Mark Davidson
email: mdavid@spectelresearch.com

Classical charged particle models derived from complex shift methods

Keywords: electron model, Kerr-Newman, Kerr-Schild, complex spacetime, C 4, double copy, complex oset, emergent quantum mechanics, string theory, ER=EPR, quantum gravity

Extended charged objects embedded in complex space-time are proposed using the double-copy or complex shift method. Most of the objects studied are 3D strings in dierent shapes. The most detailed case studied is a static-charged open string. It is found that this can be interpreted purely electromagnetically. It exhibits the same relation between charge, mass, angular momentum, and magnetic moment as the Dirac equation and the Kerr-Newman metric. The spin of the particle is purely electromagnetic, as is the mass. A gyromagnetic ratio of 2 is obtained. The elds in this case are multi-valued, and the singular part of the elds can be arranged to be on an unphysical Riemann sheet with a judicious selection of Riemann cut surfaces. The calculations of mass and angular momentum are done numerically using multi-precision algorithms included as a Python script. The mass calculation agrees with the measured electron mass. Particles for knotted or linked strings in 3 space dimensions are also proposed, and these inherit the topological invariants of the knot or link they are based on. A liquid drop model with complex shift is also discussed. The multi-valued behavior of the solutions, related to that of the Kerr-Newman metric, can be thought of as the origin of the Einstein-Rosen bridge, and the conjectures that this is the origin of quantum entanglement, ER=EPR, is therefore supported in this theory. So we have here a classical theory that has some properties of quantum mechanics. Hopefully it can oer a new phenomenological application of string theory as a semiclassical model for elementary particles, nuclei, and solitons in condensed matter, uids, and gases.

INTRODUCTION 2 1 Introduction

Constructing purely electromagnetic particles is an ongoing eort in physics.

Electrons and other fundamental charged particles tend to have both spin and magnetic dipole moments. Here we consider only classical models which have both of these. Point charges, of course, have divergent electrostatic energy. Line charges also have innite energy, but the divergence is weaker than for the point charge. It is known that if classical electromagnetism is analytically continued into complex space-time then the point charge is still divergent [START_REF] Lynden-Bell | A magic electromagnetic eld[END_REF][START_REF] Pekeris | The electromagnetic eld of a Kerr-Newman source[END_REF][START_REF] Adamo | The Kerr-Newman metric: A Review[END_REF], but including gravity yields the Kerr-Newman metric which has nite mass and angular momentum. This technique is sometimes called the complex-shift or double-copy method and it yields many exact solutions to general relativity that have a Kerr-Schild form of metric [START_REF] Bah | Kerr-Schild double copy and complex worldlines[END_REF][START_REF] Newman | Heaven and its properties[END_REF][START_REF] Newman | Classical, geometric origin of magnetic moments, spinangular momentum, and the Dirac gyromagnetic ratio[END_REF]. The idea of this paper is to apply the complex shift method to charged strings and other extended structures in complex space-time. Open and closed strings are considered, as is a liquid drop model. We study only static solutions. Most of these are multi-valued and they can have nite electromagnetic energy on at least one Riemann sheet. Although the elds can be nite, they have branch point singularities making them multivalued. Because the stress-energy tensor is nite, we ignore gravitational eects and study only the electromagnetic elds in a at spacetime.

There are several modications to electromagnetism that allow for nite electromagnetic energy, the Born-Infeld theory [START_REF] Born | Foundations of the New Field Theory[END_REF] which is nonlinear, and the Bopp-Podolsky theory [START_REF] Bopp | Eine lineare Theorie des Elektrons[END_REF][START_REF] Podolsky | A Generalized Electrodynamics Part INon-Quantum[END_REF] which adds higher-order derivative couplings to the eld equations. Both of these theories modify the Maxwell equations and are interesting, but they are not considered in this paper.

Classical charged particle models have historically been constructed for 2D surfaces, like charged spherical shells, or for 3D blobs of charge [START_REF] Pearle | Classical Electron Models[END_REF][START_REF] Coleman | Classical Electron Theory from a Modern Standpoint[END_REF][START_REF] Rohrlich | Classical Charged Particles, 3rd Edition[END_REF][START_REF] Bialynicki-Birula | Classical Model of the Electron. Exactly Soluble Example[END_REF][START_REF] Boyer | Classical model of the electron and the denition of electromagnetic eld momentum[END_REF][START_REF] Rohrlich | Comment on the preceding paper by T[END_REF]. The complex-shifted extended-charge models considered here tend to have magnetic moments as well as electric charge and hidden angular momentum.

The limiting case of a zero-length open string yields the Kerr-Newman solution, which has a g factor of 2 when the gravitational eld is included, the same as an ideal Dirac electron [START_REF] Burinskii | The Dirac Kerr-Newman electron[END_REF]. The nonzero length strings that are considered here also have a g factor of 2 to high precision, but they don't require the gravitational eld to be included. They can also have anomalous magnetic moments. Thus it seems they may be a better classical model of the electron and other particles than the spherical shell models or other classical extended charge models.

In general, the solutions we consider can have nite total electromagnetic energy as well as nite electromagnetic angular momentum calculated from the Poynting vector. Since the underlying theory is relativistically covariant, the usual 4/3 problem can be understood and tolerated if the boost to moving frames is done properly [START_REF] Boyer | Classical model of the electron and the denition of electromagnetic eld momentum[END_REF][START_REF] Rohrlich | Comment on the preceding paper by T[END_REF][START_REF] Campos | Comment on the 4/3 problem in the electromagnetic mass and the Boyer-Rohrlich controversy[END_REF][START_REF] Bialynicki-Birula | Classical Model of the Electron. Exactly Soluble Example[END_REF]. The stability of these particles is still an issue and may require some form of binding tension or negative pressure to oppose the electromagnetic repulsion. As they have electromagnetic charge, mass, angular momentum, and magnetic moment, there is the opportunity to search for an explanation for the ne structure constant which is proportional to the charge squared divided by the angular momentum of an electron in this because the forces on the shell are not in the radial direction.

In this paper, natural units for particle and atomic physics shall be employed so that c = 1, ϵ 0 = 1, µ 0 = 1, ℏ = 1, k e = 1/4π, electron mass =1, and e = √ 4πα. We deal with the microscopic elds E and B here exclusively. In the Appendix, a Python software script is listed which can run on a desktop computer and reproduce all of the numerical calculations presented in this paper for the open string.

2

A discussion of some complex manifold methods in physics

Complex manifold techniques in general relativity have a long history. The majority of these works utilize a Hermitian metric. Einstein himself was one of the rst [START_REF] Einstein | A Generalization of the Relativistic Theory of Gravitation[END_REF][START_REF] Einstein | A Generalization of the Relativistic Theory of Gravitation, II[END_REF][START_REF] Einstein | A Generalized Theory of Gravitation[END_REF]. Many others have followed this path [START_REF] Brown | On the Complex Structure of the Universe[END_REF][START_REF] Chamseddine | Hermitian Geometry and Complex Space-Time[END_REF][START_REF] Das | Complex Space-Time and Classical Field Theory. I[END_REF][START_REF] Debergh | On the Poincaré Algebra in a Complex Space-Time Manifold[END_REF][START_REF] Esposito | Complex General Relativity[END_REF][START_REF] Kaiser | Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis[END_REF][START_REF] Mantz | Hermitian Gravity and Cosmology[END_REF][START_REF] Rauscher | The ontological basis of quantum theory, nonlocality and local realism[END_REF][START_REF] Witten | Space-Time and Topological Orbifolds[END_REF]. A second method for dealing with complex space-time utilizes analytic continuation of elds from the real axes to complex coordinate axes. This technique was pioneered in general relativity too [START_REF] Newman | Maxwell's equations and complex Minkowski space[END_REF][START_REF] Newman | Heaven and its properties[END_REF][START_REF] Adamo | The Kerr-Newman metric: A Review[END_REF], but it has been applied to other elds as well. The idea that elementary particles might be related to the Kerr-Newman metric solution and string theory has been championed by Burinskii [START_REF] Burinskii | The Dirac Kerr-Newman electron[END_REF][START_REF] Burinskii | Stringlike structures in the real and complex Kerr-Schild geometry[END_REF]. Newman has also speculated about this [START_REF] Newman | Classical, geometric origin of magnetic moments, spinangular momentum, and the Dirac gyromagnetic ratio[END_REF]. Complex space methods have been proposed and applied in Bohmian and stochastic mechanics [START_REF] Yang | Extending Quantum Probability from Real Axis to Complex Plane[END_REF]. The wormholes of general relativity are related to complex space-time by the Kerr-Schild metric construction. The wormhole's origin is the Riemann cut in the Maxwell potentials in complex space-time.

These have been proposed as sources of quantum entanglement ER=EPR [START_REF] Maldacena | Cool horizons for entangled black holes[END_REF].

It has been proposed that complex space-time can account for quantum non-locality [START_REF] Rauscher | The ontological basis of quantum theory, nonlocality and local realism[END_REF], a concept championed early on by the radical physicists of the Fundamental Fysiks Group [START_REF] Toben | Space-time and Beyond: Toward an Explanation of the Unexplainable[END_REF][START_REF] Kaiser | How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival[END_REF]. It has also recently been proposed that complex space-time can provide a novel explanation for observed galaxy rotation without dark matte [START_REF] Sen | Galaxy Rotation Curve Anomaly and Complex Spacetime[END_REF][START_REF] Sen | Non-gravitational Eects of the Metric Field over Complex Manifolds[END_REF].

Stability is another issue that needs to be addressed in these models. The Kerr-Newman solution is stable obviously because of general relativity and gravity. The string shapes we consider, which can be knotted and linked as well as open, may not be stable unless some tension is added along the string. We partially address this in a later section. Most of the cases we consider here are examples of what Wheeler called Charge without charge [START_REF] Misner | Classical physics as geometry[END_REF]. They deserve this moniker because the charge is located at coordinates that have non-zero imaginary parts so that there will be no charge on the real subspace. Developing an equation of motion for these objects in the presence of an external electromagnetic eld is a non-trivial task.

