Semidefinite Relaxation-Based Optimization of Multiple-Input Wireless Power Transfer Systems

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by  Hans-Dieter Lang

IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1
Semidefinite Relaxation-Based Optimization of
Multiple-Input Wireless Power Transfer Systems
Hans-Dieter Lang, Student Member, IEEE, and Costas D. Sarris, Senior Member, IEEE
Abstract—An optimization procedure for multi-transmitter they converge to the global optimum, despite their significant
(MISO) wireless power transfer (WPT) systems based on tight computational cost.
7 semidefinite relaxation (SDR) is presented. This method ensures Recently, the fundamental physical limits of such MISO
physicalrealizabilityofMISOWPTsystemsdesignedviaconvex
1 WPTsystemsweredeterminedusingconvexoptimization[6].
optimization—a robust, semi-analytical and intuitive route to
0
ThemaximumachievablePTEaswellasallrelevantelectrical
optimizing such systems. To that end, the nonconvex constraints
2
requiring that power is fed into rather than drawn from the parameters (such as currents, resistive and reactive loading,

n system via all transmitter ports are incorporated in a convex etc.)foroptimaloperationwerederivedinclosedform.These
a semidefinite relaxation, which is efficiently and reliably solvable closed-form results, which are briefly reviewed in Sec. II, are
J by dedicated algorithms. A test of the solution then confirms
highly valuable, very intuitive and physically meaningful.
4 that this modified problem is equivalent (tight relaxation) to the
However, it still possible that insufficiently constrained
2 original (nonconvex) one and that the true global optimum has
been found. This is a clear advantage over global optimization optimization of WPT systems may lead to solutions where

] methods (e.g. genetic algorithms), where convergence to the true power is drawn from the system via some transmitter ports,
C global optimum cannot be ensured or tested. Discussions of instead of being fed into it. Such solutions are mathematically
O numericalresultsyieldedbyboththeclosed-formexpressionsand
valid optima, whose performance (high PTE) is owed to the
therefinedtechniqueillustratetheimportanceandpracticability
. “recycling” of power among transmit ports suppressing the
h of the new method. It, is shown that this technique offers a
t rigorous optimization framework for a broad range of current totaltransmitpower.Yet,theyareundesirableandimpractical
a
and emerging WPT applications. from a physical point of view.
m
Index Terms—Convex Optimization, Multiple Transmitters, Hence, the formulation of alternative techniques for the op-
[
Power Transfer Efficiency, Semidefinite Programming, Semidef- timizationofMISOWPTsystemsisstronglymotivated,where

1 inite Relaxation, Wireless Power Transfer. such operating conditions are avoided from the beginning.
v As will be discussed in detail, adding such transmit power
4
I. INTRODUCTION constraints to the original optimization problem is not trivial,
4
3 WIRELESS power transfer (WPT) systems have come astheyarenonconvex;therebybreakingconvexityoftheentire
7 a long way, since their origins over a century ago [1] optimizationproblem.Asaresult,theasssociatedoptimization
0 and their more recent rediscovery [2]. Among many other ad- problem is unsolvable in general, as the computational cost
1. vancementsanddevelopments,multi-transmitterWPTsystems grows rapidly when increasing the number of transmitters.
0 have been investigated, both theoretically and experimentally In the following, a MISO WPT optimization strategy based
7 [3]–[6]. Leveraging the additional design parameters offered on tight semidefinite relaxation is presented, whereby the
1 bytheuseofmultipletransmitters,thesemultiple-inputsingle- nonconvex problem is reformulated into a convex form with
:
v output (MISO) WPT systems have been shown to outperform slightlylooserconstraints,whichcanbeefficientlyandreliably
Xi their single-input single-output (SISO) counterparts. A key solved by dedicated algorithms. A simple test of the solution
aspect of the improved performance of MISO over SISO then proves that this relaxation is tight, i.e. fully equivalent
r
a WPT systems is the natural resilience of the former to the to the original problem and that the true global optimum was
effects of transmitter-receiver misalignment, which are known found. With this framework, these nonconvex transmit power
toseverelycompromisethepowertransferefficiency(PTE)of constraintscanbeefficientlyincludedintheoptimizationpro-
SISOWPTsystems.Ontheotherhand,theincreaseindegrees cess,ensuringthepracticalrealizabilityoftheproposedMISO
of freedom of such systems also makes it challenging to find WPTsystems.Notably,thisnewoptimizationframeworkisfar
their optimum operating mode and estimate their maximum more general, powerful and versatile than the one previously
achievable performance. In such cases, global optimization provided in closed form [6], while retaining its mathematical
methods such as Genetic Algorithms (GA) are commonly rigor.
used, see for example [7]; however, there is no guarantee that This paper begins with a brief introduction and a review
of the closed-form expressions [6]. Then, it is shown in
ManuscriptreceivedSeptemberxx,2016;revisedMonthxx,2017;accepted detail how the additional constraints are incorporated into
Monthxx,2017.Dateofpublication:Monthzz,2017.Currentversion:00
the optimization procedure, how its semidefinite relaxation
The authors are with the Electromagnetics Group of the Department of
ElectricalandComputerEngineering,UniversityofToronto,ON,M5S3G4, is derived and how the optimization result can be tested
Canada,(e-mail:[e-mail]). for tightness/equivalence to the original nonconvex problem.
Color versions of one or more of the figures in this paper are available
Finally, simulation results demonstrate the practicality of the
onlineathttp://ieeexplore.ieee.org.
