A dynamic router for microgrid applications: Theory and experimental results

Control Engineering Practice 27 (2014) 23–31

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

A dynamic router for microgrid applications: Theory and experimental results Victor Ramirez a,n, Romeo Ortega a, Olivier Bethoux b, Antonio Sánchez-Squella c a

Laboratoire des Signaux et Systémes, Supélec – 3 rue Joliot Curie, Plateau de Moulon, 91192 Gif-sur-Yvette, France LGEP – CNRS/SUPELEC, 11 rue Joliot Curie, Plateau de Moulon, 91192 Gif sur Yvette, France c Departamento de Ingeniería Eléctrica, Universidad Técnica Federico Santa María, Avda. Vicuña Mackema 3939, San Joaquín, Santiago, Chile b

art ic l e i nf o

a b s t r a c t

Article history: Received 25 February 2013 Accepted 5 February 2014 Available online 13 March 2014

Efficient regulation of the energy transfer between generating, storage and load subsystems is a topic of current practical interest. A new strategy to achieve this objective, together with its corresponding power electronics implementation, was recently proposed by the authors. The device is called dynamic energy router (DER) because, in contrast with current practice, the regulation of the direction and rate of change of the power flow is done without relying on steady-state considerations. In this paper it is shown that, unfortunately, the DER becomes non-operational in the (unavoidable) presence of losses in the system. Hence, we propose a new DER that overcomes this problem. Experimental evidence of the performance of the original and new DER is presented. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear control Power electronics Passive systems Energy management Microgrids

1. Introduction Achieving efficient transfer of electric energy between multidomain subsystems that can generate, store, or consume energy is a central problem in modern microgrid systems (Fraghani, 2010). As an example consider the case of a hybrid vehicle containing fuel cells, supercapacitors, battery and electric motors. Depending on the operation regime (in the example, essentially determined by fuel consumption considerations), energy must be transferred between the various units—that, in the sequel will be refer to as multiports—according to some energy-management policies. The energy exchange between the multiports is achieved interconnecting them through power converters, which are electronically switched circuits capable of adjusting the magnitudes of the port variables, voltage or current, to a desired value. To achieve the desired energy-management policy, it is a common practice to assume that the system operates in steady state and then translates the power demand (flow sense and magnitude) for each multiport into current or voltage references— see, e.g., Malo and Griñó (2007), Thounthong, Raël, and Davat (2005), Choi, Howze, and Enjeti (2006) and Schenck, Lai, and Stanton (2005). These references are then tracked with control loops, usually proportional plus integral (PI). Since the various multiports have different time responses, it is often necessary to n

Corresponding author E-mail addresses: [email protected] (V. Ramirez), [email protected] (R. Ortega), [email protected] (O. Bethoux), [email protected] (A. Sánchez-Squella). http://dx.doi.org/10.1016/j.conengprac.2014.02.005 0967-0661 & 2014 Elsevier Ltd. All rights reserved.

discriminate between quickly and slowly changing power demand profiles. For instance, due to physical constraints, it is not desirable to demand quickly changing power profiles to a fuel cell unit. Hence, the peak demands of the motors are usually supplied by the supercapacitors, whose time response is fast. To achieve this objective, a steady-state viewpoint is again adopted, and the current or voltage references to the multiports are passed either through low-pass or high-pass linear time-invariant (LTI) filters. The steady-state approach currently adopted in practice can only approximately fulfill the desired objectives of energy transfer and slow-versus-fast discrimination of the power demand. In particular, during the transients or when fast dynamic response is required, the delivery of demanded power in response to current or voltage references and the time response action of the filters might be far from satisfactory. Following the principles of control-by-interconnection (Ortega & van der Schaft, 2008) a new strategy to dynamically control the energy flow between lossless multiports, together with its corresponding power electronics implementation with standard circuit topologies was proposed in Sánchez-Squella, Ortega, Griñó, and Malo (2010). The device was called Duindam–Stramigioli Dynamic Energy Router (DS-DER) because, on one hand, it is inspired by the conceptual energy discrimination idea proposed in the context of walking robots in Duindam and Stramigioli (2004). While, on the other hand, in contrast to current practice, it does not rely on steady-state considerations. The DS-DER generates, via a nonlinear transformation, the references (voltages or currents) of all multiports that, under the assumptions of perfect tracking, ensures instantaneous energy transfer among multiports.

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As explained in Sánchez-Squella et al. (2010), the flow direction and rate of change of the energy transfer are regulated by means of some free parameters, which in the simplest two multiport case reduces to a single scalar function of time. The selection of these parameters is rather intuitive, and replaces the time-consuming task of selecting the LTI filters that (approximately) enforce the desired time scale separation between the multiports. Another feature that distinguishes the DS-DER with current practice is that, since the references of all interconnected multiports are generated in a centralized manner, information exchange among them is required, which is the operating scenario in some modern energy transfer applications, e.g., in microgrids. A key assumption for the correct operation of the DS-DER proposed in Sánchez-Squella et al. (2010) is that both, the multiports and the DER itself, are ideal lossless devices. Unfortunately, in this paper it is shown that in the presence of dissipation—which is, obviously, unavoidable in applications—the energy of the DER monotonically decreases leading to an improper behavior, and eventually total dysfunction, of the DS-DER. There are three objectives of this paper: (1) Propose a modified DER that overcomes this fundamental problem, with guaranteed stability properties. (2) To show that adding to the DS-DER an external energy source to compensate for the losses and an outer-loop PI provides excellent experimental results. As usual with simple engineering intuition-based control schemes, no theoretical basis for the performance improvement of this DER can be provided. (3) Compare via experiments the performances of the DS-DER, the new DER and the DS-DER plus battery configuration.