Riemann-Silberstein Vector

It is convenient to work with the Riemann-Silberstein vector given by

F(x)) = E(x) + iB(x) (1)
If the elds are time-dependent, the Maxwell equations in the absence of charges on the real subspace become simply [START_REF] Bialynicki-Birula | The role of the Rie-mannSilberstein vector in classical and quantum theories of electromagnetism[END_REF] i ∂F ∂t = ∇ × F

In the static case, we nd ∇ × F = 0 and consequently we can write F as a gradient. In most of the models we consider here, there is no charge on the real subspace, and it follows that at all real points x where the gradient exists we have

∇ • F(x) = 0 (3)
and consequently

∇ • E(x) = ∇ • B(x) = 0 (4)
The electromagnetic energy density is given by

E = |F| 2 /2 = E 2 + B 2 /2 (5)
the momentum density we assume is equal to the Poynting vector given by

P = (F * × F) /2i = E × B (6)
and the angular momentum of the electromagnetic elds by the integral

J = ˆr × PdV ol (7)

A static line charge in complex space

A straight-line charge with uniform charge per unit length is a good starting point. In the complex-shift method, the arena for physics is complex Minkowski space or C 4 . Electromagnetism in complex coordinates can be described by two equivalent methods. The rst utilizes the Riemann-Silberstein (RS) complex vector [START_REF] Bialynicki-Birula | The role of the Rie-mannSilberstein vector in classical and quantum theories of electromagnetism[END_REF], and the second utilizes a 4D complex Faraday tensor and is manifestly covariant. For simplicity, we start with the RS approach. We introduce a complex potential function ψ = ϕ + iχ where ϕ is the electrostatic potential and χ the magnetostatic potential in a vacuum away from charges. We consider a static line charge oriented along the z-axis. We also choose the imaginary shift to be a constant in the z-direction. We write

ψ L (x) = 1 4π ˆL/2 -L/2 ρ(s) (x -sẑ -iaẑ) 2 ds ( 8
)
where L is the length of the line charge, ϱ(s) the charge per unit length, and iaẑ is an imaginary vector representing how far o the real axis the charge is shifted, and in which direction so that a has units of length. So long as ρ(s) is real, there will be no apparent magnetic monopole charge far from this source.

The observation or eld point x is real-valued. The Riemann-Silberstein vector is given by

F(x) = -∇ψ L (x) (9)
If we take the limit of a → 0 we get the ordinary line charge which has divergent electrostatic energy. We shall search for systems for which the total energy and angular momentum are nite. The magnitude of the angular momentum will be proportional to the electric charge squared. The proportionality factor can be compared to the ne structure constant, and we dare to hope that some special class of these systems will give an explanation for its value. This model has two parameters, L and a, with dimension length. Dimensional considerations require that the total angular momentum be a function of the ratio L/a and not of L and a independently.

4.1 The Kerr-Newman limit

In the limit L → 0 we get the familiar point particle with complex shift which yields the Kerr-Newman solution in general relativity [START_REF] Lynden-Bell | A magic electromagnetic eld[END_REF][START_REF] Newman | Classical, geometric origin of magnetic moments, spinangular momentum, and the Dirac gyromagnetic ratio[END_REF][START_REF] Pekeris | The electromagnetic eld of a Kerr-Newman source[END_REF][START_REF] Adamo | The Kerr-Newman metric: A Review[END_REF]. The complex potential, in this case, is simply given by

ψ 0 (x) = q 4π (x -iaẑ) 2 (10)
For the Kerr-Newman metric, the angular momentum J KN and the mass m KN are determined by solving the Einstein eld equations and examining the metric behavior far from the source. It is found that a = J KN /m KN c. For an electron a = ℏ/(2m e c) = λ e /(4π) where λ e is the Compton wavelength of the electron. In our unit system, a = 1/2. This well-studied potential function has a ring singularity perpendicular to the z-axis with a radius a centered at z = 0. Because of the square root, it is a two-sheeted function. The cut can be chosen to be across the face of the ring singularity, and so long as we don't pass through this ring in analytic continuation, the function remains single-valued. This choice of the cut surface is not unique, and any deformation of it which is bounded by the ring is also a possible cut surface. If we pass through the cut though, the function exhibits its double-valued nature and we enter another Riemann sheet. This multi-valued behavior is the origin of the wormhole in general relativity which is popular in science ction as a stargate. It is also the origin of the Einstein-Rosen bridge [START_REF] Einstein | The Particle Problem in the General Theory of Relativity[END_REF], and it has been proposed that this multi-connectedness of space-time is the origin of quantum entanglement [START_REF] Maldacena | Cool horizons for entangled black holes[END_REF]. The integral of the electrostatic energy for the potential function in [START_REF] Pearle | Classical Electron Models[END_REF] is innite, but when the metric tensor is calculated using general relativity, the resulting mass of the solution as measured in the far eld multipole expansion is nite. It has also been proposed that elementary particles might be small Kerr-Newman solutions modied by interaction with quantum elds [START_REF] Burinskii | Microgeon with a Kerr metric[END_REF][START_REF] Burinskii | Spinning Particle as KerrNewman Black Hole[END_REF][START_REF] Newman | Classical, geometric origin of magnetic moments, spinangular momentum, and the Dirac gyromagnetic ratio[END_REF]. The possibility that quantum mechanics itself might be derivable in the context of complex space-time and classical eld theory has been explored [START_REF] Davidson | The Lorentz-Dirac equation in complex space-time[END_REF][START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF], and this paper is an extension of that exploration.

4.2 What about the mean value theorem and the maximum principle for harmonic functions?

The Kerr-Newman metric and the solutions that we consider here are derived from complex potential functions that satisfy Laplace's equation, that are multivalued, and that have branch-cut singularities near the center of the particle.

In order to use the mean value theorem in this case one must choose a boundary for the domain of the theorem which does not include any branch point.

Therefore, the boundary can never enclose the entire particle or soliton, and so the various elds can fall o at large distances on the physical Riemann sheet without violating the maximum principle. This is clearly true for the Kerr-Newman case, and it is also true for the cases we consider here as well. A non-constant single-valued static potential satisfying Laplace's equation everywhere must have innite energy because of the maximum principle. But, when the potential is multi-valued, nite energy is possible. So multi-valuedness is critical to most of the solutions we obtain here. This complicates the identication of the physical domain for volume integration over eld-derived quantities like energy, momentum, and angular momentum.

Evaluation of the complex potential function

Let us consider the simplest case of constant ϱ(s) in [START_REF] Bopp | Eine lineare Theorie des Elektrons[END_REF]. The indenite integral is of the form (up to a constant multiplier of q/4πL):

I(s) = ˆds α + βs + γs 2 (11)
This integral can be done analytically. The result is a multi-valued function whose Riemann-cut surfaces are quite complicated. A better way to proceed is to complete the square of the quadratic under the square root rst:

α + βs + γs 2 = γ * (s + β/(2 * γ)) 2 + α -β 2 /(4 * γ) (12)
If we change integration variables to u(s) = s + β/(2 * γ) then du = ds and the integral becomes a contour integral in the complex u plane

I(s) = 1 √ γ ˆκ du u 2 + α-β 2 /(4 * γ) γ ( 13
)
where κ is a smooth curve joining the two endpoints u(-L/2) and u(L/2).

Dene D = α -β 2 /(4 * γ) γ (14)
The parameters {α, β, γ} are determined by α + βs + γs 2 = (x -sẑ -iaẑ) 2 , and so

(x -sẑ -iaẑ) 2 = r 2 ⊥ + (z -s -ia) 2 , r 2 ⊥ = x 2 + y 2 α + βs + γs 2 = r 2 ⊥ + z 2 -a 2 -i2za -2zs + i2as + s 2 (15) α = r 2 ⊥ + z 2 -a 2 -i2za β = -2z + i2a γ = 1 (16) α = r 2 ⊥ + 1 4 β 2 (17)
From this formula, we see that

D = r 2 ⊥ (18) u(s) = (s + β/2) = (s -z + ia) (19)
This integral I(s) becomes

I(s) = 1 √ γ asinh(u(s)/ √ D) = ±asinh(u(s)/r ⊥ ) (20)
for all values of the eld position vector x. Note the ambiguity in the sign that comes from the √ γ term.

The inverse hyperbolic sine function asinh(ω) has branch points in the complex ω plane at ω = ±i. It is customary to draw the two Riemann cuts along the segments from +i to +i∞ and from -i to -i∞. Therefore, I(s) has branch points whenever u(s) = ±ir ⊥ . If a > 0 then the +i branch point maps into a ring in space dened by z = s and r ⊥ = a. The Riemann cut from this point maps into the disk in space bounded by this ring. Let Asinh denote the standard principal value function for asinh. It is discontinuous across this planar cut, and Asinh(u(s)/r ⊥ )will be discontinuous across the cut disk. The Riemann cut curve is not at all unique, but for any other choice the cut surface would be curved and with a larger area than the minimum planar cut across the disk.

Choosing a dierent cut surface could change the mass and angular momentum calculation. The complex potential function is then given by

ψ L (x) = 1 4π q L S(x) (asinh(u(L/2)/r ⊥ ) -asinh(u(-L/2)/r ⊥ )) (21)
S is a sign function that has values ±1. The sign function is determined by the physical constraint that at large radius the potential must approach a monopole electric charge. It turns out that this requires a for large r ⊥ that we must choose the principal value form

ψ L (x) = - 1 4π q L (Asinh(u(L/2)/r ⊥ ) -Asinh(u(-L/2)/r ⊥ )) , r ⊥ > a or |z| > L/2 (22)
If we use this formula for all values of r ⊥ and z, which is equivalent to choosing the cut planes across the two singular rings at z = ±L/2, then the electrostatic energy is divergent at the z-axis for -L/2 < z < L/2, and as r ⊥ → 0. The resulting electrostatic energy is innite, and so is the angular momentum. We have learned that to avoid this divergence one can choose the cut surface dierently, as shown in Figure 1 on page 9. This deformation of the Riemann cuts does not aect at all the elds at large values of r, but it greatly aects the near eld and makes the total electromagnetic energy and angular momentum nite and calculable. We choose the cylindrical shape to facilitate the integration, but we discuss the non-uniqueness of the cut surface further on in this paper.

