Analysis of Coupling Dynamics for Power Systems with Iterative Discrete Decision Making Architectures

Transcription

1 Analysis of Coupling Dynamics for Power Systems with Iterative Discrete Decision Making Architectures Zhixin Miao Department of Electrical Engineering, University of South Florida, Tampa FL USA 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 impacting 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 algorithm 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. Introduction Introduction of numerous smart buildings, distributed energy sources and energy storages poses challenges 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, 26

2 process. These privacy-preserving decision making architectures for microgrids and power systems have been proposed in the literature [, 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 [, 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 + )-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

3 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.. Type Primal-dual decomposition based decision making Consider a two-area power system economic dispatch problem. P rob minimize f (P ) + f 2 (P 2 ) (a) subject to λ : P P 2 = D (b) λ 2 : P 2 + P 2 = D 2 d P 2 d (c) (d) where P i notates output active power from Area i, f i (P i ) is the cost related to power generation, P 2 is the tie-line flow from Area to Area 2, λ i 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 P rob with the two power balanced equality constraints relaxed is as follows. L(P, P 2, P 2, λ, λ 2 ) = f (P ) + λ ( D P + P 2 ) + f 2 (P 2 ) + λ 2 ( D 2 P 2 P 2 ) (2) Given the two price signals, can the each area determine its generation P and import P 2? This question will be examined by looking at Area. For a given λ, for this objective function f (P )+λ ( D P +P 2 ), 3

4 if P 2 has no limit imposed, there is no solution unless λ =. The objective function can go. Therefore, this problem is considered not feasible. We would like to treat P 2 differently than P and P 2 to help the problem solving. If we treat P 2 as given, similar as λ and λ 2 are given, then the dual problem becomes: d (λ, P 2 ) = min P {f (P ) + λ ( D P + P 2 )} (3) d 2 (λ 2, P 2 ) = min P 2 {f 2 (P 2 ) + λ 2 ( D 2 P 2 P 2 )} (4) The above two problems should have solutions. We now consider the dual problem P rob 2 with P 2 given. P rob 2 maximize d (λ, P 2 ) + d 2 (λ 2, P 2 ) (5a) over λ, λ 2 (5b) For this problem, we end up having a solution dependent on P 2. Let us notate this solution of P rob 2 as a function of P 2 : g(p 2 ). g(p 2 ) is a dual problem of the dual problem. Since the dual problem is a concave function over λ i, the dual s dual problem should be a minimization problem over P 2. Therefore, the dual s dual problem or the primal-dual problem P rob 3 can be written as follows. P rob 3 min P 2 ( max λ + max λ 2 ) ( ) min {f (P ) λ P } + λ D + P 2 P ( ) min {f 2 (P 2 ) λ 2 P 2 } + λ 2 ( D 2 P 2 ) P 2 (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 variablep 2. Hereinafter, we will notate this virtual tie-line flow a different name: π. The subgradient of the line flow is (λ λ 2 ). Since the primal problem is a minimization problem, therefore, in the update procedure, for a positive 4

5 gradient, π should be reduced. The updating procedure is presented as follows. π k+ = π k α(λ k λ k 2) (7) where α >. For a given virtual tie-line flow π k, 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. λ k = 2a ( D + π k ) + b K f (8) λ k 2 = 2a 2 ( D 2 π k ) + b 2 K f 2 (9) where a i, b i are coefficients of a generator quadratic cost function (f i (P i ) = a i P 2 i + b i P i + c i ), f i 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 Modeling as continuous dynamics We now proceed to give an approximate continuous model for the above mentioned iterative procedure. Assuming that f = f 2 (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 π k+ = ( 2α(a + a 2 ))π k + π, () where π = α(2a D 2a 2 D2 + b b 2 ). The power references are determined by the prices. Therefore, P k = (π k + D ) K 2a f k () 5

6 Considering that the frequency measurement of the previous step is taken in the price calculation, the k + step power reference is modified as P k+ = (π k+ + D ) K 2a f k (2) Substituting π k+ and π k by P k+ and P k, we find P k+ = [ 2α(a + a 2 )]P k K α(a + a 2 ) a f k + P (3) where P = 2α(a + a 2 ). Using forward Euler method, we can express the derivative at k-step is k P P k+ P k. (4) τ where τ is the step size. Therefore, the discrete equation can now be approximated by a continuous dynamic equation. τ P = 2α(a + a 2 )P K α(a + a 2 ) a f + P (5) 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 ref i = K 2a i + τ s f i (6) where τ = τ 2α(a +a 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 C (P ) + C 2 (P 2 ) subject to: P + P 2 = D + D 2 (7a) (7b) 6