The paper is organized as follows. In Section 2, the energymanagement problem is formulated, and the classical procedure used for its solution is reviewed. In Section 3 we briefly review the DS-DER reported in Sánchez-Squella et al. (2010) and reveal its limitation in the presence of dissipation. Section 4 contains the new DER that overcomes this obstacle. Experimental results for the DS-DER and the new DER are presented in Sections 5 and 6, respectively. In Section 7 we present two ad hoc modifications to the DS-DER that were proposed to render it operative in spite of the presence of dissipation, the experimental evaluation of their performance is also given in that section. The paper ends with some concluding remarks and future research in Section 8.

2.1. The multiports It is assumed that the multiports, denoted by Σj, j A N ≔f1; …; Ng, have as port variables the terminal voltages and currents, which we denote as vj ðtÞ; ij ðtÞ A Rmj , respectively, see Fig. 1. It is also assumed that the multiports satisfy the energy-conservation law: Stored Energy ¼ Supplied Energy  Dissipated Energy: The following scenario is considered. (i) The stored energy is represented by a scalar function H~ j : Rnj -R, whose argument xj ðtÞ A Rnj is the state vector of the multiport. In an electrical circuit, xj(t) consists of electric charges in the capacitors and magnetic fluxes in the inductors. (ii) The power delivered by, or demanded from, the external environment is defined as P j ðtÞ ¼ vj> ðtÞij ðtÞ;

ð1Þ

with energy given by its integral. (iii) The dissipated power is a non-negative function denoted as dj : R þ -R þ . For instance, the power dissipated in a linear resistive element R is given by dðtÞ ¼ Ri2R ðtÞ; where R 40 is the value of the resistor and iR(t) is the current flowing through it. With this notation the energy-conservation law, in power form, becomes H_ j ðtÞ ¼ P j ðtÞ  dj ðtÞ;

ð2Þ

where H j ðtÞ≔H~ j ðxj ðtÞÞ. Integrating (2), and using (1), yields Z t Z t vj> ðsÞij ðsÞ ds  dj ðsÞ ds: H j ðtÞ  H j ð0Þ ¼ 0

0

Since dj ðtÞ Z 0, we have Z t vj> ðsÞij ðsÞ ds; H j ðtÞ  H j ð0Þ r

ð3Þ

0

reflecting the fact that the energy stored in the system cannot exceed the energy supplied from the environment, the difference being the dissipation. Notice that, in order to be able to treat multiports with sources, we have not assumed that the energy function is positive definite —or bounded from below. For instance, the dynamics of an ideal battery is given by x_ b ðtÞ ¼ ib ðtÞ 0 vb ðtÞ ¼ H~ ðxÞ; b

2. Formulation of the energy transfer problem In this section the mathematical formulation of the problem is given. We consider a system composed of N multiports interconnected, via (switch-regulated) power converters, to exchange energy according to a pre-specified energy-management policy.

where ðÞ0 denotes differentiation, and H~ b ðxb Þ ¼ V b xb is the (unbounded) energy, with V b A R þ the voltage of the battery. Clearly Z t H b ðtÞ ¼ V b ib ðsÞ ds; 0

which is the energy extracted from the battery. ~ If the energy function HðxÞ is positive definite, from (3) we obtain Z t  vj> ðsÞij ðsÞ dsr H j ð0Þ; 0

Fig. 1. Representation of a subsystem, such as fuel cell or battery, as a multiport, denoted by Σj, with port variables vj(t) and ij(t).

stating that the energy extracted from the multiport is bounded (by the initial energy), which is the usual characterization of passive systems (Ortega, Loria, Nicklasson, & Sira-Ramirez, 1998).

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

2.2. Standard energy management procedure

yields the power balance

The typical procedure to achieve the energy transfer is as follows (Choi et al., 2006; Malo & Griñó, 2007; Schenck et al., 2005; Thounthong et al., 2005). Assume that at a given time t 0 Z 0 a demand P ⋆ j of power is requested from multiport Σj. Measuring the voltage vj ðt 0 Þ, the power demand is then transformed into a constant current reference i⋆ j , solving the instantaneous power relation:

H_ T ðtÞ ¼ dI ðtÞ  ∑ dj ðtÞ:

⋆ > P⋆ j ¼ vj ðt 0 Þij :

ð4Þ

This current reference is imposed to the controller regulating the switches of the corresponding power converter, usually a PI loop, to drive to zero the current error ij ðtÞ i⋆ j . In this way, the desired energy-transfer objective is achieved asymptotically—provided the presumed steady-state behavior did not change. The following observations regarding the aforementioned strategy are in order. First, regulation towards the desired current value i⋆ j is, of course, not instantaneous, and during the transient the voltage vj(t) will change. Consequently, the actual power extracted (or supplied) to the multiport Σj will, in general, not coincide with P ⋆ j . Second, the strategy is intrinsically decentralized and neglects the loading effects that appear due to the interconnection of the multiports. To partially overcome this drawback, a second supervisory level control is added to achieve the coordination between the multiports power demands—whose successful operation relies on time-scale separation assumptions that are, partially, enforced via filtering. Both shortcomings are, to a certain point, palliated by the DER.

