. The slower the discrete decision making, the system is more stable.

Comparing the frequency response of Type-1 and Type-2 architectures, we also confirm this important

finding: the particular consensus algorithm works as a secondary frequency control with economic dispatch.

Type-2 decision making process can bring frequency back to the nominal frequency.

5. Conclusion

In this paper, the continuous dynamic models for iterative decision making processes are developed.

The developed models are used together with a power system dynamic model to determine the hybrid

system dynamic stability. Such stability issues cannot be identified should either one of the dynamics is

not considered. This paper demonstrates the continuous dynamic model derivation step and linear analysis

of the integrated power system and decision making system. The analysis identifies the key stability issue

for this type of hybrid systems. The closed-loop system poles due to turbine-governor, primary frequency

control and the decision making dynamics will move to the right half plane when the frequency deviation

gain is increased. Slower decision making process leads to a more stable system. Time-domain simulation

in PST has been conducted to validate the claims.

11

References

[1] W. Zhang, W. Liu, X. Wang, L. Liu, and F. Ferrese, “Online optimal generation control based on constrained distributed

gradient algorithm,” 2014.

[2] V. R. Disfani, L. Fan, L. Piyasinghe, and Z. Miao, “Multi-agent control of community and utility using lagrangian

relaxation based dual decomposition,” Electric Power Systems Research, vol. 110, pp. 45–54, 2014.

[3] Z. Miao and L. Fan, “Primal-dual decomposition-based privacy-preserving decision making architectures for economic

operation and frequency regulation,” 2015.

[4] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation. Prentice Hall Inc., Old Tappan, NJ (USA),

1989.

[5] F. L. Alvarado, J. Meng, C. L. DeMarco, and W. S. Mota, “Stability analysis of interconnected power systems coupled

with market dynamics,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 695–701, 2001.

[6] L. Chen and K. Aihara, “Stability and bifurcation analysis of differential-difference-algebraic equations,” Circuits and

Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 48, no. 3, pp. 308–326, 2001.

[7] R. Lasseter, S. Jalali, and I. Dobson, “Dynamic response of a thyristor controlled switched capacitor,” IEEE Trans. Power

Delivery, vol. 9, pp. 1609–1615, 1994.

[8] R. Rajaraman, I. Dobson, R. H. Lasseter, and Y. Shern, “Computing the damping of subsynchronous oscillations due to

a thyristor controlled series capacitor,” IEEE transactions on power delivery, vol. 11, no. 2, pp. 1120–1127, 1996.

[9] A. Nedic and A. Ozdaglar, “Distributed subgradient methods for multi-agent optimization,” Automatic Control, IEEE

Transactions on, vol. 54, no. 1, pp. 48–61, 2009.

[10] M. Klein, G. Rogers, and P. Kundur, “A fundamental study of inter-area oscillations in power systems,” IEEE Trans.

Power Syst., vol. 6, pp. 914–921, Aug. 1991.

[11] J. Chow, G. Rogers, and K. Cheung, “Power system toolbox,” Tech. Rep. [Online]. Available:

http://www.ecse.rpi.edu/pst/PST.html

List of Figures

1 The two-area system: physical topology and the Type-a information exchange architecture. . 14

2 The block diagram of the entire system. G1 represents the discrete decision making dynamics

while G2 represents the power system dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Root loci for Type-1 system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14




12

7 System dynamic responses with Type-1 primal-dual decision making. τ = 2,K = 300..

Clockwise from upper left: a) Generators’ speeds in pu; b) Generators’ power based on the

system power base (100 MW); c) Generators’ turbine governor unit power based on the

machine power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius. . . . . . . . . 16

8 The Lagrangian multipliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 System dynamic responses with Type-1 primal-dual decision making. τ = 2,K = 500. Clock-

wise from upper left: a) Generators’ speeds in pu; b) Generators’ power based on the system

power base (100 MW); c) Generators’ turbine governor unit power based on the machine

power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius. . . . . . . . . . . . . . 17

10 The Lagrangian multipliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

11 Comparison of the three scenarios for Type 1 primal-dual decision making. . . . . . . . . . . 18

12 System dynamic responses with Type-2 consensus decision making. τ = 2,K = 500.. Clock-




13 The Lagrangian multipliers and frequency deviation measurements. . . . . . . . . . . . . . . . 19

14 System dynamic responses with Type-2 consensus decision making. τ = 5,K = 500. Clock-




15 The Lagrangian multipliers and frequency deviation measurements. . . . . . . . . . . . . . . . 20

16 Comparison of the three scenarios for Type 2 consensus-based decision making. . . . . . . . 21

13

Gen 1

Gen 2

Gen 3

Gen 4

Load 1 (9.76 pu)

Load 2 (17.65 pu)

λ2

Agent 1 Agent 2

P1ref P2ref

Local measurements Local

measurements

P3ref P4ref

Figure 1: The two-area system: physical topology and the Type-a information exchange architecture.

G2

G1

-

-

DPref1

DPref2

Df1

Df2

Figure 2: The block diagram of the entire system. G1 represents the discrete decision making dynamics while G2 representsthe power system dynamics.