It is useful to calculate the value of ψ L on the positive z-axis. We nd

ψ L (zẑ) = - q 4πL ln L/2 -z + ia -L/2 -z + ia , z > L/2 (23)
The large z expansion of ( 23) is

ψ L (zẑ) = q 4πz + iaq 4πz 2 + (L 2 /12 -a 2 )q 4πz 3 + O(1/r 4 ) (24)
From this expression, we can calculate the full multipole expansion of the elds.

Multipole expansion for complex line charge

The complex potential function satises Laplace's equation, therefore it can be expanded in a spherical harmonic series in analytic domains. Using azimuthal symmetry, we have the following complex multipole expansion for large r:

ψ L (x) = ∞ l=0 B l r l+1 P l (cos(θ)) (25)
Along the positive z-axis, θ = 0, and P l (cos(θ)) = 1, where P l is the Legendre polynomial, and where θ is the usual polar angle in spherical coordinates. The B l are the multipole moments, real values are electric moments and imaginary values are magnetic moments. If the potential has azimuthal symmetry as it does for the line charge, then the harmonic expansion coecients B L are determined by the values on the positive z-axis. The asymptotic power series for large r for the complex line charge is found from ( 24) to be:

ψ L (x) = q 4πr + iaqP 1 (cos(θ)) 4πr 2 + (L 2 /12 -a 2 )qP 2 (cos(θ)) 4πr 3 + O(1/r 4 ) (26)
All of the even Legendre-order terms in this series are real and therefore electric, and all the odd terms are imaginary and magnetic, and in the limit a → 0 all the odd terms vanish.

The rst term is the usual Coulomb term. The next term is a magnetic dipole term and it is exactly the same as in the Kerr-Newman solution. The third term is an electric quadrupole term. It can be seen that in the limit L → 0 it approaches the Kerr-Newman result. But, the interesting fact is that if L = √ 12a then the electric quadrupole moment vanishes. This would be good for modeling a real electron because it is strongly believed that the electron has zero electric quadrupole moment, but unfortunately this value gives the wrong number for the spin and mass of the electron calculated numerically. The higherorder multipole terms are not zero with this value of L either. We nd in the limit L → 0, the usual result for the Kerr-Newman electromagnetic eld [START_REF] Lynden-Bell | A magic electromagnetic eld[END_REF] which has a non-zero electric quadrupole is obtained

ψ 0 (x) = 1 4π q r ∞ n=0 ia r n P n (cos (θ)) (27)

Electromagnetic elds for complex line charge

We can now derive the electromagnetic elds starting with the Riemann-Silberstein vector. We use the principal value function outside the cylinder region dened by the inequalities r ⊥ > a or |z| > L/2

F(x) = -∇ψ L (x) = ∇ 1 4π q L (Asinh(u(L/2)/r ⊥ ) -Asinh(u(-L/2)/r ⊥ )) (28)
Inside the cylindrical region, we use analytic continuation utilizing the cut surfaces in Figure 1 on page 9. The r ⊥ → 0 singularity from the left term now cancels the singularity of the right term for all z ̸ = 0. The way this works is that as we continue into the interior half of the cylinder through the left ring, the left term changes sign, but the right term does not. This results in the cancellation of their singularities. The right half cylinder is just the mirror image of this, and cancellation occurs there too. Now we use the chain rule in cylindrical coordinates (r ⊥ , z, ϕ). We can start with the principal value function outside the cylinder and continue inside after dierentiating.

∇Asinh(u/r ⊥ ) = 1

(u/r ⊥ ) 2 + 1 ∇ u r ⊥ = 1 (u/r ⊥ ) 2 + 1 ∇u r ⊥ - u r⊥ r 2 ⊥ ( 29
)
Using u(s) = (s -z + ia) we have ∇u(s) = -ẑ and the eld expression becomes (for points x outside the cylindrical region)

F(x) = 1 4π q L -ẑ -u(L/2) r⊥ /r ⊥ u(L/2) 2 + r 2 ⊥ - -ẑ -u(-L/2) r⊥ /r ⊥ u(-L/2) 2 + r 2 ⊥ ( 30
)
The real part of F(x) gives the electric eld, and the imaginary part gives the magnetic eld. It is convenient to dene F + (x) and F -(x) by

F ± (x) = ± 1 4π q L -ẑ -u(±L/2} r⊥ /r ⊥ u(±L/2) 2 + r 2 ⊥ (31)
F = F + + F - (32)
Assume that a is positive without loss of generality. For z = ±L/2, and r ⊥ < a we nd that F ± is discontinuous across the disk of radius a centered on the z-axis if we analytically continue from one side of the disk to the other without going through it by continuing around its rim and back again to the other side.

We can determine the elds in the vicinity of the charged string by analytic continuation. This reveals two singularity rings of radius |a| centered on the z-axis at the points z = ±L/2.

The square root function of a complex variable √ ω has a branch point at ω = 0. It is customary to draw the Riemann cut from along the negative real axis {ω ∈ R, ω < 0}. The convention for dening the principal value P √ ω of √ ω is that whenever the real part of ω is positive, then the real part of P

√

ω is also positive. The discontinuity across the Riemann cut is then 2i P |ω| since the square root must change sign when crossing the cut to land on the other Riemann sheet. In order to obtain nite values for the electromagnetic mass, it is necessary to deform the cut as shown in Figure 1 on page 9. After doing this deformation, if we approach the z-axis anywhere along it, the limit of the potential is nite so long as we haven't crossed any part of the cut surface.

Four Riemann sheets

In [START_REF] Newman | Maxwell's equations and complex Minkowski space[END_REF] we see that F has four Riemann sheets since F + and F -each have two. Now consider that the physical Riemann sheet at spatial innity might be any one of these four. Two of them have eld lines that describe particles of opposite charge. These could be interpreted as particles and antiparticles. The existence of antiparticles is normally considered a prediction of relativistic quantum mechanics. On the other two sheets, the elds give innite energy.

Calculation of (hidden) angular momentum and electromagnetic energy of complex line charge

We now calculate the electromagnetic angular momentum of the elds about this charged string using [START_REF] Born | Foundations of the New Field Theory[END_REF]. This integral must be a function of a/L based solely on dimensional grounds. A plot of spin J vs L/a is shown in Figure 2 on page 12. The graph is surprisingly linear considering the complexity of the integral for electromagnetic angular momentum.

We nd the following linear curve t to the data in 2

J(a/L) = A + B(a/L) (33)
A = 3.400917E -05; B = 1.146253E -02 (34)
This is equivalent to the following extremely simple formula with an accuracy of about 6 decimal places:

J(a/L) = απ 2 a L = e 2 8 a L (35)
since in our units e 2 = 4πα.

The L=0 limit gives the elds of the Kerr-Newman metric. Since the angular momentum and the mass of that solution is nite in general relativity, it means that the eects of gravity prevent a divergence of the present string solution as L tends to zero. The electric quadrupole moment is still non-zero and in fact quite large for the Kerr-Newman case [START_REF] Rosquist | Gravitationally induced electromagnetism at the Compton scale[END_REF].

The electron's spin is |J| = 1/2 in our units. We note that the magnetic dipole moment does not depend on L as can be seen from ( 26). The volume integral of the angular momentum was calculated using the multi-precision capabilities of Python (see Appendix A). It was found that the double-exponential or tanh-sinh quadrature [START_REF] Mori | Discovery of the Double Exponential Transformation and Its Developments[END_REF][START_REF] Bailey | Tanh-Sinh High-Precision Quadrature[END_REF] was critical in obtaining good convergence.

This algorithm performs extremely well on integrands that are singular at the boundary of the integration domain. In the present instance, the singularities occur at r ⊥ = 0 or |a|, and z = ±L/2. One must break up the integration domains into sub-domains that have these singular points on the boundaries only.

The precision value for L/a that was found to give the correct electron spin is:

L/a = 0.0229254974781 → |J| = 1/2 (36)
In order to have zero electric quadrupole moment, we would need to have L/a = √ 12 which makes the spin angular momentum much too small to describe an electron or any other particle.

The electromagnetic mass density is given by ( 5). This value will depend on both the a and L parameters in our linear string model.

M = 1 2 V E 2 + B 2 dV ol (37)
A numerical calculation of this value for the L/a which gave the correct value of ℏ/2 for the angular momentum [START_REF] Toben | Space-time and Beyond: Toward an Explanation of the Unexplainable[END_REF] gives a correct mass of the electron to about 4 millielectronvolts accuracy for the case a = 1/2. This value of a corresponds to a magnetic moment which is the same as a Dirac electron, i.e.. a = J e /m e since in our units J e = 1/2 and m e = 1. The multiple-precision calculated values are J = 0.5000000

M = 0.999999993093 (38)
M E = 1 2 V E 2 dV ol = 0.502865 (39)
M B = 1 2 V B 2 dV ol = 0.497134 (40)
This is a stunning result. The electrostatic mass is equal to the true electron mass with tiny error. Moreover, it appears that in the high precision limit, M ass = 2 * J

Next we consider the electromagnetic Lagrangian which is the volume integral of the Lagrangian density E 2 -B 2 . For this graph, the value of a was held xed and L was varied. This shows slight nonlinearity as can be seen in Figure 4 on page 15

Note the scale on the vertical axis. The result is basically quite constant.