3. The Duindam–Stramigioli DER

3.1. The interconnection system In the DER the various power converters interconnecting the multiports are grouped together. It then defines a dynamical system with state ξ A RnI , energy function H~ I : RnI -R þ and N port variables vIj ðtÞ; iIj ðtÞ A Rmj that, being a physical system, also satisfies the energy conservation law N

N

j¼1

3.2. The dissipation obstacle of the DS-DER The basic assumption of the DS-DER proposed in SánchezSquella et al. (2010) is that the energy dissipated in the multiports and the interconnection system is negligible, that is, dI ðtÞ;

dj ðtÞ  0:

In this case, the power balance of each multiport becomes H_ j ðtÞ ¼ vj> ðtÞij ðtÞ:

ð7Þ

Consider, for simplicity, the case of two lossless multiports. Assume that at time t Z 0 it is desired to instantaneously transfer energy from multiport Σ1 to multiport Σ2. That is, to make H_ 1 ðtÞ 4 0;

H_ 2 ðtÞ o 0:

ð8Þ

The DS-DER of Sánchez-Squella et al. (2010) generates the current 1 references i⋆ j ðtÞ for both multiports to ensure (8), and is defined by #" " ⋆# " # 0 αðtÞv1 ðtÞv2> ðtÞ v1 ðtÞ i1 ¼ ; ð9Þ 0  αðtÞv2 ðtÞv1> ðtÞ v2 ðtÞ i⋆ 2 where α : R þ -R is a designer-chosen function. Substituting the current expressions of (9) into (7), and setting ij ðtÞ ¼ i⋆ j ðtÞ, yields H_ 1 ðtÞ ¼ αðtÞjv1 ðtÞj2 jv2 ðtÞj2

In this section we briefly review the DS-DER of Sánchez-Squella et al. (2010) and show its limitation in the presence of dissipation.

H_ I ðtÞ ¼ ∑ vIj> ðtÞiIj ðtÞ  dI ðtÞ;

25

ð5Þ

j¼1

where H I ðtÞ≔H~ I ðξðtÞÞ and dI : R þ -R þ is the dissipation. The multiports and the power converter device are coupled via the power-preserving interconnection: #" # " # " vIj ðtÞ 0  I mj ij ðtÞ : ð6Þ ¼ I mj 0 iIj ðtÞ vj ðtÞ

H_ 2 ðtÞ ¼  αðtÞjv1 ðtÞj2 jv2 ðtÞj2 :

ð10Þ

Clearly, if αðtÞ 4 0, then (8) is satisfied. The energy direction can be inverted setting αðtÞ o 0, when energy flows from Σ2 to Σ1. Therefore, as shown in Sánchez-Squella et al. (2010), αðtÞ controls the direction and rate of change of the energy flow, obviating the need of the LTI filters used in standard practice. Unfortunately, in the presence of dissipation the performance of the DS-DER (9) is asymptotically degraded and, eventually, ceases to be functional. Indeed, because of the skew-symmetry of the matrix in the right hand side of (9), multiplying (9) on the left by the row vector ½v1> ðtÞ v2> ðtÞ, and setting ij ðtÞ ¼ i⋆ j ðtÞ, yields v1> ðtÞi1 ðtÞ þ v2> ðtÞi2 ðtÞ ¼ 0:

ð11Þ

Replacing (11) in (5), and using (6), leads to H_ I ðtÞ ¼  dI ðtÞ;

ð12Þ

consequently, after convergence of the current tracking errors, the energy of the DS-DER monotonically decreases, leading to an improper behavior.

See Fig. 2. Defining the energy of the overall system N

H T ðtÞ ¼ ∑ H j ðtÞ þ H I ðtÞ; j¼1

4. A new DER with losses compensation To overcome the limitation mentioned in the previous section we propose to design the DER taking into account the presence of the dissipation in the interconnection subsystem dI(t), which we assume is measurable. Notice that, since the DER is composed of the power converters, a good estimate of the resistive elements is available.2

Fig. 2. Overall interconnected system for N ¼ 3.

1 The DER (9) is meant for current-tracking applications. See Sánchez-Squella et al. (2010) for a voltage-tracking version. 2 It should be underscored, however, that there are losses in power converters stemming from the switching devices that are hard to estimate.

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To streamline the problem formulation we define N mappings F j : Rr -Rmj , where r≔∑N j ¼ 1 mj , and the vectors v≔colðv1 ; …; vN Þ; FðvÞ≔colðF 1 ðvÞ; …; F N ðvÞÞ:

where δj : R þ -R are functions, chosen by the designer that, besides meeting the desired power dispatch policy, should satisfy N

∑ δj ðtÞ ¼ dI ðtÞ:

The mappings Fj(v) define the current references as

j¼1

i⋆ j ðtÞ ¼

Given the clear geometric interpretation of the DER, it is our contention that a suitable selection of the coefficients δj ðtÞ is simpler than the choice of the LTI filters in standard practice—see Sánchez-Squella et al. (2010). If the multiport voltages are bounded away from zero, that is jvj ðtÞj Zϵ 4 0, the coefficients δj ðtÞ may be selected as follows. Fix the desired power of N-1 multiports P ⋆ j ðtÞ, and assign to the N-th multiport the task of compensating for dI(t).3 That is, define

