A tool for stability and power sharing analysis of a generalized class of
droop controllers for high-voltage direct-current transmission systems
Daniele Zonetti, Romeo Ortega and Johannes Schiffer

arXiv:1609.03149v2 [] 20 Mar 2017

Abstract
The problem of primary control of high-voltage direct current transmission systems is addressed in this paper, which contains
four main contributions. First, to propose a new nonlinear, more realistic, model for the system suitable for primary control
design, which takes into account nonlinearities introduced by conventional inner controllers. Second, to determine necessary
conditions—dependent on some free controller tuning parameters—for the existence of equilibria. Third, to formulate additional
(necessary) conditions for these equilibria to satisfy the power sharing constraints. Fourth, to establish conditions for stability
of a given equilibrium point. The usefulness of the theoretical results is illustrated via numerical calculations on a four-terminal
example.

I. INTRODUCTION
For its correct operation, high-voltage direct current (hvdc) transmission systems—like all electrical power systems—must
satisfy a large set of different regulation objectives that are, typically, associated to the multiple time—scale behavior of the
system. One way to deal with this issue, that prevails in practice, is the use of hierarchical control architectures [1]–[3].
Usually, at the top of this hierarchy, a centralized controller called tertiary control—based on power flow optimization
algorithms (OPFs)—is in charge of providing the inner controllers with the operating point to which the system has to be
driven, according to technical and economical constraints [1]. If the tertiary control had exact knowledge of such constraints
and of the desired operating points of all terminals, then it would be able to formulate a nominal optimization problem and
the lower level (also called inner-loop) controllers could operate under nominal conditions. However, such exact knowledge
of all system parameters is impossible in practice, due to uncertainties and lack of information. Hence, the operating points
generated by the tertiary controller may, in general, induce unsuitable perturbed conditions. To cope with this problem
further control layers, termed primary and secondary control, are introduced. These take action—whenever a perturbation
occurs—by promptly adjusting the references provided by the tertiary control in order to preserve properties that are essential
for the correct and safe operation of the system. The present paper focuses on the primary control layer. Irrespectively of
the perturbation and in addition to ensuring stability, primary control has the task of preserving two fundamental criteria: a
prespecified power distribution (the so-called power sharing) and keeping the terminal voltages near the nominal value [4].
Both objectives are usually achieved by an appropriate control of the dc voltage of one or more terminals at their point of
interconnection with the hvdc network [2], [5], [6].
Clearly, a sine qua non requirement for the fulfillment of these objectives is the existence of a stable equilibrium point for
the perturbed system. The ever increasing use of power electronic devices in modern electrical networks, in particular the
presence of constant power devices (CPDs), induces a highly nonlinear behavior in the system—rendering the analysis of
existence and stability of equilibria very complicated. Since linear, inherently stable, models, are usually employed for the
description of primary control of dc grids [3], [6], [7], little attention has been paid to the issues of stability and existence
of equilibria. This fundamental aspect of the problem has only recently attracted the attention of power systems researchers
[8]–[10] who, similarly to the present work, invoke tools of nonlinear dynamic systems analysis, to deal with the intricacies
of the actual nonlinear behavior.
The main contributions and the organization of the paper are as follows. Section II is dedicated to the formulation—under
some reasonable assumptions—of a reduced, nonlinear model of an hvdc transmission system in closed-loop with standard
inner-loop controllers. In Section III a further model simplification, which holds for a general class of dc systems with short
lines configurations, is presented. A first implication is that both obtained models, which are nonlinear, may in general have
no equilibria. Then, we consider a generalized class of primary controllers, that includes the special case of the ubiquitous
voltage droop control, and establish necessary conditions on the control parameters for the existence of an equilibrium point.
This is done in Section IV. An extension of this result to the problem of existence of equilibria that verify the power sharing
property is carried out in Section V. A last contribution is provided in Section VI, with a (local) stability analysis of a known
equilibrium point, based on Lyapunov’s first method. The usefulness of the theoretical results is illustrated with a numerical
example in Section VII. We wrap-up the paper by drawing some conclusions and providing guidelines for future investigation.
D. Zonetti and R. Ortega are with the Laboratoire des Signaux et Systémes, 3, rue Joliot Curie, 91192 Gif-sur-Yvette, France.

daniele.zonetti@gmail.com, romeo.ortega@lss.supelec.fr
J. Schiffer is with the School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK, j.schiffer@leeds.ac.uk