DigitalObjectIdentifierXX.YYYY/TMTT.2017.abcdef proposed method and the importance of including nonconvex
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 2
transmit power constraints to obtain practically realizable B. The MISO WPT System Model
MISO WPT systems. Let the (unloaded) impedance matrix Z of the MISO WPT
Remarks on the notation: Italicized thin letters represent
system be partitioned according to
scalars, bold small letters refer to vectors, and bold capital
letters are matrices; vT stands for the transpose of v, while
vH stands for its Hermitian (conjugate transpose). Real and  Zt ztr
imaginary parts of complex quantities are marked by () and Z= CN×N , (2)
0 ∈
() , respectively, i.e. α = α +jα . The complex con·jugate  
is· 0d0enoted by a bar (overlin0e). Th0e0 symbols ( ) and  zTtr zr 
 
( ) are used to denominate positive (semi-) de(cid:31)finit(cid:23)eness an≺d where the subscripts t and r refer to the transmitter and
(cid:22)
negative (semi-) definiteness of matrices, respectively. For receiver parts, respectively, and tr stands for the elements
vectors,theystandforelementwisepositive(nonnegative)and coupling the former to the latter.
negative(nonpositive)entries.Astar()? markstheoptimized The diagonal of Z and z refer to the loss resistances and
· t r
arguments that lead to the optimal solution of an optimization self-reactances of the transmitters and receiver, respectively.
problem. Typically, z ,z > 0 (inductive) when considering systems
n00,n r00
Remarks on the terminology: ‘Program’ is a synonym for of magnetically coupled loops or coils. Likewise, the off-
‘optimization problem’, commonly used in the context of diagonalentriesofZ andthecouplingvectorsz containthe
t tr
mathematical optimization [8]. Further, in this paper, ‘feasi- mutual inductances jωM . Generally, each M is com-
n,m n,m
bility’ will refer to satisfying constraints within a program; plex, due to retardation effects when the electrical distances
i.e. a feasible solution satisfies all required constraints. between the transmitter(s) and/or receiver are not very small.
For physical reasons, impedance matrices have to be
II. PRELIMINARIES positive-real [11]: The real parts of all matrices and subma-
A. Power Transfer Efficiency (PTE) trices have to be positive-definite, i.e. Z0,Z0t (cid:31)0 and zr0 >0.
Positivedefinitenessofamatrixmeansthatthequadraticform
The central figure of merit when optimizing WPT systems
of such a matrix is always positive:
is the power transfer efficiency (PTE) [3]–[6], [9], [10]: the
ratio of the power PL transferred to the load RL to the total iHZ0i>0 ∀i∈CN (3)
transmit power provided by the source P
t This mathematical property represents the fact that there is
PL PL RL always some amount of loss in the system. Positive semidef-
η = = = . (1)
P P +P R +R initeness ( 0) implies nonnegative ( 0) quadratic forms
t l L l L
(cid:23) ≥
and for negative (semi-) definiteness the opposite signs and
P isthetotalpowerabsorbedbythesystem,duetodissipation
l
directions apply.
andradiation,modeledbythelossresistanceR ,asillustrated
l
The process of adding reactances to each of the transmitter
in Fig. 1.
and receiver nodes as well as a resistive load to the receiver
P will be referred to as loading of the WPT system, where Zˆ is
l
R the resulting loaded impedance matrix:
l
Zˆ =Z+jX+R . (4)
L
P η R P
t L L
The real-valued diagonal matrix X contains the load reac-
tances (positive and negative values referring to inductive
and capacitive elements, respectively). The matrix R is zero
L
Transmitter(s) PowerTransferSystem Receiver(load) everywherebutatthelastdiagonalentry,correspondingtothe
receiver, where the load resistance R >0 is located.
L
Fig.1. IllustrationofgeneralpowertransfersystemsandthePTEasdefined
Indetail,thevoltagesandcurrentsv,iofthewhole(loaded)
in (1): The ratio of the power transferred to the load, PL, and the total
transmitted power, Pt. Loss (e.g. due to conduction losses and radiation) is WPT system are related according to
modeledbytheresistanceRl.
Zˆ
Note that, within the context of this paper, PTE refers only
totheelectromagneticpowertransferefficiencyofthesystem. v Z z X 0 0 0 i
t = t tr +j t + t
AnypracticalrealizationofaWPTsystemwillhaveanend-to- v =0 z zT z 0}|x 0 R { i
(cid:20) r (cid:21) (cid:20) tr r(cid:21) (cid:20) r(cid:21) (cid:20) L(cid:21)!(cid:20) r(cid:21)
end efficiency which is limited by this PTE; signal generation
and impedance matching on the transmitter side as well as v Z X R i
L
matching and rectification will lower the overall performance, (5)
| {z } | {z } | {z } | {z } |{z}
but are not considered here. Further, R is only practically where,similarlytotheimpedancematrices,thesubscripttand
L
realizable from a certain resistance level on. However, in this r refer to the transmitter and receiver voltages and currents,
paper the maximum achievable performance in view of the respectively.SincetheloadresistanceR isactuallypartofthe
L
electromagnetic PTE is of central interest, which includes the impedance matrix, KVL states that the corresponding receiver
question of the optimal load resistance. voltage is zero, i.e. v =0, as illustrated in Fig. 2.
r
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 3
LoadedWPTS: Zˆ=Z+jX+RL and the resulting PTE is given by
1
UnloadedWPTS: Z η = . (8)
max
P +1
C TX C l,min
i 1 RX r
1 Zt zr In order to only have to deal with real variables in the fol-
i
v1 ztr vrr=0 lowingsteps,thecomplextransmittercurrentsit areseparated
into real and imaginary parts i0t and i0t0, respectively:
RL c = i0t . (9)
t (cid:20)i0t0(cid:21)
C2 CN 1 To satisfy KVL at the receiver end, i.e. ensuring v =0 as
− r
in (5) and illustrated in Fig. 2, the following linear constraint
i2 iN−1 (affine [8] in ct) has to hold:
Fig.2. Loop-basedMISOv2WPTsystemmvoNde−l:1Corestructurewithunloaded (cid:20)zz0t0t0rrTT −zz0t0trT0rT(cid:21)ct+rR2L (cid:20)zzˆˆr0r00(cid:21)=0, (10)
impedance matrix Z (obtained via simulation or measurements), reactive where zˆ = z + R + jx is the loaded receiver self-
cvonmapdodneednttso,htheereloxande=d i−m(pωeCdann)c−e1m,artersiixstZiˆv.eNlooatdeRthLataantdthveolrteacgeeivseoruercneds, impedancre.SincrethereLceiverrreactancexr isafreeparameter,
sinceRL ispartofZˆ,thevoltageatthecorrespondingportiszero,vr =0. it can always be chosen so as to automatically satisfy the
second constraint for the optimal transmitter currents c?,
t
toTfiondmoapxtiimmiazlevtohletapgeersfovrmanandcceuorrfensutschi (sryesatlemansd, tihmeagaiimnariys x?r =−zr00− R2L z0t0rT z0trT c?t . (11)
r
parts) as well as loading components x = xT,x T = (cid:2) (cid:3)
t r Thus, the second row of the constraints (10) can be removed.