F j ðvðtÞÞ;

jAN;

and they should meet two different objectives. First, to ensure the desired power dispatch, they should satisfy > P⋆ j ðtÞ ¼ vj ðtÞF j ðvðtÞÞ;

where P ⋆ j : R þ -R is the power that we want to extract (or provide) to the j-th multiport—this information comes from a higher level energy dispatch system. Second, to compensate for the dissipation in the DER, the mappings should satisfy N

∑

j¼1

vj> ðtÞF j ðvðtÞÞ ¼

N1

⋆ P⋆ N ðtÞ≔dI ðtÞ  ∑ P j ðtÞ:

ð15Þ

j¼1

dI ðtÞ:

ð13Þ ij ðtÞ-i⋆ j ðtÞ,

Indeed, from (5) and (6), it is clear that if then the energy of the DER is regulated at a constant value, i.e., H_ I ðtÞ-0— avoiding the discharge phenomenon of the DS-DER. A geometric interpretation of the new DER formulation and the DS-DER is given in Figs. 3 and 4, respectively. Given a voltage vector v and the dissipation dI, the set F in Fig. 3 defines the admissible vectors F(v), that satisfy (13). In the figure are shown ⋆ two particular choices, which correspond to P ⋆ 1 o 0 and P 1 4 0. In the case of the DS-DER (9) that, using the notation above, takes the form " # αv1 jv2 j2 ; FðvÞ ¼  αv2 jv1 j2 the set F is orthogonal to v, as shown in Fig. 4. A possible choice for the mappings Fj(v) is given by 2 F j ðvÞ ¼ δj Π N k ¼ 1;k a j jvk j vj ;

ð14Þ

For j ¼ 1; …; N  1, let P⋆ j ðtÞ δj ðtÞ≔ N Π k ¼ 1 jvk ðtÞj2 while we fix

! N 1 ⋆ δN ðtÞ≔ N d ðtÞ  ∑ P ðtÞ : I j Π k ¼ 1 jvk ðtÞj2 j¼1 In this way, we (asymptotically) ensure the desired power dispatch, i.e., ij> ðtÞvj ðtÞ-P ⋆ j ðtÞ, while at the same time regulate the energy of the DER. Clearly, this strategy simply reduces to F j ðvj ðtÞÞ ¼

P⋆ j ðtÞ jvj ðtÞj2

vj ðtÞ;

ð16Þ

which is the solution of the time-varying version of Eq. (4), that corresponds to the current i⋆ j ðtÞ of smallest amplitude that delivers the desired power. It should be underscored that, besides the somehow minor fact that we are now generating time-varying references for the currents, another fundamental difference between the proposed DER and current practice is that all references are generated simultaneously.

5. Experimental results of the DS-DER 5.1. Implementation and model of a two-port DER

Fig. 3. Geometric interpretation of the new DER for N ¼2, with ei A R2 the i-th Euclidean basis vector.

To test experimentally the performance of the DS-DER we considered the configuration studied in simulation in SánchezSquella et al. (2010). That is, we implemented two multiports Σ1 and Σ2, which are two supercapacitors, interconnected via the DER as shown in Fig. 5. The energy functions of the supercapacitors are Cj H~ j ðvj Þ ¼ v2j ; 2

j ¼ 1; 2;

ð17Þ

where Cj are their capacitances, and vj(t) their voltages. Their dynamics are described by C j v_ j ðtÞ ¼ 

1 v ðtÞ þij ðtÞ; RC j

j ¼ 1; 2;

ð18Þ

where ij(t) are the currents, and RC is a parallel resistor. The power electronics scheme shown in Fig. 6 implements a two-ports DER. The port variables, ðvi ðtÞ; ii ðtÞÞ; i¼ 1,2, are indicated on both sides of the bidirectional converter. Applying Kirchhoff's laws over the different switched states of the circuit, and assuming

Fig. 4. Geometric interpretation of the Duindam–Stramigioli DER for N ¼2.

3 This policy is adopted just for simplicity, being possible also to distribute dI(t) among all (or some) of the multiports.

α(t)

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

Fig. 5. Interconnection of the multiports, chosen as leaky supercapacitors.

27

0.01 0.008 0.006 0.004 0.002 0 −0.002 −0.004 −0.006 −0.008 −0.01 0

1

2

3

4

5

6

7

8

9

10

11

Time (s) Fig. 8. Time evolution of αðtÞ, which controls the energy direction and exchange rate.

Fig. 6. Power electronics configuration to implement a two-port DER.

The nominal value of the DC link voltage is v⋆ C ¼ 20 V, and the initial voltage condition of v1 ðtÞ and v2 ðtÞ is 10 V. It is well-known that, for a suitable operation of this kind of power electronic device, the voltage vC(t) should not decrease below a certain level (Erickson & Maksimovic, 2004), which in this case is about 17.5 V. All the parameters of the experimental implementation are shown in Table 1. 5.2. Energy management policy The following energy management scenario for the DS-DER was considered.

 In the interval 0 r t o 1 s, there is no energy exchange between ⋆ the multiports, which corresponds to P ⋆ 1 ¼ P 2 ¼ 0 W.