Notation. For a set N = {l, k, . . . , n} of, possibly unordered, elements, we denote with i ∼ N the elements i = l, k, . . . , n.
All vectors are column vectors. Given positive integers n, m, the symbol 0n ∈ Rn denotes the vector of all zeros, 0n×m
the n × m column matrix of all zeros, 1n ∈ Rn the vector with all ones and In the n × n identity matrix. When clear from
the context dimensions are omitted and vectors and matrices introduced above are simply denoted by the symbols 0, 1 or
I. For a given matrix A, the i-th colum is denoted by Ai . Furthermore, diag{ai } is a diagonal matrix with entries ai ∈ R
and bdiag{Ai } denotes a block diagonal matrix with matrix-entries Ai . x := col(x1 , . . . , xn ) ∈ Rn denotes a vector with
entries xi ∈ R. When clear from the context it is simply referred to as x := col(xi ).
II. NONLINEAR MODELING OF HVDC TRANSMISSION SYSTEMS
A. A graph description
The main components of an hvdc transmission system are ac to dc power converters and dc transmission lines. The power
converters connect ac subsystems—that are associated to renewable generating units or to ac grids—to an hvdc network. In
[11] it has been shown that an hvdc transmission system can be represented by a directed graph1 without self-loops, where
the power units—i.e. power converters and transmission lines—correspond to edges and the buses correspond to nodes.
Hence, a first step towards the construction of a suitable model for primary control analysis and design is then the definition
of an appropriate graph description of the system topology that takes into account the primary control action.
We consider an hvdc transmission system described by a graph G ↑ (N , E), where n = c + 1 is the number of nodes, where
the additional node is used to model the ground node, and m = c+t is the number of edges, with c and t denoting the number
of converter and transmission units respectively. We implicitly assumed that transmission (interior) buses are eliminated via
Kron reduction [12]. We further denote by p the number of converter units not equipped with primary control—termed
PQ units hereafter—and by v the number of converter units equipped with primary control—that we call voltage-controlled
units, with c = p + v. To facilitate reference to different units we find it convenient to partition the set of converter nodes
(respectively converter edges) into two ordered subsets NP and NV (respectively EP and EV ) corresponding to P Q and
voltage-controlled nodes (respectively edges). The incidence matrix associated to the graph is given by:


Ip
0
BP
Iv
BV  ∈ Rn×m ,
(II.1)
B= 0
>
0
−1
−1>
v
p
where the submatrices BP ∈ Rp×t and BV ∈ Rv×t fully capture the topology of the hvdc network with respect to the
different units.
B. Converter units
For a characterization of the converter units we consider power converters based on voltage source converter (VSC)
technology [13]. Since this paper focuses on primary control, we first provide a description of a single VSC in closed-loop
with the corresponding inner-loop controller. In hvdc transmission systems, the inner-loop controller is usually achieved
via a cascaded control scheme consisting of a current control loop whose setpoints are specified by an outer power loop
[14]. Moreover, such a control scheme employs a phase-locked-loop (PLL) circuit, which is a circuit that synchronizes an
oscillator with a reference sinusoidal input [15]. The PLL is thus locked to the phase a of the voltage vac,i (t) and allows,
under the assumption of balanced operation of the phases, to express the model in a suitable dq reference frame, upon
which the current and power loops are designed, see [16], [17] for more details on this topic. For these layers of control,
different strategies can be employed in practice. Amongst these, a technique termed vector control that consists of combining
feedback linearization and PI control is very popular, see [17]–[19] for an extensive overview on this control strategy. A
schematic description of the VSC and of the overall control architecture, which also includes, if any, the primary control
layer, is given in Fig. 1. As detailed above, the inner-loop control scheme is based on an appropriate dq representation of
the ac-side dynamics of the VSC, which for balanced operating conditions is given by the following second order dynamical
system [17]:
Li I˙d,i = −Ri Id,i + Li ωi Iq,i − dd,i vC,i + Vd,i
(II.2)
Li I˙q,i = −Li ωi Id,i − Ri Iq,i − dq,i vC,i + Vq,i
where Id,i ∈ R and Iq,i ∈ R denote the direct and quadrature currents, vC,i ∈ R+ denotes the dc voltage, dd,i ∈ R and
dq,i ∈ R denote the direct and quadrature duty ratios, Vd,i ∈ R and Vq,i ∈ R denote the direct and quadrature input voltages,
Li ∈ R+ and Ri ∈ R+ denote the (balanced) inductance and the resistance respectively. Moreover, the dc voltage dynamics
can be described by the following scalar dynamical system:
Ci v̇C,i = −Gi vC,i + ii + iC,i ,

ii := dd,i Id,i + dq,i Iq,i ,

(II.3)

1 A directed graph is an ordered 3-tuple, G ↑ = {N , E, Π}, consisting of a finite set of nodes N , a finite set of directed edges E and a mapping Π from
E to the set of ordered pairs of N .