Running a curve-tting algorithm reveals the following functional relationship

ˆ E 2 -B 2 dvol = A 1 + B 1 * J + C 1 /J 2 (43)
A 1 = 1.146244E -02; B 1 = 5.515069E -08; C 1 = -4.587288E -08 [START_REF] Burinskii | Spinning Particle as KerrNewman Black Hole[END_REF] This can be approximated by the simple constant result independent of J that appears to be limiting behavior of the integral as the precision tends to innity (α is the ne structure constant)

Figure 4:

Volume integral of E 2 -B 2 ˆ E 2 -B 2 dvol ≈ π 2 α = e 2 8 ( 45
)
The extreme simplicity of these results has caused this author to worry if perhaps the numerical integrations were somehow giving wrong results, but persistent checking has not revealed an error.

Anomalous Magnetic moment correction

The anomalous magnetic moment of the electron is currently reported to have the experimental value [START_REF] Hanneke | Cavity control of a single-electron quantum cyclotron: Measuring the electron magnetic moment[END_REF] a e = g -2 2 = 0.001159652181643

In order to incorporate this into our model, we must increase the complex oset a from 1/2 to (1 + a e ) /2. We must simultaneously increase L by this same factor in order to maintain the ratio L/a in order that J doesn't change. Dimensional analysis mandates in this situation that the electrostatic mass varies in proportion to 1/a, and since a is increased by this anomalous factor, the mass will be decreased by it

M anomalous = m e 1 + a e = 0.998841691053m e (47)
or m e -M anomalous = 591.894651eV

And so the agreement is worse than without the anomalous g factor correction where the electrostatic mass was essentially exactly the electron mass. This has a positive implication though. The mass decit introduced by the anomalous g factor can be made up by a non-electromagnetic term in the particle's Lagrangian, such as the string term to improve the stability of the model. Alternatively, the anomalous g factor may be due to other eects which are not included in this simple model.

Straightforward application for the muon and tau leptons

It is trivial to apply this model to the muon and tau leptons. All we have to do is adjust the value of a for the mass of each lepton a muon = m e m µ a e ; and a tau = m e m τ a e

where a e = 1/2 in our units, and m µ is the mass of the muon, and m τ the mass of the tau. The L values would scale by these same factors to maintain the ratio a/L and thus J. This would automatically give the Dirac relations between mass, angular momentum, and magnetic moment for each lepton. Their anomalous magnetic moments would then allow for a non-electromagnetic term in the Lagrangian to help to stabilize the particle, as for the electron.

Stability of string models

For the detailed calculations of the line charge, we found that the electromagnetic Lagrangian was approximately constant if a was held xed [START_REF] Newman | Classical, geometric origin of magnetic moments, spinangular momentum, and the Dirac gyromagnetic ratio[END_REF], and this suggests that in this purely electromagnetic model, there is no tendency for the string to stretch in this case, but this is not a sti equilibrium point. If the length L were to change, there would be no restoring force to bring it back from a change. This reminds us of geons [51], but gravity is not playing a role in stability here.

To stabilize the string models, we might choose to add an internal tension force to the Lagrangian borrowed from string theory. As a candidate for complex space-time embedding, the most logical choice is the Polyakov action [START_REF] Polyakov | Quantum geometry of bosonic strings[END_REF][START_REF] Witten | Space-Time and Topological Orbifolds[END_REF][START_REF] Chamseddine | Hermitian Geometry and Complex Space-Time[END_REF].

It might also be of interest in the future to consider the much less well-known Stueckelberg-Horwitz-Piron (SHP) action [START_REF] Suleymanov | Second quantization of a covariant relativistic spacetime string in SteuckelbergHorwitzPiron theory[END_REF] since this theory generally has fewer problems with relativistic covariance than more conventional approaches [START_REF] Davidson | Relativistic quantum mechanics[END_REF]. We only consider bosonic strings here, but we keep in mind that fermionic properties, such as the g factor of 2, can arise from the complex shift technique as in the Kerr-Newman electron results. There is no requirement (yet) for 26 space-time dimensions in this theory, as there is in standard bosonic string theory because we are not quantizing the strings, and in fact. it is the author's hope that more features usually attributed to quantum mechanics may emerge from this classical theory.

An action principle

The complex form of the Polyakov action for a string was treated in [START_REF] Witten | Space-Time and Topological Orbifolds[END_REF][START_REF] Chamseddine | Hermitian Geometry and Complex Space-Time[END_REF]. The Hermitian metric approach in these theories can be applied to a purely bosonic string as in the case here. For the electromagnetic eld, we could choose to work either with the Riemann-Silberstein vector or the manifestly covariant Faraday tensor. Presumably, they are equivalent. The action for the combined system is the sum of the string action plus the eld action plus an interaction term. The interaction term between the electromagnetic eld and the charged string is the tricky part here because the coordinates of the string are complex. Since there is no actual charge on the real subspace for the string models, we plausibly can write for the real-valued action [START_REF] Chamseddine | Hermitian Geometry and Complex Space-Time[END_REF]

I = ˆdσdσg µν ∂ σ X(σ, σ)∂ σ X(σ, σ) -ˆd4 x 1 4 F αβ F αβ (50)
F αβ here is the real Faraday tensor calculated by the complex-shift method for currents that are o the real subspace using the complexied forms for the Liénard-Wiechert potentials [START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF]. The background metric g here is derived from the Minkowski metric in 4D. σ is the complex conjugate of σ.

g µν = g µν = 0 ( 52
)
where η is the real Minkowski metric and the Greek subscripts {µ, µ, ν, ν}are Lorentz indices.

In this action, the Faraday tensor F must be expressed as a functional of the string's curve X(σ, σ) utilizing the complexied Liénard-Wiechert potentials plus an additional free-eld solution. The functions to vary would be the string's world-sheet and the part of the vector potential contribution to F which is not due to the string current. In this way, we can avoid the problem of an interaction term of the form j µ A µ which would need to be evaluated at a complex spacetime point. This requires a nonlocal action [START_REF] Marnelius | Action Principle and Nonlocal Field Theories[END_REF][START_REF] Eliezer | The problem of nonlocality in string theory[END_REF][START_REF] Heredia | Nonlocal Lagrangian elds: Noether's theorem and Hamiltonian formalism[END_REF]. So basically, it's pretty dicult to work with.

The problem with introducing a j µ A µ term in the action is that there is not a perfect generalization of the Dirac delta function to complex spaces which preserves analyticity. There is, however, an imperfect one due to Lindell [START_REF] Lindell | Time-Domain Green Function Corresponding to a Time-Harmonic Point Source in Complex Space[END_REF][START_REF] Lindell | Delta function expansions, complex delta functions and the steepest descent method[END_REF][START_REF] Brewster | Generalized delta functions and their use in quantum optics[END_REF][START_REF] Brewster | Generalized Delta Functions and Their Use in Quasi-Probability Distributions[END_REF], and perhaps a local action could be developed using this. The complexshift method does not use a Hermitian metric. Consider the point particle.

Its potential is q/ (x -o) 2 and when o is complex the result is no longer real. So you can't use a Hermitian metric to evaluate (x -o)

2

. Rather you must use analytic continuation. The elds generated by the string this way are generally multi-valued. This situation, for point particles, was considered in [START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF] where a new mathematical technique called generalized analytic continuation was applied to the Liénard-Wiechert potentials in complex Minkowski space.

The result of this is that one can imagine wave-particle duality emerging from it, although the concept is a bit of a metaphysical leap.

Dependence on the choice of the cut surface

For the cut surface found in Figure 1 on page 9, the elds are discontinuous across the curved cylindrical cut surface. Because of this, if we make an innitesimal change in the surface, it will change the result of calculations for total mass and angular momentum. However, it is found numerically that the changes in these are in the same ratio as the linear slope relating mass and angular momentum so that the mass versus angular momentum curve won't change, and of course, deforming the cut won't aect the large distance moments like the magnetic moment either. Innitesimal changes in the shape of the planar cuts at z=0 do not aect the mass or angular momentum either because the two densities are continuous in z there for all values of r > 0. So the main results that we obtain should allow some variation in the shape of the cut surface before it leads to noticeable eects. The arbitrariness of the cut surface, at least for small deviations acts a bit like a gauge invariance of this theory.

Compatibility with the Kerr-Schild metric

A Kerr-Schild metric in general relativity can be expressed in the form

g µν = η µν + ϕk µ k ν (53)
where η µν is the background metric, which in our case is the Minkowski metric. ϕ is a scalar function and k µ a null congruence. The Einstein eld equations take the form

R µν - 1 2 Rg µν + Λg µν = 1 4π F µα F α ν + 1 4 g µν F αβ F αβ (54)
A concise necessary condition that must be satised by the electromagnetic elds to allow a Kerr-Schild metric is presented in [START_REF] Bah | Kerr-Schild double copy and complex worldlines[END_REF]. The condition is that the elds in the background metric must satisfy the following constraint.