Root Locus

Real Axis (seconds−1)

Imag

inar

y A

xis

(sec

onds

−1 )

−3 −2 −1 0 1 2 3 4 5

−10

−5

0

5

10

τ=20τ=2

EM dynamics

Droop dynamics

Decision making dynamics

Figure 3: Root loci for Type-1 system.

14

Root Locus


Imag

inar

y A

xis

(sec

onds

−1 )

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−3

−2

−1

0

1

2

3

τ=20τ=2

decision making dynamics

K=675

K=3519


−2 −1 0 1 2 3

−10

−5

0

5

10

Root Locus


Imag

inar

y A

xis

(sec

onds

−1 )

τ=2τ=20

EM oscillations

droop control


15

−1 −0.5 0 0.5−1.5

−1

−0.5

0

0.5

1

1.5

Root Locus


Imag

inar

y A

xis

(sec

onds

−1 )

−1 −0.5 0 0.5−1.5

−1

−0.5

0

0.5

1

1.5

Root Locus


Imag

inar

y A

xis

(sec

onds

−1 )

τ=20τ=2

K=1500K=12200


0 50 1000.9985

0.999

0.9995

1

1.0005

spee

d

0 50 100−0.2

0

0.2

0.4

0.6

Rel

ativ

e an

gles

time(s)

0 50 1005

6

7

8

9

Pel

ect

0 50 100

0.7

0.8

0.9

1

time(s)

Ptg

1

Figure 7: System dynamic responses with Type-1 primal-dual decision making. τ = 2,K = 300.. Clockwise from upper left:a) Generators’ speeds in pu; b) Generators’ power based on the system power base (100 MW); c) Generators’ turbine governorunit power based on the machine power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius.

16

0 20 40 60 80 1000

2

4

6

π

0 20 40 60 80 10010

15

20

25

λ

0 20 40 60 80 100−2

−1

0x 10

−3

time(s)

∆ f

tie−line power flow

λ1

λ2

Figure 8: The Lagrangian multipliers.

0 50 1000.99

0.995

1

1.005

spee

d

0 50 100−0.2

0

0.2

0.4

0.6

Rel

ativ

e an

gles

time(s)

0 50 1005

6

7

8

9

Pel

ect

0 50 1000.4

0.6

0.8

1

time(s)

Ptg

1

Figure 9: System dynamic responses with Type-1 primal-dual decision making. τ = 2,K = 500. Clockwise from upper left: a)Generators’ speeds in pu; b) Generators’ power based on the system power base (100 MW); c) Generators’ turbine governorunit power based on the machine power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius.

17

0 20 40 60 80 1000

5

π

0 20 40 60 80 10010

15

20

25

λ

0 20 40 60 80 100−5

0

5x 10

−3

Time (s)

∆ f

tie−line power flow

Figure 10: The Lagrangian multipliers.

0 10 20 30 40 500.9985

0.999

0.9995

1

1.0005

Time (s)

Spe

ed (

pu)

τ=2, K=300

τ=2, K=500

τ=5, K=500

Figure 11: Comparison of the three scenarios for Type 1 primal-dual decision making.

18

0 50 1000.9995

1

1.0005

1.001

1.0015

1.002

spee

d

0 50 100−0.2

0

0.2

0.4

0.6

Rel

ativ

e an

gles

time(s)

0 50 1005

6

7

8

9

Pel

ect

0 50 1000.5

0.6

0.7

0.8

0.9

1

time(s)

Ptg

1

Figure 12: System dynamic responses with Type-2 consensus decision making. τ = 2,K = 500.. Clockwise from upper left: a)Generators’ speeds in pu; b) Generators’ power based on the system power base (100 MW); c) Generators’ turbine governorunit power based on the machine power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius.

0 20 40 60 80 10016

16.5

17

17.5

18

λ

0 20 40 60 80 100−5

0

5

10

15x 10

−4

Time (s)

∆ f

Figure 13: The Lagrangian multipliers and frequency deviation measurements.

19

0 50 1000.9995

1

1.0005

1.001

1.0015

1.002

spee

d

0 50 100−0.2

0

0.2

0.4

0.6

Rel

ativ

e an

gles

time(s)

0 50 1005

6

7

8

9

Pel

ect

0 50 1000.5

0.6

0.7

0.8

0.9

1

time(s)

Ptg

1

Figure 14: System dynamic responses with Type-2 consensus decision making. τ = 5,K = 500. Clockwise from upper left: a)Generators’ speeds in pu; b) Generators’ power based on the system power base (100 MW); c) Generators’ turbine governorunit power based on the machine power base (900 MW); d) Gen 1-3’s angles relative to Gen 4 in radius.

10 20 30 40 50 60 70 80 90 10015

16

17

18

19

20

λ

0 20 40 60 80 100−5

0

5

10

15x 10

−4

Time (s)

∆ f

Figure 15: The Lagrangian multipliers and frequency deviation measurements.

20

5 10 15 20 25 30 35 40 45 50

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

1.0012

1.0014

1.0016

Time (s)

Spe

ed (

pu) τ=2, K=500

τ=5, K=500

Figure 16: Comparison of the three scenarios for Type 2 consensus-based decision making.

21

analysis of coupling dynamics for power systems with...

Documents