Reactive Power Compensation in Mechanical Systems

Transcription

1 Reactive Power Compensation in Mechanical Systems Carlos Rengifo, Bassel Kaar, Yannick Aoustin, Christine Chevallereau To cite this version: Carlos Rengifo, Bassel Kaar, Yannick Aoustin, Christine Chevallereau. Reactive Power Compensation in Mechanical Systems. The 2n Joint International Conference on Multiboy System Dynamics - IMSD212, May 212, Stuttgart, Germany. <hal > HAL I: hal Submitte on 11 Oct 212 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 The 2 n Joint International Conference on Multiboy System Dynamics May 29 June 1, 212, Stuttgart, Germany Reactive Power Compensation in Mechanical Systems Rengifo Carlos, Kaar Bassel, Aoustin Yannick, Chevallereau Christine Faculty of Electronical Engineering Universia el Cauca Calle 5 No 4-7, Popayan, Colombia caferen@unicauca.eu.co L UNAM, IRCCyN, UMR, CNRS 6597 CNRS, École Centrale e Nantes 1, rue e la Noë, BP 9211, 44321, Nantes, France [firstname.lastname]@irrcyn.ec-nantes.fr ABSTRACT In this paper the problem of energy consumption in mechanical systems is approache from an electrical engineering point of view. To achieve this objective classical concepts in electrical networks theory like apparent power, reactive power an power factor have been extene to mechanical systems. This paper focus on the role of springs in mechanical systems to avoi power oscillations between joint actuators an loas. Such oscillations are a major problem because they unnecessarily increases the mean-square value of joint torques an by consequence Joule effect losses in the actuators. The minimization of these oscillations is known as "reactive power compensation". The main points illustrate in this paper are the funamental limitations on reactive power compensation an the negative effect on the energy consumption of the harmonic content of the reference trajectory. 1 Introuction In rotational mechanical systems like robot joints, the instantaneous power elivere by a motion actuator is given by the prouct between the joint torque an the joint velocity. If the mechanical loa introuces a phase shift between these two variables, the sign of the instantaneous power is not constant. As a consequence the flow of energy between the actuator an the loa is biirectional. For a passive loa, it implies that a part of the receive energy is store an subsequently forware to the actuator. This phenomenon entails two main problems. The first one is that most actuators o not have energy recovery capabilities so this forware energy is lost by Joule effect. The secon one is that the mean-square value of the torque require to prouce a given motion is unnecessarily incremente because of the aitional transfer of energy from the actuator to the loa. In the same way as in electrical networks capacitors are use to compensate phase shifts between voltage an current create by inuctive loas, we show that springs play the same role in mechanical systems. Thus, in both electrical an mechanical systems, phase shift compensation between inputs an outputs is a funamental issue for the improvement of the energy transfer between a source an a loa. Despite of this similarity, efficient power transmission in mechanical systems is far from being evient. The main ifficulties arise from the non-sinusoial nature of joint robotic motions an the non linear ynamics present in most mechanical systems. It is important to note that in the case of nonlinear systems, phase shift compensation between torque an velocity oes not necessarily guarantee an uniirectional flow of energy. Moreover, only in the case of linear systems excite with sinusoial inputs, phase-shift compensation guarantees an efficient energy transfer between source an loa [2]. The objective of this paper is to show the applicability of recent theoretical avances [3], [5], [6] in powerfactor compensation of nonlinear electrical networks excite with non-sinusoial signals for the minimization of energy consumption in mechanical systems. To achieve this objective it has been necessary to generalize classical concepts in electrical engineering like power factor, apparent power an reactive power. This paper is organize as follows. In section 2 the problem statement an the assumptions for the rest of the paper are presente. In section 3 mathematical operators for the root-mean-square value of a perioical signal an for the active power are introuce. In section 4 reactive power compensation is formulate as an optimization problem using two criteria, the mean-square value of the joint torque an the so-calle "power