F ν µ A ν -χA µ = 0, F µν = ∂ µ A ν -∂ ν A µ ( 55
)
where χ is a scalar function. If χ ̸ = 0 then it follows that A is a null vector. In general, this is a dicult condition to test for. Our charged line model provides us with E and B elds from which we can construct the Maxwell-Faraday tensor F µν and see if it satises [START_REF] Marnelius | Action Principle and Nonlocal Field Theories[END_REF]. I've tested the condition numerically. It is found to be satised in the limit L → 0 as it should because this is the Kerr-Newman limit, but for nonzero values of L, it does not allow an exact Kerr-Schild metric solution. The numerical method used the following approach. First, pick a point x in 3 space. Next, calculate F µν there for the charged line. Next, nd the eigenvectors which satisfy the equation

F µ ν l ν = λl µ (56)
These produce a null tetrad with two degenerate real and null eigenvectors and two complex conjugate null ones. Choose one of the real eigenvectors and do the eigenvector calculation in a neighborhood of x making sure that the resulting function l µ (x) varies smoothly in this neighborhood. This smoothness condition is critical because as there are two real null eigenvectors if the computer algorithm selects the wrong one for a neighboring point, then there will be a discontinuity which will prevent taking derivatives of A µ . We write

A µ (x) = χl µ (x) (57)
We determine χ by the condition

χ = A 0 (x)/l 0 (x) (58)
where A 0 (x) is the known electric potential. Then the 3-vector potential to test is

A test (x) = χl(x) (59)
and so B test = ∇ × A test , and if this agrees with the magnetic eld that we started with, then we have met the condition for a Kerr-Schild metric. This was all done numerically, and it was found that there is a small dierence between the magnetic elds, so a Kerr-Schild metric solution cannot be exact, but it is approximate, and the deviation could in principle be added as a small perturbation to the Kerr-Schild metric. It is noteworthy that there are two null elds to choose from, and they produce vector potentials that have a handedness or chiral property that distinguishes them, although the magnetic and electric elds are identical. These two solutions must therefore dier by a gauge transformation. This handedness feature could perhaps be related to the two chiral states of the Dirac equation.

5

Other models

A curve in complex space with uniform charge density

Let's now consider a curved string parameterized by arclength (of its real part)

s. Let x s denote the real part of the curve, and let the total length be L with charge per unit length constant and given by q/L. We shall use the Frenet-Serret apparatus to describe this curve in terms of tangent, normal, and binormal vectors. In the static case, we can write ψ as

ψ(x) = ˆL 0 ρ(s) (x -x 0 (s)) 2 ds (60)
x 0 (s) = X re (s) + it(s) t(s) + in(s)n(s) + ib(s) b(s)

ˆL 0 ρ(s)ds = q

where X re (s) is a real curve in space, t(s), n(s), and b(s) are the tangent, normal, and binormal unit vectors for this curve which is parametrized by arclength s. The complex shift is characterized by the functions t, n, and b, and where ρ(s)

is the charge per unit length along the curve. Closed as well as open strings are possible. All knots and links are possible. In general, we would expect such a structure to be time varying. This would require a complex Minkowski space to be completely general. The elds would need to be calculated by using Liénard-Wiechert potentials suitably generalized, and this is highly non-trivial.

Alternatively, We can write the coordinates of the line charge in of the form

x(s) = X re (s) + iX im (s) (63)
where X re and X im are independent functions of s. In the case of a closed loop, we could obviously generalize this further by including a line-current contribution to the electromagnetic potential. This model is not generally invariant under parity, and so it can have a chirality property.

A simple case of a ring of charge with a complex shift along the axis of the ring

The simplest case is the following source geometry

x 0 (s) = Rcos(θ(s))x + Rsin(θ(s))ŷ + iaẑ (64) θ(s) = s R , ρ(s) = q 2πR (65) ψ(x) = 1 4π ˆ2πR 0 q/2πR (x -x 0 (s)) 2 ds (66)
When we shrink the ring diameter to zero we obtain the Kerr-Newman source. The integral can be done analytically in terms of Appel hypergeometric functions. Along the z-axis, the result is trivial.

ψ(x)| x=y=0 = 1 4π q R 2+ (z -ia) 2 (67)
Doing a large-z Taylor expansion we nd

4πψ(x)| = q/z + qia/z 2 -q(2a 2 + R 2 )/2z 3 + O(1/z 4 ) (68)
We have azimuthal symmetry here, and since the potential satises Laplace's equation, we then know from this that the Legendre expansion for all angles is

4πψ(x) = q/r + qia/r 2 P 1 (cos(θ)) -q(a 2 + R 2 /2)/r 3 P 2 (cos(θ)) + O(1/r 4 ) (69)
So we can see from this that the monopole and dipole terms are the same as for the Kerr-Newman case and the line charge above, but the electric quadrupole moment, that is (a 2 + R 2 /2)/4π, is dierent, and unfortunately it can never vanish, so this simple ring charge cannot describe an electron with zero electric quadrupole moment.

Another simple case of a barbell of two charges with complex shift

Consider two charges, each with half the charge, separated by a distance L and with the same complex shift a parallel to the direction of the separation. For this case, which also has azimuthal symmetry, we get 4πψ(x) = q/r +qia/r 2 P 1 (cos(θ))-q((a 2 -L 2 /4))/r 3 P 2 (cos(θ))+O(1/r 4 ) (70)

We see that the electric quadrupole moment here is proportional to (a 2 -L 2 /4), and this vanishes if L = 2a. We could replace the two charges on the end of the barbell with short line-charges, and this would allow us to have a large but nite mass and angular momentum.

A composite string model that can possibly describe an electron

We would like to nd models that give correct values for four parameters of the electron: mass, spin, magnetic dipole, and electric quadrupole. The simple line charge had two parameters, L and a. The ring had two parameters R and a. If we get lucky, we might nd a simple model which gives the correct value for all four parameters with just three degrees of freedom, and this would amount to a calculation of the ne structure constant. However, there is no obvious way to guess what such a structure might be, and so we can in the meantime consider composite systems with four free parameters to adjust. Here we will consider one such model of this type with four degrees of freedom. For simplicity, we will consider models with azimuthal symmetry.

Consider the model of a ring of charge with radius R with a complex shift utilizing the Frenet apparatus as in [START_REF] Brewster | Generalized delta functions and their use in quantum optics[END_REF] with a constant charge density ρ as well as constant real-valued shift parameters t, n, and b resulting in the potential integral in (66), but with a more complicated x 0 (s) of the form x 0 (s) = Rcos(θ(s))x + Rsin(θ(s))ŷ + it t(θ(s)) + inn(θ(s)) + ib b(θ(s)) (71)

t(θ) = -sin(θ)x + cos(θ)ŷ (72) n(θ) = -cos(θ)x -sin(θ)ŷ (73) b(θ) = ẑ (74)
The far eld expansion on the z-axis is found to be

4πψ(z) = q/z + iqb/z 2 + q(t 2 + n 2 -2b 2 -R 2 -i2Rn)/(2z 3 ) + O(1/z 4 ) (75)
This expression has a monopole charge, a magnetic dipole given by qb, and an electric and magnetic quadrupole term. The electric quadrupole vanishes if the following condition is satised:

t 2 + n 2 -2b 2 -R 2 = 0 (76)
The magnetic quadrupole moment is -qRn, and this can vanish if n = 0, and therefore, it is possible to eliminate both quadrupole moments with this model.

The binormal shift b is determined by the dipole moment alone. In the limit of R → 0 we approach the Kerr-Newman solution once again, and therefore the electromagnetic mass will diverge if we don't include metric curvature. Obviously, for R → ∞ , the electromagnetic mass goes to zero. Consequently, somewhere in between these limits, it will equal the mass of the electron. If we require that both quadrupole moments be zero, then we can still get the correct mass, but will have no more parameters to adjust for the angular momentum, so either it just happens to be the correct one for the electron, i.e.

ℏ/2, or we would have to allow one of the quadrupole moments to be nonzero.

If the correct spin of the electron was derived this way, it would amount to a mathematical derivation of the ne structure constant, although we consider this lucky outcome to be very unlikely. Presumably, it would be more popular to set the electric quadrupole to zero and allow a nonzero magnetic quadrupole.

Then we should be able to adjust the parameters to give the correct spin. So it seems likely that this model can describe an electron as a purely electromagnetic object. It does have a magnetic quadrupole moment though, which might be a way to test this picture further. Finding the parameters t, n, b, and R is beyond the scope of this paper. Dierent sets of parameters could be used for other charged particles, and so we have here a fairly general-purpose classical charged model.

A spherical shell model which has only a charge monopole and a magnetic dipole

Consider a charged spherical shell of radius a, and no charge except on this.

Assume azimuthal symmetry. The solution for Laplace's equation in spherical coordinates is:

ψ(x) = ∞ l=0 A l r l + B l r l+1 P l (cos(θ)) (77)
Inside the shell, the B l terms must vanish, and outside the A l terms must vanish.

Outside the shell, we write

ψ(x) = q 4πr + iq 4π a 1 r 2 P 1 (cos(θ)), r > a (78)
Inside the shell we apply continuity requirements for Maxwell's equation to obtain

ψ(x) = iqa 1 4π r -2/a 3 P 1 (cos(θ)), r < a (79)
The magnetic dipole moment is M = qa 1 . This model does not have any internal chirality. It was analyzed and angular momentum and energy were calculated in [START_REF] Giulini | Electron spin or classically non-describable two-valuedness[END_REF], section 3.1. Inside the shell, the magnetic eld is constant and directed along ẑ, the symmetry axis, and the electric eld is zero. This ensures that the radial component of the magnetic eld is continuous across the shell. With these elds, we can calculate the mass, the angular momentum, and the magnetic dipole moment. Since the electric eld is zero inside the shell, the Poynting vector is zero there too, and so the hidden angular momentum comes only from the elds outside of the shell. The elds are all single-valued, and therefore the calculations of mass and angular momentum are straightforward.

This model gives zero for all higher multipole moments which is consistent with a Dirac particle. There are two free parameters, the radius of the sphere a and the magnetic dipole moment controlling parameter a 1 . We would like to t the mass, angular momentum, and magnetic dipole of the particle. If a 1 is chosen to give the measured dipole moment, then this leaves only the one parameter a to t the mass and the angular momentum. The electrostatic energy calculated from [START_REF] Giulini | Electron spin or classically non-describable two-valuedness[END_REF] is

E = q 2 8πa + M 2 4πa 3 = q 2 8πa 1 + 2a 2 1 a 2 (80)
The electromagnetic angular momentum is

J = q 2 a 18π 3a 1 a 2 = q 2 6π a 1 a (81)
For an electron, this relation becomes

ℏ 2 = e 2 6π a 1 a (82)
But e 2 = 4πα and ℏ = 1 in our units. Therefore

a 1 a = 3 4α ≈ 102.77699 (83)
and then the mass of the electron, which is 1 in our units, is given by

me=1= α 2a 1 + 2 3 4α 2 (84)
and from this we can solve for a. In our units, the classical electron radius is simply α.

where λ e is the Compton wavelength of the electron. We can see from ( 80) and ( 83) that practically all the mass of the electron in this model is due to the magnetic eld.