7 where C i (P i ) is the cost of generation, P i is the power generation at Area i and D i is the load consumption at Area i. The dual problem is described in (8). max λ min C (P ) + C 2 (P 2 ) + λ( D P + D 2 P 2 ) (8) P,P 2 The above problem can be converted to a consensus problem by introducing λ and λ 2 for each area. λ should be equal to λ 2. Therefore, the optimization problem is converted to a maximization problem with a consensus constraint. max λ,λ 2 min C (P ) + λ ( D P ) + C 2 (P 2 ) + λ 2 ( D 2 P 2 ) P,P 2 s.t. λ = λ 2 (9) The consensus algorithm that utilizes a stochastic matrix to conduct weighted averaging only guarantees consensus of multiple λ i. 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. λ λ 2 k+ = A λ where A is a stochastic matrix. For the test two-area system, we select A = λ 2 k K f (2) f 2 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 λ = 2a P + b (2) λ 2 = 2a 2 P 2 + b 2 (22) 7

8 The iteration procedure (2) is now expressed in terms of the power commands: P P 2 k+ + b 2a b 2 2a 2 = A P P 2 k + b 2a b 2 2a 2 K f 2a f 2 2a 2 (23) The difference equation is now converted to a continuous dynamic equation. K 2a P = (τs A + I) f (24) K P 2 2a 2 f 2 }{{} G (s) The gain matrix G 2 (s) defines the transfer function matrix from the frequency deviation to the power commands. G (s) = K 2τs(τs + ) 2τs+ 2a 2a 2 2a 2τs+ 2a 2 (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- 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.. This system comes from the classical two-area four-machine power system [] 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 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 is 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

9 represented by a two-generator system.the two rotor angles are expressed as: δ = M s 2 ( P m + T δ 2 ) + D s + T (26) δ 2 = M 2 s 2 ( P m2 + T 2 δ ) + D 2 s + T 2 (27) where M, D, T are inertia constants, damping and synchronizing coefficients. T = T 2. Rearranging the equations, we have δ = (M s 2 + D s + T ) P m + T P m2 (M s 2 + D s + T )(M s 2 + D s + T ) T T 2. (28) The transfer function matrix G 2 (s) that defines the relationship from the power command to the speed deviations due to the power system dynamics is expressed in (3). f = G 2 (s) f 2 P ref P ref 2 (29) where G 2 (s) = G tg (s)s ω [(M s 2 +D s+t )(M 2 s 2 +D 2 s+t 2 ) T T 2 ] M s 2 + D s + T T T 2 M 2 s 2 + D 2 s + T 2 (3) where G tg is the turbine-governor transfer function representing the relationship from the power order P ref to the mechanical power P m. In addition, the droop control has to be included. Therefore, the diagonal components of G 2 (s) have to be modified to include the droop control. The entire system block diagram is obtained and shown in Fig Root loci To examine the stability of the closed-loop system, the open-loop gain matrix G 2 G will be examined. G 2 G 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 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

10 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.-.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.. Power System Toolbox [] is selected as the dynamic simulation platform. The power system and Type decision making architecture are shown in Fig.. The discrete decision making will take place every 2 seconds or every 5 seconds. The power commands from Agent 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 consists of Gen and Gen 2 and Load. 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 7.27 pu, 7. pu, 7.6 pu and 7. pu. Assume that in Area the two generators are having the same quadratic cost functions:.5p 2,.5P 2 2 and in Area 2 the two generators are also having the same quadratic cost functions P 2 3 and P 2 4. The the total load is 27.4 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 and Gen Type 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 = 3, Figs τ = 2, K = 5, Figs. 9-. τ = 5, K = 5

11 Oscillations at.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 5 gain. Oscillations are then damped. The machine speeds for Gen for the three scenarios are compared in Fig.. 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 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 = 5, Figs τ = 5, K = 5, Figs The comparison of the two scenarios is presented in Fig. 6. It is observed that when τ = 2 seconds,.5 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- 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.