Fig. 7. Photograph of the implemented test bench. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Value

L1 ; L2 ; Lb R1 ; R2 ; Rb C1 ; C2 CC RC Vb Switching frequency

195,193,210 μH 0.05 Ω 58 F 1.05 mF 1 MΩ 10 V 20 kHz



a sufficiently fast sampling time, the average dynamics of the DER interconnected to the multiports are given by L1

di1 ðtÞ ¼  R1 i1 ðtÞ  vC ðtÞu1 ðtÞ þ v1 ðtÞ dt

L2

di2 ðtÞ ¼  R2 i2 ðtÞ  vC ðtÞu2 ðtÞ þ v2 ðtÞ dt

CC

dvC ðtÞ ¼ u1 ðtÞi1 ðtÞ þ u2 ðtÞi2 ðtÞ; dt

and this remains until t ¼ 4:4 s, at a power rate P ⋆ 1 ¼ 100 W.

 For t 4 4:4 s a quick reversal of the energy flow is desired, 

Table 1 Parameters of the power electronics implementation. Component

 At t ¼ 1 s a slow transfer of energy from Σ1 to Σ2 is requested, remaining unchanged until t ¼ 7:4 s, now at a power rate P⋆ 2 ¼ 100 W. For t 4 7:4 s another quick reversal of the energy flow is desired, remaining unchanged until t ¼ 10:05 s, but now at half the preceding power value, that is P ⋆ 1 ¼ 50 W. Finally, for t 4 10:05 s the energy flow is instantaneously stopped until the end of the trial at t ¼ 11 s.

A profile of the function αðtÞ that implements this strategy is shown in Fig. 8. Notice that the first slope is smaller than the second and third, reflecting the desire to carry out a slower energy transfer. On the other hand, the fourth slope, at t¼ 10.05 s, is very large to implement a quick stop in the energy transfer. The numerical values of αðtÞ are computed, using (10) with the nominal voltages, to deliver the desired powers indicated above. 5.3. Current tracking via input–output linearization

ð19Þ

where i1 ðtÞ and i2 ðtÞ are the inductors currents, vC(t) is the voltage in the DC link, R1 and R2 , are the series resistances of the inductors, and u1 ðtÞ; u2 ðtÞ A ð0; 1Þ are the duty cycles of the switches, which are the control signals. The overall dynamics is hence described by the fifthorder system (18) and (19). The experimental setup is shown in Fig. 7. From left to right, we can see a black box with PWM modulator, behind it the MOSFETS of the DER. To the right a box with the currents sensor and next to it the three inductors (round shaped) and the two supercapacitors (blue). The battery lies below a big fuel cell in the back.

The current references for the DS-DER are defined in (9), which can be rewritten as, " ⋆ # " # αðtÞv1 ðtÞv22 ðtÞ i1 ðtÞ ¼ : αðtÞv2 ðtÞv21 ðtÞ i⋆ 2 ðtÞ The design is completed adding a control strategy for the system (18) and (19) that ensures the tracking of the current references. The problem of controller design for power converter systems has been extensively studied in the power electronics and control literature, see e.g., Ortega et al. (1998), Erickson and Maksimovic (2004), Hernandez-Gomez, Ortega, Lamnabhi-Lagarrigue, and Escobar (2010), and references therein. To remain as close as possible to the ideal tracking situation, it was assumed that the dynamics of the loads are perfectly known, and an input–output linearization of the whole system was implemented. Towards this end, we take

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

100 80 60 40 20 0 −20 −40 −60 −80 −100

Power (W)

Power (W)

28

0

1

2

3

4

5

6

7

8

9

100 80 60 40 20 0 −20 −40 −60 −80 −100 0

10

100

200

300

400

800

900

1000

600

700

800

900

1000

600

700

800

900

1000

0.5

0

Error (A)

Error (A)

700

1

0.5

−0.5

0 −0.5 −1

−1

−1.5

−1.5

−2 0

1

2

3

4

5

6

7

8

9

10

0

100

200

300

400

21.5 21

Voltage (V)

20.5 20 19.5 19 18.5 18 17.5 0

1

2

3

4

5

6

7

8

9

10

20 19 18 17 16 15 14 13 12 11 10

0

100

200

300

500

Fig. 10. Experimental results of the DS-DER (30)–(32) in a long time window: (a) P 1 ðtÞ; P 2 ðtÞ; (b) i~1 ðtÞ; i~2 ðtÞ; (c) vC(t).

To complete the design the new inputs wj(t) are taken as PI controllers around i~j ðtÞ, that is, Z t ~i ðsÞ ds; j ¼ 1; 2; wj ðtÞ ¼  kpj i~j ðtÞ  kij ð22Þ j

as system outputs the tracking errors ~i 1 ðtÞ ¼ i1 ðtÞ  αv1 ðtÞv2 ðtÞ 2 ~i 2 ðtÞ ¼ i2 ðtÞ þ αv2 ðtÞv2 ðtÞ; 1

400

Time (s)

Time (s) Fig. 9. Experimental results of the DS-DER (30)–(32) in a short time window: (a) P 1 ðtÞ; P 2 ðtÞ; (b) i~1 ðtÞ; i~2 ðtÞ; (c) vC(t).