3 factor". In section 5 a geometrical interpretation of the power factor is given. In section 6 an optimality conition vali for the two criteria is euce. In section 7 two numerical examples are presente. One of them illustrates the funamental limitations on reactive power compensation an the other one shows the negative effect it can have the harmonic content of the reference trajectory in energy consumption. Last Section is evote to conclusions an perspectives. 2 Problem statement The problem aresse in this paper is how to optimize the energy transfer between a motion actuator an a mechanical loa. In robotic systems, for example, the loa correspons to the mechanical structure of the robot an the motion actuator to an electric, hyraulic, pneumatic or any other type of evice suppling the joint torque necessary to prouce the esire motion. The compensator system can be a torsional spring or any other elastic element capable of storing energy. The mechanical loa is suppose to be functioning as the feeback system presente in Figure 1. q a(t) is the esire motion for the actuate joints, Σ l is the ynamical system representing the mechanical loa, Σ c (θ) is a non-issipative passive system calle mechanical compensator an Ω is a given close loop controller Γ c Compensator Σ c q a(t) + Ω Γ + Γ l Σ l q a (t) Controller Loa Figure 1. Close loop mechanical system. For the close loop system of Figure 1 the following assumptions will be mae A1: The reference perioic motion q a(t) is a vector of smooth signals with a common perio. A2: The controller Ω ensures the convergence of q a (t) to q a(t). A3: The close loop system is consiere to be functioning for a long time before t =. For t, q a (t) is consiere to be converge to q a(t). In such a case, it is sai that the system has reache the steay state. A4: The mechanical loa Σ l is suppose to be a passive ynamical system [7]. Uner the assumptions A1 an A2 passivity implies that the average power elivere by the actuator in a cycle is nonnegative 1 To Γ T l (t) q a(t) t (1) A5: The mechanical compensator is compose by non-issipative passive elements. Uner the assumptions A1 an A2, it implies 1 To Γ T c (t) q a (t) t = (2) 3 Mathematical notation Given two perioical vector value signals x(t) IR n an y(t) IR n with a common funamental perio, the application of the binary operator <, > to x(t) an y(t) gives a real quantity efine as x(t), y(t) 1 To x T (t) y(t) t (3)

4 Using this operator, assumptions A4 an A5 can be written as Γ l (t), q a(t) an Γ c (t), q a(t) =. The value x(t), x(t) correspons to the mean-square value of the vector signal x(t) x(t), x(t) = 1 To x T (t) x(t) t, (4) an by consequence x(t), x(t) is the root-mean-square (rms) value of x(t) IR n. In the sake of simplicity, instea of x(t), x(t), the rms value is enote as follows x(t) x(t), x(t) (5) 4 Optimization criteria In this section two ifferent criteria for the minimization of steay state energy consumption are presente. Optimization will be mae with respect to θ, a vector containing the parameters of the compensator system Σ. For example, if Σ is a torsional spring, θ is its stiffness. The first criterion is given by Γ 2 1 To Γ T (t) Γ(t) t (6) Despite of the wiesprea utilization of (6) as a performance inex for trajectory generation in robotic systems, its main inconvenient is the ifficulty to assert when a given Γ 2 is small enough for a given motion. A particular value of Γ 2 coul be consiere small for certain motions but not for others. This fact oes not allow a proper comparison between motions with ifferent. It woul be "unfair" to compare slow an fast motions just in terms of Γ 2 even if they are applie to the same system. The other criterion we present in this section is known in electrical engineering as power factor [4]. The equivalent of the efinition presente in [4] for mechanical systems is p f = P S (7) with P Γ, q a n S Γ i q a i Γ i an q a i being the i-th component of the vectors Γ IR n an q a IR n. The scalar quantities P an S, respectively known as active power an apparent power [2], satisfy the Cauchy-Schwarz inequality As Γ = Γ l + Γ c (see Figure 1), active power can be rewritten as P = Γ l, q a + Γc, q a S P S (9) Uner the assumption A4 the term Γ c, q a is zero an by consequence active power P oes not epen on the compensator system P = Γ l, q a (11) Uner the assumption A5 the term Γ l, q a is a nonnegative quantity. By consequence the inequality (9) becomes P S (12) The above inequality implies that power factor is a quantity between an 1. The main avantage of powerfactor is that is a normalize quantity. When this quantity is close to zero, most part of the energy transfere to the loa is store an subsequently forware to the actuator. This phenomenon entails two main problems. The first one is that most actuators o not have energy recovery capabilities so this forware energy is lost by Joule effect. The secon one is that the mean-square value of the torque require to prouce a given motion is unnecessarily incremente. Conversely, an unitary power factor implies that for all time t power goes from the actuator to the loa. These aspects will be explaine in the next section. (8) (1)