12 References [] W. Zhang, W. Liu, X. Wang, L. Liu, and F. Ferrese, Online optimal generation control based on constrained distributed gradient algorithm, 24. [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., pp , 24. [3] Z. Miao and L. Fan, Primal-dual decomposition-based privacy-preserving decision making architectures for economic operation and frequency regulation, 25. [4] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation. Prentice Hall Inc., Old Tappan, NJ (USA), 989. [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. 6, no. 4, pp , 2. [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 , 2. [7] R. Lasseter, S. Jalali, and I. Dobson, Dynamic response of a thyristor controlled switched capacitor, IEEE Trans. Power Delivery, vol. 9, pp , 994. [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., no. 2, pp. 2 27, 996. [9] A. Nedic and A. Ozdaglar, Distributed subgradient methods for multi-agent optimization, Automatic Control, IEEE Transactions on, vol. 54, no., pp. 48 6, 29. [] M. Klein, G. Rogers, and P. Kundur, A fundamental study of inter-area oscillations in power systems, IEEE Trans. Power Syst., vol. 6, pp , Aug. 99. [] J. Chow, G. Rogers, and K. Cheung, Power system toolbox, Tech. Rep. [Online]. Available: List of Figures The two-area system: physical topology and the Type-a information exchange architecture The block diagram of the entire system. G represents the discrete decision making dynamics while G 2 represents the power system dynamics Root loci for Type- system Root loci for Type- system Root loci for Type-2 system Root loci for Type-2 system

13 7 System dynamic responses with Type- primal-dual decision making. τ = 2, K = 3.. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius The Lagrangian multipliers System dynamic responses with Type- primal-dual decision making. τ = 2, K = 5. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius The Lagrangian multipliers Comparison of the three scenarios for Type primal-dual decision making System dynamic responses with Type-2 consensus decision making. τ = 2, K = 5.. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius The Lagrangian multipliers and frequency deviation measurements System dynamic responses with Type-2 consensus decision making. τ = 5, K = 5. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius The Lagrangian multipliers and frequency deviation measurements Comparison of the three scenarios for Type 2 consensus-based decision making

14 Gen Gen 3 Gen 2 Gen 4 Load (9.76 pu) Load 2 (7.65 pu) Local measurements P ref P 2ref λ 2 Local measurements P 3ref P 4ref Agent Agent 2 Figure : The two-area system: physical topology and the Type-a information exchange architecture. DP ref DP ref 2 G G Df 2 Df Figure 2: The block diagram of the entire system. G represents the discrete decision making dynamics while G 2 represents the power system dynamics. Root Locus Imaginary Axis (seconds ) 5 5 EM dynamics Droop dynamics τ=2 τ=2 Decision making dynamics 3 2 Real Axis (seconds 2 ) Figure 3: Root loci for Type- system. 4

15 Root Locus Imaginary Axis (seconds ) 2 3 τ=2 τ=2 2 decision making dynamics K=675 K= Real Axis (seconds ) Figure 4: Root loci for Type- system. Root Locus τ=2 τ=2 Imaginary Axis (seconds ) 5 5 droop control EM oscillations Real Axis (seconds ) Figure 5: Root loci for Type-2 system. 5

16 Root Locus Root Locus.5 τ=2.5 τ=2 Imaginary Axis (seconds ).5.5 K=5 Imaginary Axis (seconds ).5.5 K= Real Axis (seconds ) Real Axis (seconds ) Figure 6: Root loci for Type-2 system speed.9995 P elect Relative angles.4.2 P tg time(s) 5 time(s) Figure 7: System dynamic responses with Type- primal-dual decision making. τ = 2, K = 3.. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius. 6

17 6 4 tie line power flow π 2 λ λ x 3 λ f time(s) Figure 8: The Lagrangian multipliers. speed P elect Relative angles.4.2 P tg time(s).4 5 time(s) Figure 9: System dynamic responses with Type- primal-dual decision making. τ = 2, K = 5. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius. 7

18 5 tie line power flow π λ x 3 f Time (s) Figure : The Lagrangian multipliers..5 τ=2, K=5 τ=5, K=5 Speed (pu) τ=2, K= Time (s) Figure : Comparison of the three scenarios for Type primal-dual decision making. 8

19 speed P elect Relative angles.4.2 P tg time(s).5 5 time(s) Figure 2: System dynamic responses with Type-2 consensus decision making. τ = 2, K = 5.. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius λ x 4 f Time (s) Figure 3: The Lagrangian multipliers and frequency deviation measurements. 9

20 speed P elect Relative angles.4.2 P tg time(s).5 5 time(s) Figure 4: System dynamic responses with Type-2 consensus decision making. τ = 5, K = 5. Clockwise from upper left: a) Generators speeds in pu; b) Generators power based on the system power base ( MW); c) Generators turbine governor unit power based on the machine power base (9 MW); d) Gen -3 s angles relative to Gen 4 in radius λ x 4 f Time (s) Figure 5: The Lagrangian multipliers and frequency deviation measurements. 2

21 τ=5, K=5. Speed (pu) τ=2, K= Time (s) Figure 6: Comparison of the three scenarios for Type 2 consensus-based decision making. 2