An extended uid drop model with complex shift

We apply the complex shift method to the model in [START_REF] Bialynicki-Birula | Classical Model of the Electron. Exactly Soluble Example[END_REF]. The particle there was a soliton in an inviscid charged uid. The relativistic stress-energy tensor is

T µν = 1 4π F µ λ F λν + (ρ + p) u µ u ν -g µν - 1 16π F αβ F αβ + p ( 86
)
The ow is assumed to be adiabatic, and the pressure as a function of the uid density η is taken to be p = -κη 6/5 (87) where this form was selected to allow a simple analytic solution, although other possibilities might also lead to interesting models. The pressure is negative and grows more negative the denser the uid is. This leads to a cohesive force that balances the electrostatic repulsion. The resulting static electric eld is found in [START_REF] Bialynicki-Birula | Classical Model of the Electron. Exactly Soluble Example[END_REF] to be

E = qr 4π b 2 + r 2 -3/2 , b = (4π/3) 1/2 e 2 /6κ 5/2 (88) with potential ψ = q 4π b 2 + r 2 -1/2 (89)
Now we make the complex shift along the z-axis so that z origin → ia to obtain

ψ = q 4π b 2 + x 2 + y 2 + (z -ia) 2 -1/2 (90)
We can analyze the multipole expansion by setting x = y = 0 and calculating the Taylor-Laurent expansion at large z

ψ| r=z ẑ = q 4π b 2 + (z -ia) 2 -1/2 (91) ψ| r=z ẑ = q 4π 1 z + ia z 2 - b 2 + 2a 2 2z 3 - i 3ab 2 + 2a 3 2z 4 + O( 1 z 5 ) (92)
From this, we have the usual monopole and magnetic dipole terms. The electric quadrupole term is nonzero except in the limit a = b = 0. The magnetic octapole term can be zero if 3ab 2 + 2a 3 =0. This liquid drop model might be interesting for modeling nuclei.

6

Can wave-particle duality be understood in complex space-time models without formally quantizing them?

Arguments supporting this idea were presented in [START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF][START_REF] Davidson | The Lorentz-Dirac equation in complex space-time[END_REF]. First consider the electromagnetic eld produced by a moving charge in conventional, i.e. real space-time. Assuming the particle is moving on a timelike curve, an observer at some instant sees the particle at exactly one point, which involves a retarded time calculation that makes the apparent position dependent on the past trajectory of the particle. There is no wave-particle duality here. The electromagnetic force is our window into reality. The Liénard-Wiechert potentials can be viewed as dening the properties of this window. Let's suppose that our classical charged particle is describable by a Hamilton-Jacobi (HJ) theory. It's well known that quantum mechanics can be transformed into HJ theory via Bohmian mechanics, and the converse is also true. Any HJ classical particle model can be transformed into a wave equation with a suitable potential function that may be nonlinear. This is true for multi-particle systems as well. Now suppose that each particle is moving in complex Minkowski space-time and is guided by some HJ equations. This problem was studied in [START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF] for a particular Bohmian system. The retarded time calculation is no longer unique. The apparent location of the source can now be multi-valued. If you allow a partition of unity of the particle trajectory and perform an independent analytic continuation of each partition, then you can obtain wave-particle duality. I called this procedure generalized analytic continuation in [START_REF] Davidson | Bohmian Trajectories for KerrNewman Particles in Complex Space-Time[END_REF]. The Liénard-Wiechert potentials, when utilized in complex space-time, give us a blurry coke-bottle view of reality, but they produce more than just an apparent distortion because they determine how a particle interacts electromagnetically with all other particles. Given the multitude of apparent positions that a single charged particle can have due to this eect, the natural question of causality and locality obviously needs to be addressed. Modern interpretations of quantum mechanics are moving towards the inclusion of time-symmetric formulations [START_REF] Bopp | Time Symmetric Quantum Mechanics and Causal Classical Physics ?[END_REF], and of course, quantum entanglement, in general, has suggested the possibility of nonlocal connections in physics. Since complex space-time is intimately connected with classical general relativity and with electromagnetism, one is led to wonder whether it could be possible that classical general relativity with electromagnetism could quantize itself. In other words, could quantum theory be derivable from this classical eld theory? This would be a beautiful and unexpected solution to the quantumgravity dilemma that physics currently faces, and it would be one that Einstein would probably be very pleased with. Given the results of this paper, it would also seem that string theory should play a critical role in this project.

Conclusion

The complex-shift method for generating charged electromagnetic particle-like solutions has been applied to extended charged objects in this paper, and it was found that several features of elementary particles can be phenomenologically A number of phenomena normally associated with quantum mechanics can be described qualitatively in this framework. These include charged particles with magnetic dipoles having Dirac g factors of 2 or anomalous g factors, particles with zero electric quadrupole moments, particles with internal chirality and topological symmetries related to knots and links, a possible explanation for wave-particle duality, and A qualitative understanding of the possible origins of quantum entanglement and nonlocality. These results seem to suggest a plausible path to emergent quantum mechanics from classical general relativity and electromagnetism in complex space-time. Charged string theory in complex space-time might turn out to be a theory of everything, including quantum mechanics itself. Although we can understand the possible origins of wave-particle duality, we don't have a detailed understanding of how nature might orchestrate a universe with a large multitude of Riemann sheets superimposed with one another to give us the laws of quantum mechanics. A statistical mechanics treatment is called for, perhaps one based on Adler's trace dynamics [65].

In place of Many Worlds, we have many Riemann sheets. The philosophy of emergent quantum mechanics is in sharp contrast to the prevailing ER=EPR movement which aims to derive everything, including the space-time manifold, from quantum theory. The two approaches might be dual in the sense that quantum mechanics might be emergent from classical theory while the inverse is also true.

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2: Angular momentum plot for total charge e as a function of the dimensionless ratio a/L in units of ℏ

  this error gets smaller and smaller. Let's focus on what this means. The Dirac equation relates the mass, the spin magnitude, and the magnetic dipole moment, and it is in the framework of quantum mechanics. If you know any two of these parameters, then you also know the third. The Kerr-Newman metric solution gives exactly the same relation between mass, spin, and magnetic dipole

Figure 3 :

 3 Figure 3: Total electromagnetic energy plot for a=1/2

  described in this way. The string models are particularly interesting because they can have nite values for mass and angular momentum, they can have chirality, and they can be knotted and linked in complex and diverse ways. The linear charged model perfectly predicts the electron mass to be its measured mass given only the charge, spin, and magnetic moment of the electron. The sceptical reader can reproduce this result by executing the software listed in the appendix. This result is based entirely on electromagnetism, and it gives the same relation between mass, spin, and magnetic moment that the Dirac equation gives, but without quantum theory. The spin in this case is entirely due to electromagnetic angular momentum. Nothing is rotating. It can be compared with the expectation of the spin operator in a quantum Pauli equation which behaves much like a classical angular momentum[START_REF] Giulini | Electron spin or classically non-describable two-valuedness[END_REF].

  return factor *(1/ L) * \ ( rightsign (z ,r ,L ,a) * cf ( uf (L /2 ,z ,a) ,r) * mp . matrix ([ -1 , -uf (L /2 ,z ,a )/r ]) -leftsign (z ,r ,L ,a)* cf ( uf (-L /2 ,z ,a) ,r) * mp . matrix ([ -1 , -uf (-L /2 ,z , a)/r ]) ) def E(L ,a ,z , r): return F(L ,a ,z ,r). apply ( mp . re ) def B(L ,a ,z , r): return F(L ,a ,z ,r). apply ( mp . im ) # Now define Poynting vector = E x B in the phi direction in cylindrical # coordinates . # z x r = phi unit vector equation def Poynting (L ,a ,z , r): return E(L ,a ,z ,r) [0]* B(L ,a ,z ,r) [1] -E(L ,a ,z ,r) [1]* B (L ,a ,z ,r ) [0] # Now we define the angular momentum density in the z direction ( r x P) _z # It appears that Jz is continuous across the cut boundary def Jz (L ,a ,z ,r): return r* Poynting (L ,a ,z ,r) def Esq (L ,a) : def igE (z ,r ): Ez =E(L ,a ,z ,r) [0] Er =E(L ,a ,z ,r) [1] return 2* mp . pi *r *( Ez * Ez + Er * Er ) sumE0 = mp . quad ( igE ,[0 ,L /2] ,[a , rm ], error = True , method = 'tanhsinh ', ebsabs = eabs ) sumE1 = mp . quad ( igE ,[L /2 , rm ] ,[a , rm ], error = True , method = ' tanhsinh ', ebsabs = eabs ) sumE2 = mp . quad ( igE ,[L /2 , rm ] ,[0 ,a] , error = True , method = 'tanh -sinh ', ebsabs = eabs ) sumE3 = mp . quad ( igE ,[0 ,L /2] ,[0 ,a ], error = True , method = 'tanhsinh ', ebsabs = eabs ) sumEsq = sumE0 [0]+ sumE1 [0]+ sumE2 [0]+ sumE3 [0] return twofactor * sumEsq def Bsq (L ,a) : def igB (z ,r ): Bz =B(L ,a ,z ,r) [0] Br =B(L ,a ,z ,r) [1] return 2* mp . pi *r *( Bz * Bz + Br * Br ) sumB0 = mp . quad ( igB ,[0 ,L /2] ,[a , rm ], error = True , method = 'tanhsinh ', ebsabs = eabs ) sumB1 = mp . quad ( igB ,[L /2 , rm ] ,[a , rm ], error = True , method = ' tanhsinh ', ebsabs = eabs ) sumB2 = mp . quad ( igB ,[L /2 , rm ] ,[0 ,a] , error = True , method = 'tanh -sinh ',

  The radius of the shell is

	a =	α 2	+	9 16α	≈ 77.0863985 ≈ 12.26868λe

[65] S. L. Adler, Quantum theory as an emergent phenomenon: the statistical mechanics of matrix models as the precursor of quantum eld theory, Cambridge University Press, 2004.