diag(X), R , which maximize the PTE, obtained according
L Using the objective (7) in terms of the real currents (9) in
(cid:2) (cid:3)
to its definition (1) as
combinationwiththeremainingconstraintof(10),theprogram
1iHR i iHR i to solve for optimal wireless power transfer to a specific load
η = 2 L = L . (6) R at the receiver can be given in detail as
12(iHv)0 iH(Z0+RL)i L
Aiaslosente.oTuthaebvoovleta,gtheesvPTfEollcoawnfbreomgiv(e5n),ibnuttearmresoonflythpehcyusrirceanlltys Pl,min =mcitn 12cTt (cid:20)Z00t Z00t(cid:21)ct+rR2L z0trT 0 ct+ RzrL
(cid:2) (cid:3)
meaningful as long as vr =0 is ensured. s.t. z T z T c = 2 (z +R )
0tr 0t0r t − R r0 L
r L
(12)
(cid:2) (cid:3)
C. Review of the Convex Optimization of MISO WPT Systems
The final optimization problem (12) constitutes a convex
As pointed out previously [6], the biquadratic fraction (6) quadratic program (QP) in terms of the transmitter currents
with both convex numerator and denominator, does not have ct. Such programs can efficiently and reliably be solved to
to be convex itself. However, convexity is the property that (nearly) arbitrary precision using numerical methods, such as
ensures that the objective has a single and global optimum interior-point or active-set algorithms [8]. More importantly
that can be reliably found by dedicated algorithms [8]. however, the fact that (12) is a purely equality-constrained
Note that there are numerical algorithms dedicated to deal- convex QP means that it has an analytic solution, obtainable
ing with this type of problem directly, often referring to it for example via Lagrangian duality [8]. The steps involved
as maximization of the constrained Rayleigh quotient [12], arestraightforwardand,therefore,omittedhere.Theseclosed-
[13]. However, in the following, a more direct and physically formexpressionscanthenbeoptimizedwithrespecttoRL,by
intuitive way of dealing with this problem is taken. setting its derivative to zero. In the following, the so obtained
In order to obtain a uniquely defined optimization problem, closed-form expressions of [6] are reviewed and presented in
the receiver current could be assumed to be a known constant a more physically meaningful and intuitive form.
ir R, ir = 0, (purely real, non-zero). More specifically
∈ 6
and without loss in generality, the receiver current is chosen
D. Closed-FormExpressionsforOptimalMISOWPTSystems
to be i = 2/R , leading to unit transferred power, i.e.
r L
P = 1iHR i= 1R i2 =1. Then, maximizing the PTE (6) 1) FundamentalQuantities: Theoptimaloperatingparame-
L 2 Lp 2 L r
is equivalent to minimizing the power loss ters,suchastransmittercurrentsandloadingelements,aswell
as the resulting PTE are found in terms of two fundamental
Pl = 1iHZ0i quantities with physical interpretation:
2 Minimum-loss output impedance
1 •
= 2(cid:16)i0tTZ0ti0t+i0t0TZ0ti0t0+2irz0tTri0t+i2rzr0(cid:17) . (7) zo =zo0 +jzo00 =zr−zTtrZ0t−1z0tr ∈C (13)
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 4
Mutual coupling quality factor (also called “mutual Q”) Finally, when in addition to the reactance also using the
•
optimal load resistance
U =szHtrZz0t−o01ztr ∈R (14) RL? =zo0 1+U2 (18)
the maximum achievable PTEp(the physical limit of the WPT
Sinceallimpedance(sub-)matricesarepositive-real,itfollows
system) results:
that z ,U >0.
o0
The real part of the minimum-loss output impedance z U2
o η = η = . (19)
corresponds to the (minimized) loss due to radiation and max res|RL=RL? (1+√1+U2)2
dissipation (i.e. Rl = zo0 in (1) and Fig. 1), as illustrated Note that η and R? are of the same form as for SISO
in Fig. 3, whereas the imaginary part z is the reactive max L
o00 systems [9], [14], where U is given by (15). In other words,
component to be canceled out by the receiver reactance x .
r SISO WPT systems are special cases of the more general
MISO WPT systems, since in the case of a single transmitter,
(14) collapses to (15) and real part of (13) is replaced by R ,
P t
l,min
as previously mentioned; the rest of the expressions remain
v1 i1 x1 zo xr ir the same.
Pt Z RL PL III. PROBLEMSTATEMENT
i x
N 1 N 1 A. Port Impedance Matrices (PIMs) T
− − n
v
N 1
− The total transmit power in the denominator of (1) can be
Zˆ separated into the contributions of each port n:
Fig. 3. The optimized MISO WPT system, including the minimal output 1 1
P =P +P = (iHv) = iHZˆ i
impedance zo responsible for both losses during the power transfer as well t l L 2 0 2 0
asoutputreactancetobecompensated.
1 1
= Pt,n = (vnin)0 = iHTˆni (20)
2 2
The presented mutual coupling quality factor is the natural n n
X X
extension of the well-known “mutual Q” for SISO systems where Tˆ are going to be referred to as port impedance
n
[9], [14], commonly given as
matrices (PIMs). These PIMs are interesting in many ways:
ωM They sum up to the total loss resistance matrices:
U =k QtQr = (15) •
√R R
p t r Tn =Z0 and Tˆn =Zˆ0 . (21)
Note that generally the mutual inductance M and also the n n
X X
coupling coefficient k are complex quantities. In such cases, Note that T = Tˆ , for all n except n = N, where
n n
their absolute values should be used in (15). Evidently, the Tˆ =T +R .