500

Time (s)

Time (s)

Voltage (V)

600

1.5

1

17

500

Time (s)

Time (s)

0

ð20Þ

with kpj ; kij some positive tuning gains. This yields the exponentially stable dynamics

that we want to drive to zero. Some simple calculations show that the system (18) and (19), with outputs (20) and inputs u1 ðtÞ; u2 ðtÞ, has a well-defined relative degree ð1; 1Þ and can be input–output linearized with the control law

2 d i~j

   L1  R1 i1 þ v1 αv22 v1 þ i1 þ vC L1 C1 RC1     L1 2αv1 v2 v2 i2 þ  w1 þ vC C2 RC2    L2  R2 i2 þ v2 αv21 v2  i2 þ u2 ¼ vC L2 C2 RC2     L2 2αv1 v2 v1  i1 þ  w2 ; vC C1 RC1

5.4. Effect of dissipation on the DS-DER

u1 ¼

ð21Þ

where, w1 ðtÞ; w2 ðtÞ are the new input signals. That is, the closedloop system takes the simple linear form d~i j ðtÞ ¼ wj ðtÞ; dt

j ¼ 1; 2:

dt

2

ðtÞ þ kpj

d~i j ðtÞ þkij ~i j ðtÞ ¼ 0; dt

j ¼ 1; 2:

The transient performance of the DS-DER with the input– output feedback linearizing controller (20)–(22) is depicted in Fig. 9. As seen from the figure the current tracking errors are kept small, the power transfer is done in the desired direction and requested rate of change. Moreover, the DC link voltage vC(t) is kept within reasonable values. However, we observe that since the power dissipated in the DER is not compensated, the values of the capacitors powers tend to decrease with time. The power loss trend is more clearly seen in Fig. 10, which corresponds to a much longer experimentation time. Notice that the current tracking errors are still kept small,4 however the DC 4 Actually, a similar behavior was observed for other current tracking controllers, e.g., PI-based.

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

29

Power (W)

100 50 0 −50 −100 0

1

2

3

4

5

6

7

8

9

10

11

Time (s) Fig. 11. Power electronics implementation of the new DER router with a battery as a third multiport.

link voltage decreases to a level where the device ceases to be operational. This deleterious behavior was not observed in the simulations of Sánchez-Squella et al. (2010) were a larger capacitor was used in the DC link, whose discharge time was much larger than the considered time horizon.

Fig. 12. Time evolution of P ⋆ 1 ðtÞ.

Table 2 Gains of the PI controllers. Controller

kpj

kij

u1 u2 u3

0.20224 0.20224 0.25101

0.00143 0.00143 0.00112

6. Experimental results for the new DER In this section the new DER proposed in Section 4 is tried in experiments. To compensate for the losses in the DER we add a third multiport that consists of a simple battery, whose control is fixed by the energy management policy described in Section 4.

one supercapacitor is transferred to the other—with the profiles and magnitudes specified before—while the battery provides the energy dissipated in the DER. That is, we select

6.1. Implementation and model of a three-port DER

where the dissipation in the DER is computed from its mathematical model, that is,

The power electronics scheme shown in Fig. 11 implements the three port DER with the aforementioned battery. The average dynamics of the circuit, terminated with the supercapacitors and the battery, includes now a third state and a third control signal, and are given by di1 ðtÞ ¼  R1 i1 ðtÞ  vC ðtÞu1 ðtÞ þ v1 ðtÞ dt di2 L2 ðtÞ ¼  R2 i2 ðtÞ  vC ðtÞu2 ðtÞ þ v2 ðtÞ dt di3 Lb ðtÞ ¼  Rb i3 ðtÞ vC ðtÞu3 ðtÞ þv3 ðtÞ dt dvC ðtÞ ¼ u1 ðtÞi1 ðtÞ þ u2 ðtÞi2 ðtÞ þ u3 ðtÞi3 ðtÞ; CC dt L1

ð23Þ

x_ 3 ðtÞ ¼ i3 ðtÞv3 ðtÞ ¼ V b ; where V b A R þ is the voltage of the battery. The overall dynamics of the DER interconnected to the multiports are then given by the sixth-order system (18) and (23), with v3 ðtÞ ¼ V b . 6.2. Energy management policy The system is operated keeping the multiport voltages bounded away from zero. Consequently, the current references are selected according to the simple formula (16) that, in the present scalar case, reduces to P⋆ j ðtÞ vj ðtÞ

;

j ¼ 1; 2; 3:

P⋆ 3 ðtÞ ¼ dI ðtÞ;

dI ðtÞ ¼ R1 i21 ðtÞ þ R2 i22 ðtÞ þ R3 i23 ðtÞ:

ð25Þ

ð26Þ

P⋆ 1 ðtÞ

that implements this strategy—that A profile of the function obviously mimics αðtÞ—is shown in Fig. 12. The remaining task is the design of a control strategy for the system (18) and (23) that ensures the tracking of the current references defined in (24). Notice that the only parameters needed for the definition of the current references are the DER resistors Rj, which we assume are known. 6.3. Current tracking via linear PI control

where i3 ðtÞ; V 3 are the current and voltage of the third port (the battery), respectively, and u3 ðtÞ A ð0; 1Þ is the duty cycle of the additional switch. As explained in Section 2.1, the dynamics of the battery is given by

i⋆ j ðtÞ ¼ F j ðvj ðtÞÞ ¼

⋆ P⋆ 1 ðtÞ ¼  P 2 ðtÞ;

In this case each converter switch is regulated via a PI controller, that is Z t ~i ðsÞ ds; j ¼ 1; 2; 3: uj ðtÞ ¼  kpj i~j ðtÞ  kij ð27Þ j 0

The controllers gains are selected using standard linear control techniques (Joós & Espinoza, 1999; Ogata, 2003) based on the linearization of the system around an operation point and trying to enforce a time-scale separation between the loops, and are summarized in Table 2. Using the same test bench of Fig. 7 experiments were carried out and the results are shown in Fig. 13. As seen in the figure the desired power transfer between the supercapacitors is ensured, while the battery transfers the power required to compensate for the DER losses and the DC link voltage is kept within reasonable values. We also observe that the current tracking errors are larger than the ones obtained for the DS-DER (see Fig. 9(b).) This seems to be due to the fact that a simple PI, without feedback linearization, was implemented—see the next subsection.