PLL

Fig. 1: Control architecture of a three-phase voltage source converter that interfaces an ac subsystem—characterized by a
three–phase input voltage vac,i (t)—to an hvdc network—characterized by an ingoing dc current iC,i (t). Bold lines represent
electrical connections, while dashed lines represent signal connections [16].

where iC,i ∈ R denotes the current coming from the dc network, ii denotes the dc current injection via the VSC, Ci ∈ R+
and Gi ∈ R+ denote the capacitance and the conductance respectively. For a characterization of the power injections we
consider the standard definitions of instantaneous active and reactive power associated to the ac-side of the VSC, which are
given by [20], [21]:
Pi := Vd,i Id,i + Vq,i Iq,i , Qi := Vq,i Id,i − Vd,i Iq,i ,
(II.4)
while the dc power associated to the dc-side is given by:
PDC,i := vC,i ii .

(II.5)

We now make two standard assumptions on the design of the inner-loop controllers.
?
Assumption 2.1: Vq,i = Vq,i
= 0,

∀t ≥ 0.

Assumption 2.2: All inner-loop controllers are characterized by stable current control schemes. Moreover, the employed
schemes guarantee instantaneous and exact tracking of the desired currents.
Assumption 2.1 can be legitimized by appropriate design of the PLL mechanism, which is demanded to fix the dq
transformation angle so that the quadrature voltage is always kept zero after very small transients. Since a PLL usually
operates in a range of a few ms, which is smaller than the time scale at which the power loop evolves, these transients can
be neglected.
Similarly, Assumption 2.2 can be legitimized by an appropriate design of the current control scheme so that the resulting
closed-loop system is internally stable and has a very large bandwidth compared to the dc voltage dynamics and to the outer
loops. In fact, tracking of the currents is usually achieved in 10 − 50 ms, while dc voltage dynamics and outer loops evolve
at a much slower time-scale [1].

Under Assumption 2.1 and Assumption 2.2, from the stationary equations of the currents dynamics expressed by (II.2),
?
?
i.e. for I˙d,i
= 0, I˙q,i
= 0, we have that
d?d,i =


1
?
?
?
−Ri Id,i
+ Li ωi Iq,i
+ Vd,i
,
vC,i

d?q,i =


1
?
?
−Li ωi Id,i
− Ri Iq,i
,
vC,i

(II.6)

?
?
where Id,i
and Iq,i
denote the controlled dq currents (the dynamics of which are neglected under Assumption 2.2), while
?
Vd,i denotes the corresponding direct voltage on the ac-side of the VSC. By substituting (II.6) into (II.3) and recalling the
definition of active power provided in (II.4), the controlled dc current can thus be expressed as

i?i =

? ?
? 2
? 2
Vd,i
Id,i − Ri (Id,i
) − Ri (Iq,i
)
P ? − Di?
= i
,
vC,i
vC,i

(II.7)

where
? ?
Pi? := Vd,i
Id,i ,

 ? 2

? 2
Di? := Ri (Id,i
) + (Iq,i
)

(II.8)

denote respectively the controlled active power on the ac-side and the power dissipated internally by the converter. We then
make a further assumption.
Assumption 2.3: Di? = 0.
Assumption 2.3 can be justified by the high efficiency of the converter, i.e. by the small values of the balanced threephase resistance R, which yield Di? ≈ 0. Hence, by replacing (II.7) into (II.3) and using the definitions (II.8), we obtain the
following scalar dynamical system [21]:
Ci v̇C,i = −Gi vC,i +

?
Vd,i
I ? + iC,i
vC,i d,i

(II.9)

with i ∼ EP ∪ EV , which describes the dc-side dynamics of a VSC under assumptions 2.1, 2.2 and 2.3. By taking (II.9) as a
point of departure, we next derive the dynamics of the current-controlled VSCs in closed-loop with the outer power control.
If the unit is a PQ unit, the current references are simply determined by the outer power loop via (II.4) with constant
?
active power Pjref and reactive power Qref
j , which by noting that Vq,j = 0, are given by:
?
Id,j
=

Pjref
? ,
Vd,j

?
Iq,j
=−

Qref
j
? ,
Vd,j

(II.10)

with j ∼ EP , which replaced into (II.9) gives
Cj v̇C,j = −Gj vC,j + uj (vC,j ) + iC,j .