N N L
real part of the output impedance zo0 corresponds to the loss While Z0 and Zˆ0 are purely real-valued, symmetric
resistance on the receiver side R , whereas dividing by the •
r and positive definite as previously noted, the PIMs are
loss resistance on the transmitter side R is replaced by the
t complex-valued, Hermitian, singular and indefinite.
inverse of the real part of the transmitter impedance matrix Each T , Tˆ has exactly one positive, one negative
n n
Z0t. Thus, in terms of quality and coupling factors, the mutual • and N 2 zero eigenvalues. These eigenvalues and the
coupling quality factor (14) becomes U = kHQtkQr for correspo−nding eigenvectors can be derived analytically,
MISO systems. q as laid out in appendix A.
2) Optimal Currents, Loads and PTE: The optimal trans-
mitter currents are found as
B. Nonconvex Transmit Power Constraints
i?t =−Z0t−1(cid:18)z0tr+ zo+zRo0LU+2 jxr ztr(cid:19)ir . (16) itivTeh),ewtohtaelntrtraannssmfeitrripnogweprowiseralPwLays>Pt0.>Ho0w(esvtreirc,tlythipsosis-
not necessarily true for the transmitter powers at each port
The optimal receiver reactance is canceling the reactive
n = 1,...,N 1, since all T are indefinite, as previously
componentoftheoutputimpedance,asmentioned,x?r =−zo00. mentioned. − n
The transmitter reactances x = diag(X ) are not essential
t t As will be shown in the results section, in some cases, the
for maximizing the PTE. They can be chosen arbitrarily, as
optimal currents (16) lead to one or more transmitter powers
long as the optimal transmitter currents result from applying
being negative. This implies that via some transmitter ports,
the voltages to the loaded system.
power is drawn from the system, rather than fed into it. Such
Further,usingtheoptimalreceiverreactancex?,thesystem
r negative transmitter powers lead to a reduced net transmit
operates at the resonant PTE, given by
power (20), which has a positive effect on the PTE (1).
U2 R While mathematically correct and physically meaningful
L
η = η = . (17)
res |xr=x?r 1+RL/zo0 +U2 · RL+zo0 overall (at least from a circuit theory point of view), such
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 5
solutions prove difficult to realize in practice, as discussed in where the objective and the inequality constraints contain the
moredetailinSec.V.Therefore,itmakessensetotrytoavoid leadingprincipalM M submatricesoftherealblockmatrix
×
such solutions altogether, during the optimization process. representations:
In order to accomplish that, the constraints
Pt,n = 1iHTni 0 n=1,...,N 1 (22) Qn = 12(cid:20)TT0n0n0 −TT0n0n0(cid:21)M M (26a)
2 ≥ − ×
would have to be added to (12). They ensure nonnegative and Q0 = nQn.
transmit powers and, thus, power being fed into the system Similar to the real part of the impedance matrix and the
P
at all transmitter ports, rather than being drawn from it. PIMs, Q0 (cid:31) 0 while Qn (cid:15) 0. The equality constraints
Adding the transmitter power constraints (22), the program represent the KVL at the receiver node and the fixed receiver
(12) in terms of complex currents i becomes: currents (real and imaginary parts). The affine equality con-
straints Ac=b are actually
1
Pl,min =miin 2iHZ0i z0tr zr+RL −z0t0r 0 c= 0 (27)
1 0 1 0 0 2/R
s.t. iHT i 0 n=1,...,N 1 (cid:20) (cid:21) (cid:20) L(cid:21)
n
2 ≥ − (23) to ensure KVL at the receiver end (v =p0) as well as unit
v = zT,zˆ i=0 r
r tr r transferred power P =1, as before.
L
(cid:2) (cid:3) 2 Using the cyclic property of the trace cTQ c =
ir = 0,1 i= i
rRL tr(cTQic) = tr(QiccT) the objective and all inequality
(cid:2) (cid:3) constraints can be written in linear terms of the (quadratic)
Unfortunately, since Tn (cid:14) 0 (indefinite and not negative- current matrix C=ccT 0:
semidefinite, as required to make (22) convex), the added (cid:23)
constraints are nonconvex, rendering the whole optimization
P =min tr(Q C)
l,min 0
problem nonconvex. Thus, (23) is a nonconvex QCQP [8] C,c
(as opposed to the convex QCQP mentioned at the end of s.t. tr(Q C) 0 n=1,...,N 1
n ≥ − (28)
Sec. II-C). Note that the same would be true for constraints
Ac=b
of the form 1iHT i P , i.e. limiting the maximum
2 n ≤ t,n,max C=ccT
transmit power at each port.
Nonconvex QCQPs belong to the class of NP-hard prob-
Note that the three programs (23), (25) and (28) are math-
lems [15]. This means that there are no known algorithms
ematically fully equivalent. The non-convexity of the former
which can solve the problem for an arbitrary number of
two has been isolated in the last condition of (28), since the
transmitters.
traces are linear operations in terms of the matrices C.
IV. SEMIDEFINITERELAXATION
B. Semidefinite Relaxation in Partially Conic Form
In the following, a method is presented which incorporates
the nonconvex transmit power constraints while remaining The idea is to exchange the nonconvex constraint with a
computationally efficiently and reliably solvable. practically equivalent, yet convex one. Similar to x=y being
equivalent to both x y and x y, C = ccT is equivalent
≥ ≤
to both C ccT and C ccT. The semidefinite relaxation
A. Nonconvex QCQP (cid:23) (cid:22)
(SDR) arises from removing the second inequality constraint
Zero KVL in the imaginary part can always be achieved and using
by appropriately choosing the receiver reactance x and the
r
imaginary part of the receiver current is therefore chosen to C ccT =0 SDR C ccT 0 (29)
− −−−→ − (cid:23)
always remain zero. Thus, the actual problem size is M =
2N 1 and the unknown currents are (in real form) Implicitly, this also ensures C 0, since ccT 0.