ð24Þ

To illustrate the capabilities of the new DER to transfer the energy between the multiports we considered the same energy management scenario of the previous section, but with the following essential modification. During certain period of time the energy of

6.4. Current tracking with an approximate feedback linearizing controller In spite of the excellent performance achieved by the PI control, for the sake of completeness, an approximate input–output feedback

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

Power (W)

30

differences were observed in the behavior of the DC link voltage and the current tracking errors, which exhibited smaller deviations with respect to their desired value. Keeping tighter regulation of the DC link voltage is of paramount importance in applications where small capacitors are used and may justify the use of the, admittedly more complex, linearizing control strategy.

100 80 60 40 20 0 −20 −40 −60 −80 −100 0

1

2

3

4

5

6

7

8

9

10

Error (A)

Time (s)

7. Ad hoc modifications to the DS-DER Although the new DER showed remarkable performance that can be theoretically justified, it requires the knowledge of the losses in the DER, which are difficult to model in a switching device (see Footnote 2). For this reason it is interesting to try other practically motivated options to render operative the original DS-DER. A first attempt was the standard solution of nested PI loops to drive to zero the DC voltage error:

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

v~ C ðtÞ≔vC ðtÞ  v⋆ C;

0

1

2

3

4

5

6

7

8

9

10

Time (s)

That is, the inputs (22) are replaced by Z t ~i ðsÞ ds wj ðtÞ ¼  kpj i~j ðtÞ  kij j 0

 kpv v~ C ðtÞ  kiv

Z

Voltage (V)

v~ C ðsÞ ds;

j ¼ 1; 2;

ð29Þ

0

21.5 21 20.5 20 19.5 19

t

0

1

2

3

4

5

6

7

8

9

10

Time (s) Fig. 13. Experimental results of the new DER (24)–(27): (a) P 1 ðtÞ; P 2 ðtÞ; P 3 ðtÞ; (b) i~1 ðtÞ; i~2 ðtÞ; i~3 ðtÞ; (c) vC(t).

where we notice the addition of an outer-loop PI in the voltage errors. This new controller was experimentally tested showing a marginal improvement with respect to the control (22), in the sense that the decay to zero of the DC link voltage took a longer time. Clearly, this phenomenon is unavoidable without the inclusion of a additional energy to compensate the losses in the DER. A second alternative is to add an external regulated battery, as done in the new DER, but not to treat it as an additional multiport. Instead, the battery is regulated via nested current and voltage PIs —a configuration that is standard in applications. That is, Z t ~ i⋆ ðtÞ ¼  k ðtÞ  k v v~ C ðsÞ ds pv C iv b 0

linearization controller for the new DER was also tested. This is given by " # ⋆ dij γj 1 uj ðtÞ ¼ vj ðtÞ Rj ij ðtÞ Lj ðtÞ þ ½L ~i ðtÞ; j ¼ 1; 2; 3 ð28Þ vC ðtÞ dt vC ðtÞ j j where γ j 40 are tuning parameters. Indeed, replacing (28) in (23) yields the simple linear, exponentially stable, system d~i j ðtÞ ¼  γ j i~j ðtÞ; dt

j ¼ 1; 2; 3;

which implies that the current-tracking errors converge to zero exponentially fast, at a rate determined by γj, achieving the desired objective. The only parameters needed for the implementation of (28) are Rj ; Lj , which are in the DER, hence are reasonably well-known. On the ⋆ other hand, the control requires the terms ðdij =dtÞðtÞ. Recalling that ⋆ ij ðtÞ is defined by (24), it is clear that to compute its derivative the dynamics of the multiports, i.e., (18), must be taken into account. Besides the fact that the resulting control law becomes extremely involved, in a practical scenario the multiports dynamics are highly ⋆ uncertain. Therefore, we propose to obtain ðdij =dtÞðtÞ with an approximate differentiation filter: WðsÞ ¼

bs : τs þ1

The power transfer behavior of the new controller observed in the experiments was almost identical to the PI scheme of the previous subsection—therefore, the plots are omitted for brevity. However,

~i ðtÞ ¼ i ðtÞ  i⋆ ðtÞ b b b u3 ðtÞ ¼  kpb ði~b ðtÞÞ  kib

Z

t 0

ði~b ðsÞÞ ds:

ð30Þ

Moreover, we propose to add to the reference signals generated by the DS-DER the reference signal i⋆ b ðtÞ weighted by a switch that decides the direction of the flow of the battery current as a function of the sign of the parameter αðtÞ—we refer in the sequel to this scheme as directional DER.5 This leads to the following new definition of the supercapacitor current references: ½1  signðαðtÞÞi⋆ b ðtÞ 2 ½1 þ signðαðtÞÞi⋆ 2 b ðtÞ : i⋆ 2 ðtÞ ¼  αðtÞv1 ðtÞv2 ðtÞ  2