(II.11)
Pjref

with the new current variable uj and the dc voltage vC,j verifying the hyperbolic constraint
= vC,j uj , j ∼ EP . Hence,
ref
:= Pjref ,
a PQ unit can be approximated, with respect to its power behavior, by a constant power device of value PP,j
see also Fig. 2a. On the other hand, if the converter unit is a voltage-controlled unit, the current references are modified
according to the primary control strategy. A common approach in this scenario is to introduce an additional deviation (also
called droop) in the direct current reference—obtained from the outer power loop—as a function of the dc voltage, while
keeping the calculation of the reference of the quadrature current unchanged:
?
Id,k
=

Pkref
? + δk (vC,k ),
Vd,k

?
Iq,k
=−

Qref
k
? ,
Vd,k

(II.12)

with k ∼ EV and where δk (vC,k ) represents the state-dependent contribution provided by the primary control. We propose
the primary control law:

1
2
δk (vC,k ) = ? µP,k + µI,k vC,k + µZ,k vC,k
,
(II.13)
Vd,k
with k ∼ EV and where µP,k , µI,k , µZ,k ∈ R are free control parameters. By replacing (II.12)-(II.13) into (II.9), we obtain
Ck v̇C,k = −(Gk − µZ,k )vC,j + µI,k + uk (vC,k ) + iC,k ,

(II.14)

with the new current variable uk and the dc voltage vC,k verifying the hyperbolic constraint Pkref + µP,k = vC,k uk , k ∼ EV .
Moreover, with Assumption 2.3 the injected dc power is given by:
ref
2
PDC,k (vC,k ) = PV,k
+ µI,k vC,k + µZ,k vC,k
,

(II.15)

with
ref
PV,k
:= Pkref + µP,k ,

from which follows, with the control law (II.13), that a voltage-controlled unit can be approximated, with respect to its

(a) Equivalent circuit scheme for PQ units.

(b) Equivalent circuit scheme for voltage-controlled units.

Fig. 2: Equivalent circuit schemes of the converter units with constant power devices (CPDs), under Assumption 2.2.
power behavior, by a ZIP model, i.e. the parallel connection of a constant impedance (Z), a constant current source/sink
ref
(I) and a constant power device (P). More precisely—see also Fig. 2b—the parameters PV,k
, µI,k and µZ,k represent the
constant power, constant current and constant impedance of the ZIP model. Finally, the dynamics of the PQ units can be
represented by the following scalar systems:
Cj v̇C,j = −Gj vC,j + uj + iC,j ,
ref
0 = PP,j
− vC,j uj ,

while for the dynamics of the voltage-controlled units we have:
Ck v̇C,k = −(Gk − µZ,k )vC,k + µI,k + uk + iC,k ,
ref
0 = PV,k
− vC,k uk ,

with j ∼ EP , k ∼ EV and where vC,j , vC,k ∈ R+ denote the voltages across the capacitors, iC,j , iC,k ∈ R denote the
network currents, uj , uk ∈ R denote the currents flowing into the constant power devices, Gj ∈ R+ , Gk ∈ R+ , Cj ∈ R+ ,
Ck ∈ R+ denote the conductances and capacitances. The aggregated model is then given by:



       
CP v̇P
GP
0
vP
u
0
i
=−
+ P +
+ P ,
(II.16)
CV v̇V
0 GV + GZ vV
uV
ūV
iV
together with the algebraic constraints:
ref
PP,j
= vP,j uP,i ,

ref
PV,k
= vV,k uV,k ,

with i ∼ EP , k ∼ EV and the following definitions.
- State vectors
vP := col(vC,j ) ∈ Rp ,

vV := col(vC,k ) ∈ Rv .