− (cid:23) (cid:23)
ViaSchurcomplement,therelaxedconstraint(29)isequiv-
i0t alent to
c=ii0t0r0∈RM . (24) C−ccT (cid:23)0 ⇔ cCT c1 (cid:23)0 (30)
  (cid:20) (cid:21)
The nonconvex QCQP (23) can be formulated in standard
which is the most common way to express the semidefinite
QCQP form as follows:
(inequality) constraint. Note that another formulation of the
P =mincTQ c last constraint in (28) would be to require that the rank of
l,min 0
c the composite matrix in (30) be 1. This implicitly ensures
s.t.cTQnc≥0 n=1,...,N −1 (25) both symmetry and C = ccT, while the equality constraint
Ac=b Ac=b ensures a nonzero result.
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 6
Thus,therelaxedsemidefiniteprogram(SDP)of(28),using received power is fixed in its quadratic form directly, i.e.
the semidefinite constraint (30) is 1i2R =1 (since i =i and i =0). Let
2 r L r 0r 0r0
1 R 0
Prelax =min tr(Q C) R= L (32)
l,min C,c 0 2(cid:20) 0 RL(cid:21)M M
s.t. tr(Q C) 0 n=1,...,N 1 ×
n ≥ − (31) be the leading principal submatrix of the block form of RL,
Ac=b similar to (26). Due to the row/column reduction, R 0
(cid:23)
C c has only one non-zero entry, the Nth entry on the diagonal,
0.
cT 1 (cid:23) where RL/2 0 resides. Then, the condition that ensures
(cid:20) (cid:21) ≥
unit transferred power (and thereby fixes the real part of the
andrepresentsthedualofthewell-knownShorrelaxation[15] receiver current) becomes:
with added affine equality constraints. This problem can be 1
i2R =1 tr(RC)=1. (33)
implemented readily in Matlab using CVX [16], [17], a 2 r L ⇔
package for specifying and solving convex programs such as
Further, since the only remaining constraint is homogeneous,
thisSDP.TheactualnumericalalgorithmusedisSDPT3[18],
it can also be represented in quadratic form, without loss of
[19], which for example for a MISO-3 system (problem size generality.Letk= z ,z +R , z andK =kkT,then
M = 7) usually converges within ten to thirty iterations, in 0tr r L − 0t0r 0
just a few seconds. kTc(cid:2)=0 tr(K0C(cid:3))=0 (34)
⇔
With this relaxed program, a lower bound on the minimum
isresponsibletosatisfyKVLatthereceiverend,i.e.toensure
powerlossand,thus,byapplicationof(8),anupperboundon v =0.
r
the PTE is found: Prelax P , corresponding to ηrelax
l,min ≤ l,min max ≥ For increased numerical accuracy, the following redundant
η . If equality is achieved, the relaxation is referred to
max matrix equalities may be added
as tight [20], referring to the fact that the bounds have zero
tr(K C)=0 m=1,...,M (35)
gap.Evidently,then(31)isafullyequivalentreformulationof m
(23),(25)and(28)andsolvingthe(convex)SDPisequivalent where K = u kT + kuT, where u , is a unit column
m m m m
to solving the original (nonconvex) QCQP. In essence, this vector of zeros everywhere but at the mth position. Note that,
asalstoisfiimedplbieystthheatrethsteorfemthoevepdrocgornamstr.aint C(cid:22)ccT is naturally whTilheuKs,0i(cid:23)nco0r,pKormati(cid:15)ng0tfhoer aclolnmstr=ain1t,s...(,3M3).and (34), the
Notethat,ingeneral,relaxationswheretheconstraintswere semidefiniterelaxationof(25)inpurelyquadratic(conic)form
relaxed (but not the objective) necessarily lead to a remaining is obtained as:
feasibility problem, unless they are tight. By removing a Prelax =min tr(Q C)
l,min 0
constraint, the feasible region for solutions was enlarged and, C
therefore,thesofoundsolutioncannotbefeasiblefortheorig- s.t. tr(Q C) 0 n=1,...,N 1
n ≥ −
inal problem unless that constraint was loose, i.e. otherwise tr(K C)=0 m=0,...,M (36)
m
satisfied to begin with. The remaining feasibility problem is
tr(RC)=1
then to find “the next best” solution which is feasible to the
C 0
original optimization problem but has the smallest gap to the (cid:23)
relaxed optimum. Such problems can generally be arbitrarily Note that the programs (31) and (36) are fully equivalent.
difficult to solve, as they essentially contain the original non- 2) Duality: The Lagrangian of the semidefinite program
convexity and therefore must still be NP-hard. (36)isobtainedbyaddingtheconstraints,weightedbypenalty
Inthiscase,agoodcandidateofafeasiblesolutionisfound factors, called dual variables [8], to the objective:
directly, since the vector c? (optimal solution to c) necessar- N 1 M
−
ily satisfies the affine equality constraints (representing the L=tr Q λ Q ν K σR C +σ
KVL at the receiver end). However, that vector only satisfies " 0− n=1 n n−m=0 m m− # !
X X
the other constraints if the relaxation is tight. Methods to (37a)
find bounds on these types of inequalities are available, see
=tr [P(λ,ν) σR]C +σ (37b)
e.g. [21], but deemed not strict enough to be useful in this −
case. =tr(cid:0)QC +σ (cid:1) (37c)
As it turns out, while it is intuitive and numerically con-
venient, the SDP form with a separate vector for the affine wonhleyrepuanl(cid:0)lisλhnt(cid:1)≥r(Q0(aCnd) <the0r.efNoroetethtehavtecPto=rλP(cid:23)T 0a)n,dinthoursdearlstoo
constraints (31) is mathematically difficult to deal with, when n
Q=QT.
attempting to prove tightness.