A

Python script Break the integration region into four regions . Multiply the integrals for J , Esq , and Bsq by two to account for left + right symmetry , and a = J /m should be 1/2 for the electron in our units ( hbar =1 , mass of electron = 1) """ import scipy . constants as c; from scipy import integrate ; import time import mpmath as mp ; print ( ' Start Program at ', time . ctime () ) mp . mp . dps = 12; eabs = 1e -8; mp . pretty = True ; twofactor = 2 lda = mp . mpf (0.0229254974781) q = mp . sqrt (4* mp . mpf (c . pi ) * mp . mpf (c. alpha ) ) factor = -(q /(4* mp . pi ) ) rm = mp . inf def uf (s ,z ,a ): return s -z +1 j* a def cf (u ,r): """ Here is looping code to create spreadsheet data """ f= open ( ' JBEData . txt ', 'a ') print ( 'L ,a , rm , dps , prec , eabs = ' ,L ,a , rm ,mp . mp . dps ,mp . mp . prec , eabs ) print ( 'L ,a , rm , dps , prec , eabs = ' ,L ,a , rm ,mp . mp . dps ,mp . mp . prec , eabs , file =f) print (" a/L ,L ,a , Jcalc ,( E2calc + B2calc ) /2 ,( E2calc -B2calc ) /2 ") print (" a/L ,L ,a , Jcalc ,( E2calc + B2calc ) /2 ,( E2calc -B2calc ) /2 " , file = f) f. close () while L < Lfinish : Lnext =(1+ increment )* L Jcalc = Jst (L/ a) E2calc = Esq (L ,a) B2calc = Bsq (L ,a) f= open ( ' JBEData . txt ', 'a ') print (a /L , ', ',L , ',',a , ', ', Jcalc ,', ' ,( E2calc + B2calc ) /2 , ', ' ,( E2calc -B2calc ) /2) print (a /L , ', ',L , ',',a , ', ', Jcalc ,', ' ,( E2calc + B2calc ) /2 , ', ' ,( E2calc -B2calc ) /2 , file =f) f. close () L= Lnext """ Here is some previous output Ldafound = 0.0229254974781 L ,a , rm , dps , prec = 0.0114627487391 0.5 + inf 12 43 J = 0.499999994429 Esq = 0.502865609542 Bsq = 0.49713438355 Esq -Bsq = 5.85730092837 kev Mass = 0.999999993093 Mass Error = -0.00352955023643 ev """


J Hazra

Y Phulpin

D Ernst
email: dernst@ulg.ac.be

HVDC Control Strategies to Improve Transient Stability in Interconnected Power Systems

This paper presents three HVDC modulation strategies to improve transient stability in an interconnected power system. AC variables such as rotor speeds, voltage phasors, and tieline power flows are used as input to the controller that modifies the power flow settings through the HVDC-links. The proposed techniques are tested on the IEEE 24-Bus reliability test system and critical clearing times obtained for several contingencies are analyzed. The paper shows that HVDC modulation can lead to substantial improvement in transient stability.

I. INTRODUCTION

In the aftermath of severe power system disturbances, transient stability issues such as loss of synchronism phenomena may occur [START_REF] Pavella | Transient Stability of power systems[END_REF]. Those phenomena may have for effect to lead in a very short period of time to the loss of many synchronous generators and, in turn, endanger the stability of the whole system [START_REF] Kundur | Power System Stability and Control[END_REF]. Numerous control schemes have thus been proposed to mitigate loss of synchronism phenomena. They can be broadly classified into two groups, i.e. preventive control and emergency control. Whereas preventive control anticipates events to ensure that the power system is to withstand the most critical events, emergency control is executed immediately after the disturbances.

Much research has been carried out to design effective emergency control schemes. By way of example, an emergency generator and load shedding scheme based on the topological and dynamic characterization of stability boundary is proposed in [START_REF] Min | A new method for generation shedding and load shedding in power system emergency control[END_REF]. Reference [START_REF] Li | An on-line transient stability emergency control strategy based on PMU forecasted trajectory[END_REF] proposes an on-line emergency control strategy for generator shedding based on PMU measurements. An open-loop emergency control scheme based on generation rescheduling and/or generation tripping is developed in [START_REF] Ruiz-Vega | A comprehensive approach to transient stability control. II. open loop emergency control[END_REF]. Real-time closed-loop emergency control schemes for generation shedding are also presented in [START_REF] Ernst | Closed-loop transient stability emergency control[END_REF], [START_REF] Glavic | E-SIME -a method for transient stability closed-loop emergency control: achievements and prospects[END_REF]. These methods compute corrective actions by using equal area type criteria together with real-time measurements.

Usually, emergency control techniques involve actions on generators and/or loads to maintain stability. In addition, flexible transmission devices such as fast valving and breaking resistors [START_REF] Patel | A modified approach to transient stability enhancement with fast valving and braking resistor applications[END_REF], fast excitation controllers [START_REF] Fernandopulle | Domain of stability of a synchronous machine with excitation control-a cell mapping approach[END_REF], or tie-line reactance controllers [START_REF] Reitan | A method of improving transient stability by bang-bang control of tie-line reactance[END_REF] could also be used as means for emergency control of transient stability. In power systems where HVDClinks have been installed, a better alternative could be to use those devices as emergency control means. As AC/DC conversion involves no inertia and power settings of HVDClinks can be changed quasi-instantaneously [START_REF] Hingorani | Understanding FACTS: Concepts and Technology of Flexible AC Transmission Systems[END_REF], modulating of active power flow through the HVDC-links has been proposed to improve power system stability. By way of example, References [START_REF] Tso | Fast prediction of transient stability margin in systems with SVC control and HVDC link[END_REF]- [START_REF] Xiao-Ming | Application of HVDC modulation in damping electromechanical oscillations[END_REF] present modulation techniques for stability purpose.

When an HVDC-link replaces a conventional AC line, this alternative is even necessary, as DC transmission devices do not offer the damping abilities of AC transmission lines when they are operated in a constant power mode. In this paper, we propose to study how the power flow settings of the HVDClinks can be controlled to increase transient stability margins. Four control strategies are considered and their effectiveness is analyzed. First, the HVDC control settings are unchanged during the disturbance. Second, the power flow through the HVDC-link is modulated based on the rotor speed at the two sides of the HVDC-link. Third, the modulation is based on the phase angle difference between the inverter and converter side buses. Fourth, the modulation is based on the rate of change of the power flow between the control areas where the HVDC end buses are located. The performance obtained with the different control strategies is evaluated on the IEEE 24 bus system with two HVDC links.

The paper is organized as follows. Section II details the proposed control strategies, Section III presents the benchmark power system, Section IV presents the simulation results, and Section V concludes.

II. CONTROL SCHEME FOR HVDC-LINK MODULATION

This paper focuses on four control strategies for the HVDClink, where the inputs are based on different AC variables. It is supposed that those variables are known instantaneously and no information delay is considered. The control variables are the power flow settings P DC for each HVDC-link. The settings are subjected to limitations with respect to a maximum and minimum value of the power flow. In addition, as introduced in [START_REF] Khatir | Analysis of recovery from commutation failures in an HVDC inverter connected to a weak receiving ac system[END_REF] to reduce the risk of commutation failure and voltage instability, P DC is automatically reduced when the DC voltage at the rectifier bus is below a specified threshold value.

Paper accepted for presentation at 2009 IEEE Bucharest Power Tech Conference, June 28th -July 2nd, Bucharest, Romania 978-1-4244-2235-7/09/$25.00 ©2009 IEEE For each controllable HVDC-link, the power flow setting is derived from a specific input signal e by a proportional integral controller, which can be mathematically written as follows

P DCset = K p • e + K i e • dt, (1)
P DC = P DC0 + P DCset , (2)
where K p and K i are the proportional and integral coefficients and P DC0 is the reference value for the HVDC power flow, which is to be determined by a secondary control level with a larger time constant. We refer to References [START_REF] Pizano-Martinez | Modeling of VSC-based HVDC systems for a Newton-Raphson OPF algorithm[END_REF], [START_REF] Pilotto | Determination of needed FACTS controller that increase asset utilization of power systems[END_REF] for some practical examples of such secondary control levels.

The four control strategies to compute the HVDC settings P DCset based on different input signals e are described hereafter.

A. Strategy 1

With strategy 1, e is equal to zero, i.e. no specific power modulation control is applied to the HVDC system. Nevertheless, P DC is subjected to the limitations introduced here above.

B. Strategy 2

With strategy 2, P DCset is modulated based on the rotor speed of the generators located closest to the rectifier and inverter buses of the HVDC-link, respectively.

As emphasized in [START_REF] Eriksson | On the assessment of the impact of a conventional HVDC on a test power system[END_REF], this type of input signal may have for effect to transfer more power from the area where the machines are accelerating faster, which should mitigate this acceleration and therefore improve transient stability in the system. The error signal e is derived as follows

e = d dt (w DCr -w DCi ), (3)
where w DCr and w DCi are the rotor speeds of the rectifier and inverter buses of the HVDC-link, respectively.

C. Strategy 3

With strategy 3, P DCset is modulated based on the voltage phase angle at the end buses of the HVDC-link. The rationale of this control strategy is that the difference of voltage phase angle at the end buses of the HVDC-link may give a good image of the mechanical angles of the generators located closest to those buses. The voltage phase angles can be measured through phasor measurement units, for example.