2 i⋆ 1 ðtÞ ¼ αðtÞv1 ðtÞv2 ðtÞ 

ð31Þ

The directional DER was implemented with linear PI controllers for the two multiports, that is Z t ~i ðsÞ ds; j ¼ 1; 2: uj ðtÞ ¼  kpj i~j ðtÞ  kij ð32Þ j 0

Controller gains that are used for the experiments are kp1 ¼ kp2 ¼ 0:2151, ki1 ¼ ki2 ¼ 0:0012, kp3 ¼ 0:2511, ki3 ¼ 0:001, kpb ¼ 0:2133 and kib ¼ 0:00127. Experimental results are shown in Fig. 14, which illustrates the excellent behavior obtained with this controller that, unfortunately, cannot be theoretically analyzed. 5 The authors thank Professor Robert Griño from Catalonian Polytechnical University for this suggestion.

V. Ramirez et al. / Control Engineering Practice 27 (2014) 23–31

8. Conclusions A practical limitation of the DS-DER reported in SánchezSquella et al. (2010) was identified in this paper. Namely, due to the power-preserving nature of the DS-DER, the energy of the interconnection system—that is implemented with power electronic devices—decreases asymptotically in the presence of dissipation, rendering the DS-DER asymptotically dysfunctional, see (12). To overcome this obstacle it was proposed to drop the powerpreserving feature of the DS-DER and a new DER, that takes into account the losses, was proposed. The new DER was tested in experiments using a simple PI scheme and an (approximate) input–output linearizing controller. The performance in both cases was excellent with the latter controller achieving, at the prize of higher complexity, a better regulation of the DC link voltage. Besides the new DER, two ad hoc modifications to the DS-DER were proposed, and tested in a experimental Test bench: adding an outer-loop PI regulator for the DC link voltage, and providing energy to the DER with an external source, which is controlled with switched nested PI loops. The first alternative turned out to be inadequate, both from the energy management and voltage regulation perspectives. On the other hand, adding an external battery effectively removed the problem, but its realization is restricted to the particular implementation of the DER proposed in the paper—which clearly can be implemented with various converter topologies. This should be contrasted with the new DER, whose implementation requires only the knowledge of the losses in the DER, and is not restricted to a particular topology of the power electronic device. The new DER has recently been implemented in a realistic fuelcell based system available in our laboratory and the results will be reported elsewhere. Towards this end, novel multiport converter topologies were required. On the theoretical side, a question that remains to be addressed is the robustness of the new DER, in particular, vis-à-vis uncertainty in the dissipation function dI(t)— whose value is needed to define the references. Further experiments have shown that the actual dissipation may significantly

31

differ from the one predicted by the lumped parameter model. Hence, to enhance its robustness, an adaptive version of the DER must be worked out. Another interesting, though hard, theoretical question is the stability analysis of the directional DER. Invoking time scale separation arguments this analysis seems feasible. However, this is not the scenario that was observed in our experimentation where, to obtain good performance, both loops must operate at the same time scale. References Choi, W., Howze, J. W., & Enjeti, P. (2006). Fuel-cell powered uninterruptible power supply systems: Design considerations. Journal of Power Sources, 157, 311–317. Duindam, V., & Stramigioli, S. (2004). Port-based asymptotic curve tracking for mechanical systems. European Journal of Control, 10(5), 411–420. Erickson, R., & Maksimovic, D. (2004). Fundamentals of power electronics. New York: Kluwer Academic Publishers. Fraghani, H. (2010). The path of the smart grid. IEEE Power and Energy Magazine, 8(1), 18–28. Hernandez-Gomez, M., Ortega, R., Lamnabhi-Lagarrigue, F., & Escobar, G. (2010). Adaptive pi stabilization of switched power converters. IEEE Transactions on Control Systems Technology, 18(3), 688–698. Joós, G., & Espinoza, J. (1999). Three-phase series var compensation based on a voltage-controlled current source inverter with supplemental modulation index control. IEEE Transactions on Power Electronics, 14(3), 587–598. Malo, S., & Griñó, R. (2007). Design and construction of an electric energy conditioning system for a pem type fuel cell. In Proceedings of the 33rd annual conference of the IEEE industrial electronics society (IECON07) (pp. 1633–1638). Taipei, Taiwan. Ogata, K. (2003). Modern control engineering. New York: Prentice-Hall. Ortega, R., van der Schaft, A., Castaños, F., & Astolfi, A. (2008). Control by statemodulated interconnection of port-hamiltonian systems. IEEE Transactions on Automatic Control, 53(11), 2527–2542. Ortega, R., Loria, A., Nicklasson, P. J., & Sira-Ramirez, H. (1998). Passivity-based control of Euler–Lagrange systems. Berlin, Germany: Springer-Verlag. Sánchez-Squella, A., Ortega, R., Griñó, R., & Malo, S. (2010). Dynamic energy router. IEEE Control Systems Magazine, 30(6), 72–80. Schenck, M.E., Lai, J., & Stanton, K. (2005). Fuel cell and power conditioning system interactions. In Proceedings of the 20th annual IEEE applied power electronics conference and exposition (APEC05) (pp. 114–120). Austin, Texas, USA. Thounthong, P., Raël, S., & Davat, B. (2005). Utilizing fuel cell and supercapacitors for automotive hybrid electrical system. In Proceedings of the 20th annual IEEE applied power electronics conference and exposition (APEC05) (pp. 90–96). Austin, Texas, USA.