5 Apparent power S Reactive power Q α Active power P Figure 2. Power factor is the cosine of α. When reactive power is zeroe, then power factor equals to one. 5 Unerstaning power factor Power factor can be unerstoo through the right-angle triangle presente in Figure 2. The hypotenuse represents the apparent power, the horizontal cathetus the active power an the cosine of the angle between them is the power factor. The vertical cathetus is known as reactive power. From Figure 2 it can be seen that a reuction in the reactive power leas to an improvement of the power factor. To illustrate this iea, the close loop system presente in Figure 1 will be consiere. The mechanical loa Ω l is suppose to be a linear single actuate system Γ l = J q + f v q (J an f v being the inertia moment an the viscous friction coefficient). The compensator is suppose to be a torsional spring escribe by Γ c = k q. Uner the assumption A3, the close system can be escribe as For this system active an apparent power are given by Γ = J q a + f v q a + k qa (13) }{{}}{{} Γ l Γ c P =< J q a + f v q a + k q a, q a > S = J q a + f v q a + k q a q a Using Fourier series it can be prove that terms < q a, q a > an < q a, q a > are zero for any perioic signal q a(t). In such a case the expressions for P an S can be simplifie as with P = f v q a 2 S = J 2 q a 2 + fv 2 q a 2 + k 2 qa J k < q a, qa > q a = fv 2 q a 2 + J q a + k qa 2 q a = ( f v q a 2 ) 2 + ( J q a + k qa q a ) 2 = P 2 + Q 2 (14) (15) Q = J q a + k q a q a (16) Depening on the harmonic content of q a(t), a positive constant k minimizing Q can be foun. If k is such that J q a + k q a = for all t, then Q is zeroe an by consequence the power factor becomes unitary. In such a case, the compensate mechanical loa is Γ = f v q a, which is equivalent to a pure viscous friction element. Γ = f v q a implies that instantaneous power remains non negative for all time t an that energy flows in one irection, from the actuator to the loa. 6 Optimality conitions Firstly, we euce a conition for the maximization of the power factor with respect to θ (a vector containing the parameters of the compensator Σ). Equation (11) shows that the active power P is inepenent of θ.

6 By consequence, the minimization of the apparent power S leas to the maximization of the quotient P/S, which is efine as the power factor. Apparent power can be written in the following way S = = = = n Γ i q a i n Γ li + Γ ci q a i n Γli + Γ ci 2 q a i n Γli 2 + Γ ci Γ li, Γ ci q a i (17) Conversely, the apparent power for the uncompensate system (Γ c (t) ) is given by S u = n Γli 2 q a i (18) If we compare the expressions for S an S u, it can be conclue that the following conition guarantees S < S u, Γ ci Γ li, Γ ci < i {1,..., n}, (19) If the above inequalities are satisfie for all i, then apparent power of each actuator is ecrease an so the total apparent power S. Conition (19), however, is sufficient but not necessary. The total apparent power S coul be ecrease even if the above inequalities are satisfie for some (but not all) values of i. In the case of a single-actuate system (n = 1) conition (19) becomes both necessary an sufficient. Now we euce a necessary an sufficient conition for the minimization of the mean-square joint torque. The joint torque supplie by the actuator is ecompose as the sum of Γ l an Γ c (see Figure 1) Γ 2 = 1 To [Γ l (t) + Γ c (t)] T [Γ l (t) + Γ c (t)] t = 1 To Γ T l (t) Γ l (t) + Γ T c (t) Γ c (t) + 2Γ T l (t) Γ c (t)t Using the operators. an <, > the above equation can be rewritten as (2) If the following inequality is satisfie or equivalently Γ 2 = Γ l 2 + Γ c Γ l, Γ c (21) Γ c Γ l, Γ c <, (22) n Γ ci Γ li, Γ ci <, (23) then Γ 2 < Γ l 2. In such a case, the mechanical compensator leas to a less energy consumption in the sense of the criterion (6). Criteria (19) an (23) are equivalent only for single-actuate systems. 7 Numerical simulations In this section two numerical examples are presente. The first one illustrates the funamental limitations on reactive power compensation an the other one shows the negative effect that can have the harmonic content of the reference trajectory in energy consumption.