The original problem, when considering duality called the
primal problem, can be obtained from the Lagrangian by
minimizing the objective given that an inner maximization
C. Semidefinite Relaxation in Purely Conic (Quadratic) Form
ensures that all constraints are satisfied:
1) QuadraticFormsofAffineConstraints: Insteadofdefin- Prelax =min maxL, (38)
l,min
ing the receiver current via affine equality constraints, the C 0 λ 0
(cid:23) ν(cid:23),σ
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 7
If one or more of the constraints are not satisfied, the La- are
grangian is unbounded and diverges to + . Thus, (38) and
∞ C? 0 Primal feasibility (42a)
(36) are equivalent. (cid:23)
tr(K C?)=0 Primal eq. constraints 1,...,M (42b)
General (weak) duality [8] states that m
tr(RC?)=1 Primal equality constraint M +1 (42c)
tr(Q C?) 0 Primal ineq. constr. 1,...,N 1 (42d)
min maxL max minL. (39) n ≥ −
C 0 λ 0 ≥ λ 0 C 0 λ? 0 Dual feasibility (42e)
(cid:23) ν(cid:23),σ ν(cid:23),σ (cid:23) (cid:23)
Q? 0 Dual ineq. (semidef.) constraint (42f)
(cid:23)
tr(Q?C?)=0 Complementary slackness (CS) (42g)
where the right hand side is called the dual problem. Since
the original problem is convex and optimal feasibility can be If the relaxation is tight, rankC? = 1 or in other words
presumed for physical reasons, strong duality [8] is expected C has exactly one non-zero eigenvalue and c? is the cor-
to hold, leading to equality in (39). responding eigenvector, i.e. C? = c?(c?)T. Thus, to test
As can easily be seen from (37c), the dual problem is only tightness, either the eigenvalues of C? can be investigated,
bounded as long as Q is positive semidefinite: or, numerically more efficient, the normalized tightness error,
defined as
C? c?(c?)T
(cid:15)= − (43)
0 Q 0 (c?)Tc?
(cid:13) (cid:13)
mintr QC = (cid:23) (40) (cid:13) (cid:13)
C 0 ( otherwise can be used. As long as (cid:15) is small (within the bounds of
(cid:23) −∞
(cid:0) (cid:1) typical numerical approximation errors), this proves that the
semidefinite relaxation is tight and, therefore, provides an
Thus, the dual problem really is exact solution c? to the nonconvex QCQP (28). Numerical
experiments showed that this seems always the case (down to
typicalnormalizedtightnesserrorsaround(cid:15) 10 12)andthat
Pld,muainl =Plr,emlainx =mλa0x{σ : Q(cid:23)0} (41) C? hasonlyonenonzero(andindeedpositiv≈e)ei−genvalueand
ν(cid:23),σ c? is its corresponding eigenvector.
whichisagainanSDP,asrequired(sinceSDPsareself-dual). D. Optimizing the Receiver Load R
L
3) TestofTightness: TheKarush-Kuhn-Tucker(KKT)con- The presented method addresses optimization of the trans-
ditions [8] of optimality for the two problems (36) and (41) mittercurrentsandreceivercapacitance,butleavesthereceiver
z z
3=0 3=0
θ θ
30 !30 30 !30
planar MISO y y
x x
60 !60 60 !60
z
3=0
θ
30 !30 90 !90 90 !90
y Pt=!20 !10 0 10 20W Pt=!20 !10 0 10 20W
x
(b) (c)
60 !60
z z
3=0 3=0
θ θ
90 !90 30 !30 30 !30
Pt=!20 !10 0 10 20W y y
x x
(a)
60 !60 60 !60
coaxial MISO
90 !90 90 !90
P =!200!100 0 100 200W P =!200!100 0 100 200W
t t
(d) (e)
Fig.7. ComparisonofthetransmitpowerpatternsofWPTsystemsoptimizedviaclosed-formexpressions(13)to(19),[6]:(a)SISOreference,(b)and(c)
planarconfigurationsMISO-2pand-3p,respectively,aswellascoaxialsetupsMISO-2cand-3c,(d)and(e),respectively.Negativetransmitpowermeanthat
powerisdrawnfrom,insteadoffedinto,thesystemviathatparticulartransmitterport,referencedviathecolors.Distanceofthereceiverd=λ/10.Note
thedifferentscalingof(d)and(e)withrespecttoalltheothers.
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 8
loadresistanceR unchanged.R couldbeoptimizednumer- setups also provide enhanced performance, predominantly in
L L
ically in an outer loop (while optimizing the system for each the broadside direction. In these cases, the transmitters are
R intheinnerloop);astheoptimalPTEisaconcavefunction often being used “in parallel” (as observed when looking
L
in R . However, it has been observed that the maximum PTE at the currents), so as to minimize conduction losses. The
L
is usually not sensitive to the load resistance R , but strongly performance enhancement appears to become less significant
L
depends on x x? [22]. Thus, even using the closed-form when further increasing the number of transmitters.
r ≈ r
expression (18), results that are very close to the absolute 2) Optimal receiver capacitance: Fig. 5(b) shows the op-
optimum can be expected, as long as x is optimized. timal receiver capacitances obtained via the closed-form ex-
r
pressions. It is observed that the capacitance values remain
V. EXAMPLES fairly close to about 68.86pF, over all angles.
As mentioned in the introduction of Sec. IV, previous
Fig. 4 shows the geometrical setups of the basic SISO and
studies [22] revealed that achieving the maximum possible
MISO WPT systems under consideration. The frequency of
PTE is generally very sensitive (particularly for such single-
operation is f = 40MHz and the coils are single-turn loops
turn loop-based MISO setups) to the receiver capacitance
with radii of r = λ/100, where λ is the freespace wave-
loop values; commonly tolerances much lower than one percent
lengthattheoperatingfrequency.Theconductorsaremadeof
are obtained, to maintain a performance within 1% of the
copper(σ =5.8 107S/m)wirewithathickness(wireradius)
× optimum PTE. On the other hand, the sensitivity on the load
of r = r /10. The spatial separations of the multiple
wire loop resistance R is much smaller; often tolerances of 10% and
transmitter loops are ∆x=λ/50 and ∆z =λ/100, in the x- L
higher are obtained for the same permissible decrease in
and z-directions, respectively. The receiver loop position is in
performance.
the xz-plane (y = 0), at a distance d = 0.05λ,...,0.2λ from
3) PositiveandNegativeTransmitPowers: Fig.7showsthe
the center of the transmitter (array), as specified for each of
underlying working principles of these MISO WPT systems
the following results, and at the angle θ off broadside (off the
in greater detail, considering the individual transmit powers
z-axis). Note that all the loops are parallel to the xy-plane.