The error signal e is computed as follows

e = d dt (δ DCr -δ DCi ), (4)
where δ DCr and δ DCi are the voltage phase angle at the rectifier and inverter side buses of the HVDC-link, respectively.

D. Strategy 4

With strategy 4, P DCset is modulated based on the AC power flow in the inter-area tie lines. This control scheme supposes the definition of well-defined control areas interconnected by the HVDC-links. The input signal e is based on the sum of the active power flows between the control areas where the converter and inverter buses are located, which is represented by

P IA . e = d dt (P IA ) (5)

III. DESCRIPTION OF THE BENCHMARK POWER SYSTEM

This section first introduces the models of the different power system elements of the IEEE 24 bus system that will be used later in the simulations. Second, the numerical program chosen to simulate the system is described.

A. AC system representation

1) Generator model: In the benchmark system, the generators are synchronous machines with constant mechanical power. The electrical part of the generator is represented by a constant voltage source behind a transient reactance. This model has often been chosen in transient stability studies [START_REF] Pai | Energy Function Analysis for Power System Stability[END_REF], [START_REF] Pavella | Transient Stability of Power Systems: Theory and Practice[END_REF].

2) Load model: Load demands are considered voltage dependent in the benchmark system. In particular, the real and reactive power demand at bus i (P i and Q i , respectively) are expressed as follows

P i = P i0 (a i + b i • V i + c i • V 2 i ), (6)
Q i = Q i0 (a i + b i • V i + c i • V 2 i ), (7)
where P i0 and Q i0 are the active and reactive power demand at bus i at nominal voltage, respectively, a i , b i , and c i are the coefficients of active power demand at bus i and a i , b i , and c i are the coefficients of reactive power demand at bus i.

3) Transmission line model:

The transmission lines are represented by constant admittances.

B. HVDC-link representation

The HVDC power electronics dynamics is neglected in this study, as its time constant is much smaller than those involved in transient stability phenomena [START_REF] Ni | A simplified two-terminal dc model and its use in direct transient stability assessment[END_REF]. The power flow P DC through each HVDC-link is represented as a positive load at the rectifier bus, and negative load at the inverter bus. The parameters of those elements are function of the rectifier and inverter side AC bus voltages, rectifier and inverter side transformer tap positions, and internal HVDC parameters, i.e. converter firing angle α, inverter extinction angle γ, and current in the DC line.

Three modes of operation are designed to compute the power injections from the end buses of each HVDC-link [START_REF] Kundur | Power System Stability and Control[END_REF]. Under normal conditions, the HVDC-links operate in constant power mode when the rectifier operates in constant current control and the inverter operates in constant extinction angle control. During disturbances, when α reaches extreme values, the mode of operation changes. The control modes can be written as follows.

• Mode 1: Constant power/current control at the rectifier and constant extinction angle at the inverter. • Mode 2: Minimum firing angle control at the rectifier and constant power/current control at the inverter. • Mode 3: Minimum firing angle control at the rectifier, and constant extinction control at the inverter. This is the transition mode between Mode 1 and Mode 2.

C. Computation tool for time-domain analysis

Most transient stability analysis methods represent HVDClinks as constant power loads regardless of their response during disturbances. However, in this study, HVDC-links are treated as non-linear variable transmission devices. To incorporate their features, a specific algorithm has been developed, where load flows are sequentially solved at each simulation step to update the injections at the rectifier and inverter buses.

As emphasized in [START_REF] Mobarak | Modified load flow analysis for integrated ACDC power systems[END_REF], Newton-Raphson method may be used to solve the network equations. However, this method is computationally expensive because it needs to calculate the Jacobian at each iteration of the algorithm. To improve the computational performance, a fast decoupled solution technique is used to solve AC and DC network equations iteratively. The time-domain computation technique developed for transient stability evaluation of AC/DC system is as follows. 3) Compute the real and reactive power injections at each DC terminal. If the mismatch is greater than the predefined tolerance, go back to step 1 and solve AC equations. 4) Using the bus voltages, calculate generator currents and electrical outputs. 5) Solve generator swing equations using trapezoidal rule [START_REF] Kaw | An interactive e-book for illustrating the Trapezoidal Rule for numerical integration[END_REF] and update the voltages behind the transient reactance by changing their phase angles. 6) Implement changes in the HVDC settings based on the control scheme under investigation. 7) If the maximum simulation time has not been reached, repeat the simulation procedure from step 1.

To improve computational time, the sparsity technique introduced in [START_REF] Zollenkopf | Large Sparse Sets of Linear Equations, ch. Bifactorization-Basic computational algorithm and programming techniques[END_REF] has been used to solve the network equations.

IV. SIMULATION RESULTS

Simulations have been carried out on the IEEE 24 bus reliability test system with two bipolar HVDC-links added as shown in Fig. 1. One link is located between bus 21 and bus 11, and the other one is between bus 18 and bus 3. The main pre-fault load-flow information are provided in Tables I-III.

A. Illustrative example

A three phase to ground fault is applied to bus 16 at t=100 ms, and the fault is cleared by tripping lines 16-14, 16-17 and 16-19 at t=200 ms. As the time frame of interest in transient stability analysis is usually three to five seconds following the disturbance [START_REF] Kundur | Power System Stability and Control[END_REF], this study is limited to the first five seconds after the short-circuit inception. If the maximum angular difference between two machines goes beyond 180 degrees, the system is assumed unstable.

1) Strategy 1: Simulation results in this section are obtained with no specific control action. System responses due to the bus fault are displayed in Fig. 23, which show the power flow through HVDC-links and electromechanical swings of the generators, respectively.

It can be noticed by observing the swing curves that the generators separate into two groups in the aftermath of the disturbance. In particular, generators at buses 18, 21, and 22 get out of step together. Moreover, though P DCset remains unchanged for each HVDC-link, P DC oscillates due to the voltage swings, which induce limitations on P DC to avoid commutation failures.

2) Strategy 2: Simulation results obtained with strategy 2 are shown in Fig. 45. As introduced in Section II-B, since no generator is located at the southern end bus of the HVDC, the rotor speed of the generator at bus 13 has been associated to the end buses.

With this modulation scheme, the system damps the severe electromechanical swing among the generators and prevents the generators at buses 18, 21, and 22 from going out of step. Therefore, in this case, the system is stable. The performance of the controller is based on the proper tuning of the controller parameters. In this case, those are chosen as follows: K p = 0.01 and K i = 0.001.

3) Strategy 3: Fig. 6 shows the modulated HVDC signal and Fig. 7 presents the corresponding swing plot of all the generators in the system.

Those figures show that the system is stable when the HVDC modulation is done according to strategy 3 with K p = 0.1 and K i = 0.001.

4) Strategy 4: Fig. 8 shows P DC and Fig. 9 shows the generator angles. In this study, P IA is computed as the sum of the active power flows from bus 15 to bus 24, 14 to 11, 23 to 12, and 23 to 13.

Simulation results show that using strategy 4 is effective. In this case, K p = 0.02 and K i = 0.001.

B. Effectiveness of the control strategies

The effectiveness of the proposed control strategies is analyzed in terms of critical clearing time (CCT). This index represents the maximum short-circuit duration the power can sustain without loosing synchronism [START_REF] Kundur | Power System Stability and Control[END_REF]. As most the methods for computing CCT do not consider control actions, the CCTs are computed by using a dichotomy search.

The effects of four different contingencies have been analyzed, and simulation results are reported in Table IV every contingency considered. On the other hand, significantly larger CCTs are obtained with the alternative strategies. In particular, strategy 4 leads to the highest CCT for every contingency under consideration. The improvement is however limited (10 through 70 ms) when the short-circuits are located at buses 15 and 6. In those cases, it might be supposed that the improvement in CCT would also be limited if an AC line were in the place of each HVDC-link.

V. CONCLUSIONS This paper presents three HVDC modulation techniques to improve transient stability of interconnected power systems with integrated HVDC systems. The proposed techniques assess the power flow settings for the HVDC-links based on different AC variables. Simulation results show that HVDC modulation significantly increases the stability of the system, which has been measured in our experiences in terms of critical clearing times for different contingencies. This finding suggests that the proposed emergency control schemes could be effective for systems with HVDC links. However, before applying the proposed techniques, important issues need to be addressed. In particular, the assumption of instantaneous knowledge of the AC variables is certainly optimistic, if not naive, and the impact of communication delays/failures on the emergency control schemes should be investigated. In addition, as the performance obtained with the control strategies depends on the proportional and integral coefficients, a proper selection should be handled to optimally assess their values. Furthermore, it must be acknowledged that, whereas the test system was tested under different contingencies for which the HVDC-modulation was effective, it is unlikely to obtain the same results for any contingency in the system. It could thus be relevant to design advanced control strategies that could improve transient stability of the system for every possible contingency. Those strategies should certainly rely on coordinated control for power systems with multiple HVDClinks.

1 )

 1 Solve the load flow problem to obtain bus voltages. 2) Solve Mode 1 DC equations. (a) If α > α min , go to step 3. (b) If α ≤ α min , solve Mode 2 equations. If γ > γ min , go to step 3. (c) If γ ≤ γ min , solve Mode 3.

Fig. 1 .

 1 Fig.1. IEEE 24 bus RTS-96 system with two HVDC-links.

2 Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 . 2 Fig. 8 .Fig. 9 .

 2234567289 Fig. 2. Power flow through the HVDC-link with strategy 1.

TABLE II PRE

 II -CONTINGENCY POWER FLOWS IN THE POWER SYSTEM.

	From bus To bus	P from (MW) Q from (MVAR)
	16	17	-144	-49
	15	21	-111	-40
	15	24	98	-2
	11	14	44	10
	23	12	114	16
	23	13	40	16

  . It can be observed that the lowest CCT is obtained with strategy 1 for