7 25 Minimal amplitue for Ko > [egrees] Normalize Frequency Figure 3. Amplitue-Frequency optimality conition. Minimal motion amplitue require to obtain a positive optimal spring constant. Frequency is normalize with respect to the natural frequency of the penulum. Example 1 Consier the close-loop mechanical system presente in Figure 1 an suppose that Σ l is a single penulum system, Σ c is a series torsional spring an Ω a control law satisfying the assumption A3. The parameters of the penulum in the international units (MKS) are J =.981 (inertia), m =.1 (mass), l = 1 (length), f v =.1 (viscous friction) an g = 9.81 is the gravity force. The objective is to maximize the power factor by optimizing the stiffness of the spring. The esire motion is suppose to be given by The steay state close loop ynamics can be escribe as follows q a(t) = A sin (ω o t) (24) Γ l = J q a + m g l sin ( q a) + fv q a Γ c = k q a (25) For small motion amplitue sin ( q a) can be approximate by q a. Uner this assumption it can be shown that the optimality conition (19) leas to k < 2 ( J ω 2 o m g l ) (26) The above inequality implies that power factor can be improve using a torsional spring only when frequency motion is greater than the natural frequency of the penulum ω n = m g l/j, otherwise problem is infeasible because a negative k is require. If the reference motion oes not allow to approximate sin ( ) qa by qa the optimality conition (19) gives an upper limit for k epening on A an ω o. Unfortunately, an expression in a close form like (26) cannot be obtaine in that case. The graph presente in Figure 3 shows the minimal value of A require to obtain a non-negative upper limit for k for a given frequency ω o. If the pair (A, ω o ) is below the curve, the optimality conition (19) cannot be satisfie for any positive value of k. If (A, ω o ) is above the curve, there exist a set of positive values of k leaing to an improvement of the power factor. It is interesting to note that for frequencies higher than the natural frequency, optimization is possible for all amplitues. As seen in Section 4 if the compensate mechanical loa is seen by the actuator as pure viscous friction element, then the power factor becomes unitary. Using the equation (25) it can be conclue that Γ = f v q a can be obtaine, if an only if, there exist a constant value of k satisfying J q a + m g l sin ( q a) + k q a =, t (27)

8 1.9 Optimal Power Factor Wn 1.Wn 2.Wn 5.Wn Motion amplitue [egrees] Figure 4. Maximal power factor in a single penulum system when the reference motion is a sinusoial signal of given amplitue. The above equation can be satisfie for a constant k only if the amplitue of the esire motion is small. In such a case, k is given by k = Jω 2 o m g l. If sin ( q a) cannot be approximate by q a then an unitary optimal power factor cannot be obtaine. Figure 4 shows the maximal power factor as a function of the amplitue an frequency. From this figure it can be observe that for a fixe frequency if the amplitue of the esire motion is augmente then the maximal power factor that can be obtaine ecreases. Conversely, for a motion of fixe amplitue, greater is the frequency, greater is the maximal power factor. Example 2 Reference motion in robotic systems is often inicate by using only initial an final conitions on joint positions an velocities. This implies that an infinite number of time functions satisfying such conitions can be generate. For example, for the system of the previous example, the two perioic motions presente in Figure 5 satisfy the conitions q() = 2 o, q(t/2) = 2 o, q() =, q(t/2) =. Both reference motions have a funamental frequency equal to ω o = 1 ra/seg which is ten times the natural frequency of the penulum (ω n = m g l/j). One of the motions is efine as a single frequency sinusoial signal an the other one is obtaine by concatenating two polynomials of thir orer. Sinusoial motion: q a(t) = A sin (ω o t) (28) Polynomial motion: a 3 t 3 + a 2 t 2 + a 1 t + a, qa(t) = b 3 t 3 + b 2 t 2 + b 1 t + b, t < To 2 2 t < (29) As motions are almost ientical it coul be expecte that the joint torque curves for the polynomial an the sinusoial references be also quite similar. However, this is not always true. It epens on the transfer function Γ(s)/q a(s). If Γ(s)/q a(s) has a strong gain at high frequencies, some of the high-orer harmonic components of the polynomial motion coul have a more important gain that the funamental component. In such a case, the torques coul be quite ifferent even if motions are very similar. If the high frequency gain of Γ(s)/q a(s) is limite, the convergence of q a (t) towars q a(t) can be seriously affecte. In summary, there is trae-off between tracking an energy consumption. To illustrate this point a linearize version of