P as defined in (22): The first graph (a) shows the transmit
t,n
z z z z z power of the SISO reference system as a radial function of
the receiver position angle θ. As is well known, there are two
RX
rangesofangles(aroundapprox. 60 offbroadside)atwhich
θ θ θ θ θ ± ◦
the transmitter loop couples very weakly to the receiver loop
TX y y y y y (in fact, infinitely small loops would have points where they
do not couple at all). In those areas, the transmitter power
x x x x x
spikes to very high values, in order to still transfer unit power
(a) (b) (c) (d) (e)
(P =1W) to the receiver.
L
Fig. 4. Illustrations of the SISO WPT reference (a) and the MISO WPT
configurations under consideration: Two planar configurations MISO-2p (b)
and MISO-3p (c), with transmitters located at the positions (xi,yi,0), and 3=0
two coaxial configurations MISO-2c (d) and MISO-3c (e) with transmitters
30 !30
at(0,0,zi).Inallcasesasinglereceiverloopislocatedat(d,θ,φ=0).
For all the results in this paper, the unloaded impedance
60 !60
matrices were obtained via full-wave simulation using the
MultiradiusBridge-Current(MBC)method,acomputationally
efficient and accurate wire-based method of moments (MoM)
code with sinusoidal current elements [23], [24]. 90 !90
2max =0 0.2 0.4 0.6 0.8
A. Closed-Form Optimized MISO WPT Systems (a)
1) Maximum achievable PTE: As expected, MISO WPT
3=0
systems can provide superior performance compared to the 30 !30
SISO reference system: Fig. 5(a) shows typical PTE patterns
as a function of the receiver position angle θ at the distance
d = λ/10 to the transmitter center. Two main observations 60 !60
can be made: First, all the MISO setups outperform the SISO
system at every angle θ. However, overall, the increase in
PTE is more significant for the planar MISO systems, than
90 !90
for the coaxial ones. This is due to the fact that at angles Cr =68.82 68.84 68.86 68.88pF
off broadside (θ = 0), the distance to one of the transmitters
6 (b)
is always smaller than d. Thus, it has to be expected that
the transmitter closest to the receiver is favored over the Fig.5. ComparisonofthepatternsofthemaximumachievablePTEηmax(a)
and receiver capacitance C? (b) obtained via closed-form optimization (13)
r
others and that all (or at least most) the power is transferred to (19), for the planar and coaxial MISO WPT configurations Fig. 4 (using
to the receiver via that particular one. Second, the coaxial thesamecolorcode)aswellastheSISOreference,atthedistanced=λ/10.
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 9
z 3=0 3=0 0.05 6 3=0 0.05 6
30 !30 30 !30 0.106 30 !30 0.106
θ 0.156 0.156
y
60 !60 60 !60 60 !60
x
90 !90 90 !90 90 !90
Pt=!10 !5 0 5 10W "2max=!50 !40 !30 !20dB 106#"Cr/Cr=!60 !30 0 30 60
(a) (b) (c)
z 3=0 3=0 0.05 6 3=0 0.05 6
30 !30 30 !30 0.106 30 !30 0.106
θ 0.156 0.156
y
60 !60 60 !60 60 !60
x
90 !90 90 !90 90 !90
Pt=!10 !5 0 5 10W "2max=!50 !40 !30 !20dB 106#"Cr/Cr=!200!100 0 100 200
(d) (e) (f)
Fig.9. TransmitpowersPt,n ofclosed-formoptima(thin,dashed)andSDRresults(thick,solid)atdistanced=0.1λ(a,d),PTEreductionwithrespectto
closed-formoptimization∆ηmax indB(b,e),andnormalizeddeviationofreceivercapacitance∆Cr/Cr (c,f)oftheplanarmulti-transmitterconfigurations
MISO-2p(a-c)andMISO-3p(d-f).
The top right two graphs, in Figs. 7(b) and (c), show the
P
l
transmit power patterns of the planar MISO setups, whereas
(d) and (e) on the bottom right show the coaxial multi- Pt,1 =−12.07W
transmitter cases. In the planar cases, most of the power is
always transferred via the closest transmitter, as expected. In Pt,2 =2.94W η PL =1W RL
addition, the remaining transmitters appear to draw away a
part of the remaining power in other directions, and feed it
back into the system. A similar observation can be made for Pt,3 =11.35W
the coaxial setups: In the regions around the 60 angles,
◦
± MISO-3cWPTSystem
most of the power is inserted into the system by the first
(closest) loop, while the others are used to feed some power Pt =2.22W
back into the system. However, at angles close to broadside, (a)
mostofthepoweristransferredviathefarthestloop,withthe
remaining loops acting as directors (and also feeding power
P
l
back into the system). Depending on the distance d between Pt,1 =0W
thetransmittercenterandthereceiver,theamplitudesofthese
negative transmit powers can be quite significant. Note that
all loops are modeled with conduction loss and, therefore, Pt,2 =0W η PL =1W RL
extractingpoweralsocomesatacost,unlikeinlosslesscases;
this is already accounted for during the optimization.