9 2 Joint position Degrees Time [sec] Joint velocity 2 Degrees/sec Time [sec] Figure 5. Reference motions for the example 2. One of them is efine as a single frequency sinusoial signal (soli line) an the secon one by concatenating two polynomials of thir orer (ashe line). the penulum system will be consiere Γ = J q a + f v q a + (m g l + k) q a (3) By taking the Laplace transform of the last equation, the following transfer function is obtaine Γ(s) q a(s) = J s2 + f v s + (m g l + k) (31) This transfer function is unrealistic because the resulting steay-state gain increases inefinitely as frequency increases. In practice, close-loop steay-state gain is limite by the actuator ynamics. To obtain a more convincing transfer function the following consierations will be mae Actuator is suppose to be moele as a low-pass filter with a cutoff frequency of 1ω o (ω o being the funamental frequency of the reference motion) G a (s) = 1 (.1s + 1) 2 (32) Controller Ω (see Figure 1) is represente by a classical lea compensator [1] G c (s) = 5 1.1s + 1.1s + 1 (33) With the above consierations the steay-state gain of the transfer function Γ(s)/q a(s) ecreases as frequency increases when ω > 1ω o. (Figure (6)). For frequencies between ω o an 1ω o the gain increases as frequency increases. From this figure it can be seen that harmonic components with frequencies between 2ω o an 1ω o have a gain more than ten times larger than the gain of the funamental harmonic ω o. As consequence, even if the two motions presente in Figure 5 are very similar, the corresponing instantaneous power curves are very ifferent (see Figure 7). From this Figure it can be seen that for the sinusoial motion instantaneous power remains non-negative, an for the polynomial motion power oscillations are important. For the sinusoial motion power factor is unitary an Γ 2 =.369, for the polynomial motion power factor is.4569 an Γ 2 =.1769.

10 1 3 Steay state gain Normalize frequency [ω/ω ] o Figure 6. Steay state gain of the close loop transfer function Γ(jω)/q a(jω) when actuator ynamics is consiere. 2 Steay state power Power [J/s] Time [sec] Figure 7. Instantaneous power for the sinusoial motion (soli line) an for the polynomial motion (ashe line).

11 8 Conclusions an perspectives The problem of the steay state reactive power compensation in close loop mechanical system subject to perioic motion has been presente. Compensation is one by using non-issipative passive elements like torsional springs. Parameter selection for these elements is formulate as an optimization problem. Two ifferent performance inex are presente. One of them is the classical mean-square value of the torque an the other one, inspire from the electrical networks theory, is known as power factor. This latter is a normalize quantity between an 1 which epens on the torque an on the esire motion. It has the avantage to allow the comparison between systems with ifferent motions. Base on the iea of optimization of power factor by the use of capacitors in electrical system, the use of springs is propose to optimize the "energy" consumption in mechanical system. Since the stiffness is a positive coefficient the efficiency of this approach epens on the esire trajectory. In the example 1, the positiveness of the spring stiffness epens on the amplitue an the frequency of the motion. An unitary power factor can be obtaine when the compensate mechanical loa is seen by the actuator as a linear viscous friction element of the form Γ = f v q a. In such a case instantaneous power remains on-negative for all time t an energy flows in one irection, from the actuator to the loa. Another aspect stuie in the paper is the effect of the harmonic content of the reference motion in the power factor of the system. In the Example 2, it is shown that very similar reference motions can prouce very ifferent power factors. In the cite example, this phenomenon is explaine by the amplification of the high orer harmonic components of the reference motion. The feeback system presente in Figure 1, allows to inclue springs only in active joints. Our main perspective is then to evelop a conceptual framework allowing to stuy more general interconnections between mechanical systems an passive compensators. It woul be interest, for example, to unerstan the effect of passive arms in the energy consume by a bipeal robot. Acknowlegements Carlos F. Rengifo woul like to acknowlege an express his sincere gratitue to Universia el Cauca for the financial support given to him uring this project. REFERENCES [1] Y. Chen. Replacing a PID controller by a lag-lea compensator for a robot - A frequency-response approach. IEEE Transactions on Robotics an Automation, 5(2): , apr [2] E. Garcia-Canseco, R. Grino, R. Ortega, M. Salichs, an A.M. Stankovic. Power-factor compensation of electrical circuits. IEEE Control Systems Magazine, 27(2):46 59, April 27. [3] Dimitri Jeltsema. Moeling an Control of Nonlinear Networks: A Power-Base Perspective. PhD thesis, Delft University of Technology, 25. [4] G Kassakian, M.F Schlecht, an G.C Verghese. Principles of power electronics. Reaing MA: Aison- Wesley, [5] H. Lev-Ari an A.M. Stankovic. Hilbert space techniques for moeling an compensation of reactive power in energy processing systems. IEEE Transactions on Circuits an Systems I: Funamental Theory an Applications, 5(4):54 556, april 23. [6] R. Ortega, D. Jeltsema, an J.M.A. Scherpen. Power shaping: a new paraigm for stabilization of nonlinear RLC circuits. IEEE Transactions on Automatic Control, 48(1): , oct. 23. [7] Romeo Ortega, Julio Antonio Loría Perez, Per Johan Nicklasson, an Hebertt J. Sira-Ramirez. Passivity-base control of Euler-Lagrange systems: mechanical, electrical, an electromechanical applications. Springer, 1998.