Pt,3 =2.22W
B. MISO WPT Systems with Transmit Power Constraints MISO-3cWPTSystem
Fig. 8 compares one example (MISO-3c, with the color Pt =2.22W
scheme in accordance with the setup shown in Fig. 7(e), at (b)
θ = 18 off broadside and d = λ/10) of an optimal solution
◦ Fig. 8. Examples with and without negative transmit powers: Similar
from the closed-form expressions that involves negative trans- performance (max. PTE of ηmax = 1/2.22 45% in both cases), but
≈
mit powers (a) and compares it to its nonnegative counterpart case (a) involves harvesting power from one port, while (b) involves only
nonnegative transmit powers; the loops connected to the first two ports are
(b), to shed light on the difference between the underlying
onlyusedpassively.
mechanisms. In the first case, power is fed into the system
via two transmitter ports, while one transmitter port extracts
power. In turn, the net power supplied to the system, given by of the impedance matrix (3) always remains positive). It is
thesumofalltransmitterportpowers,ismuchsmallerthanthe implicitly assumed that the power extracted from the system
maximum transmit power (but due to the positive-definiteness is fully harvested and can be completely “reused”, i.e. fed
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 10
back into the system at 100% efficiency. Evidently, this poses To ensure practical realizability, it is important to incor-
difficultieswhentryingtorealizesuchasysteminpractice,as porate constraints on the transmit powers into the optimiza-
thepowercannotbeeasily“recycled”—atleastnotatperfect tion procedure to avoid such problems from the beginning.
efficiency. However, this is not a trivial task, as such constraints are
The nonnegative transmit power solution in Fig. 8(b) nonconvex, thereby rendering the entire optimization problem
achieves approximately the same performance by only insert- nonconvex.
ing power via the last port and using the optimally tuned This paper presents a step-by-step derivation of a convex
transmitterantennasconnectedtotheothertwoportspassively. semidefinite relaxation of the nonconvex problem, where by
This operating mode can be implemented in practice in a loosening some constraints a convex formulation is obtained,
straightforwardfashion.Additionally,withalltransmitpowers which can efficiently and reliably be solved numerically by
being positive, they also become smaller in amplitude, which dedicated algorithms. Furthermore, a simple test is given,
further simplifies the realization of such systems. which reveals whether the solved problem is equivalent to
Figs.9and11comparetransmitpowers,performancedegra- the original nonconvex problem (i.e. whether the relaxation
dation and deviation in receiver capacitances in polar form is tight) and that resulting optimum is indeed the true global
for the two planar and the two coaxial MISO WPT systems, optimum looked for. This is a clear advantage over global
respectively. Fig. 10 shows the corresponding normalized optimizationmethods,suchasgeneticalgorithms(GA),where
tightness errors, confirming that the relaxation is tight and the convergence to the true global optimum cannot be guaranteed
solutions solve the original (nonconvex) problem. or tested.
The transmit power patterns in the graphs Figs. 9 and Simulation results of some basic multi-transmitter WPT
11 (a) and (d) demonstrate that the semidefinite relaxation systems show that the new method is very practical and
does indeed lead to solutions where no power is drawn from indeed able to avoid any realizability issues due to negative
any transmitter port, as can be seen by comparing the new transmit powers. Furthermore, it is shown that the resulting
results (solid thick lines) to the closed-form solutions (thin performancegapbetweentheclosed-form(absolute)optimum
dashedlines).Notethattheoptimalconstrainedtransmitpower and the new, practically realizable solution is usually very
solutions are different from simply setting formerly negative small; typically below 1% of the predicted PTE. Lastly, plots
transmit powers to zero (this in fact would both lead to of the tightness test results reveal that in all cases under
inferior performance as well as break the normalization to considerationtherelaxationisindeedtight,i.e.thatthesolved
unit received power). The performance degradation patterns problem is equivalent to the actual nonconvex problem and
in the subfigures (b) and (e) reveal that the drop in maximum that the obtained optimum PTE is the global optimum PTE
PTE is minor, in most cases; usually much lower than 1%. satisfying all constraints.
The rightmost graphs (c) and (f) show the deviation of the
This new optimization procedure is powerful, versatile and
optimal receiver capacitances (see Fig. 5(c)), obtained from
retains the rigor of convex optimization, while ensuring the
the original closed-form expressions (11) and (13). It can be
practical realizability of the optima it produces.
seen that the optimum capacitances remain almost unchanged
in all cases, particularly at larger distances.
Lastly,Fig.10confirmsthattherelaxationsweretightinall
APPENDIXA
cases,asthenormalizedtightnesserrorsalwaysremainedvery
ANALYTICALFORMULATIONOFTHEPOSITIVEAND
small, usually1 around 10 12. Gaps in the error plots refer to
− NEGATIVEPIMPARTSBYEIGENVECTORS
cases where the closed-form expressions did not lead to any
negative transmit powers (no power drawn from the system
Let the impedance matrix of the unloaded WPT system be
via any of the transmit ports) and which, therefore, did not
have to be optimized using the relaxation method.
Z=R+jωL+jωM, (44)
VI. SUMMARY&CONCLUSION
where R and L are real-valued diagonal matrices containing
A closed-form optimization framework [6] for multi- thelossesandself-reactances(usuallyinductancesL >0,for
n
transmitter (MISO) wireless power transfer systems has been loop-based magnetically coupled systems) of the transmitters
reviewed to highlight its simplicity and underlying physical and the receiver. M=MT is a symmetric (due to reciprocity
meanings.Thispowerfulandintuitivesetofequationsisuseful of the passive system) hollow matrix containing the mutual
for analysis and design of MISO and SISO WPT systems and impedances jωM . Generally, each M is complex-
n,m n,m
generalizes previous results for SISO systems [9]. valued, due to retardation effects when the electrical distance
However, as discussed, the obtained absolute optimum between the transmitter(s) and receiver are not very small.
operating modes can result in power being drawn via some
Moreover, let
transmitter ports, rather than fed into the system. Although
analytically correct, such operating conditions are difficult to
λ ,v =eig(T ) n=1,...,N (45)
n,m n,m n
realize in practice as power cannot easily be “recycled”. { }
denote the mth eigenvalue λ and corresponding eigenvec-
1Larger errors are due to the inherent numerical difficulties of the n,m
impedancematricesatlargerdistances,particularlyslightasymmetry. tor vn,m of the N N port impedance matrix (PIM) Tn. In
×