- Network ingoing currents
iP := col(iC,j ) ∈ Rp ,

iV := col(iC,k ) ∈ Rv .

uP := col(uj ) ∈ Rp ,

uV := col(uk ) ∈ Rv .

- Units ingoing currents
- External sources ūV := col(µI,k ) ∈ Rv .
- Matrices
CP : = diag{Cj } ∈ Rp×p , CV := diag{Ck } ∈ Rv×v ,
GP : = diag{Gj } ∈ Rp×p , GV := diag{Gk } ∈ Rv×v , GZ := diag{−µZ,k } ∈ Rv×v .
C. Interconnected model
For the model derivation of the hvdc network we assume that the dc transmission lines can be described by standard,
single-cell π-models. However, it should be noted that at each converter node the line capacitors will result in a parallel
connection with the output capacitor of the converter [22]. Hence, the capacitors at the dc output of the converter can be
replaced by equivalent capacitors and the transmission lines described by simpler RL circuits, for which it is straightforward
to obtain the aggregated model [11]:
(II.17)
LT i̇T = −RT iT + vT ,

with iT := col(iT,i ) ∈ Rt , vT := col(vT,i ) ∈ Rt denoting the currents through and the voltages across the lines and
LT := col(LT,i ) ∈ Rt×t , RT := col(RT,i ) ∈ Rt×t denoting the inductance and resistance matrices. In order to obtain the
reduced, interconnected model of the hvdc transmission system under Assumption 2.2, we need to consider the interconnection
laws determined by the incidence matrix (II.1). Let us define the node and edge vectors:
 
 
 
VP
vP
iP
Vn := VV  ∈ Rc+1 , Ve := vV  ∈ Rm , Ie := iV  ∈ Rm .
0
vT
iT
By using the definition of the incidence matrix (II.1) together with the Kirchhoff’s current and voltage laws given by [23],
[24]:
BIe = 0,
Ve = B > Vn ,
we obtain:
iP = −BP vP ,

iV = −BV vV ,

Replacing iP and iV in (II.16) and vT in (II.17), leads to

 
CP v̇P
−GP
0
CV v̇V  =  0
−GV
BP>
BP>
LT i̇T

vT = BP> vP + BV> vV .

the interconnected model:
     
−BP
vP
uP
0
−BV  vV  + uV  + ūV  ,
iT
0
0
−GZ

(II.18)

(II.19)

together with the algebraic constraints:
ref
PP,j
= vP,j uP,j ,

ref
PV,k
= vV,k uV,k ,

(II.20)

with i ∼ EP , k ∼ EV .
Remark 2.4: With the choice
µP,k = 0,

nom
µI,k = dk vC
,

µZ,k = −dk ,

the primary control (II.13) reduces to:
δk (vC,k ) = −

dk
nom
? (vC,k − vC ),
Vd,k

while the injected current is simply given by
i?k =

?
Vd,k
P ref
?
nom
Id,k
= k − dk (vC
− vC,k ),
vC,k
vC,k

with k ∼ EV . This is exactly the conventional, widely diffused, voltage droop control [2], [6], [25], where dk is called droop
nom
is the nominal voltage of the hvdc system. The conventional droop control can be interpreted as an
coefficient and vC
appropriate parallel connection of a current source with an impedance, which is put in parallel with a constant power device,
thus resulting in a ZIP model. A similar model is encountered in [4] and should be contrasted with the models provided in
[3], [7], where the contribution of the constant power device is absent.
Remark 2.5: A peculiarity of hvdc transmission systems with respect to generalized dc grids is the absence of traditional
loads. Nevertheless, the aggregated model of the converter units (II.16) can be still employed for the modeling of dc grids
with no loss of generality, under the assumption that loads can be represented either by PQ units (constant power loads) or
by voltage-controlled units with assigned parameters (ZIP loads). This model should be contrasted with the linear models
adopted in [3], [7] for dc grids, where loads are modeled as constant current sinks.
III. A REDUCED MODEL FOR GENERAL DC SYSTEMS WITH SHORT LINES CONFIGURATIONS
Since hvdc transmission systems are usually characterized by very long, i.e. dominantly inductive, transmission lines, there
is no clear time-scale separation between the dynamics of the power converters and the dynamics of the hvdc network. This
fact should be contrasted with traditional power systems—where a time-scale separation typically holds because of the very
slow dynamics of generation and loads compared to those of transmission lines [26]—and microgrids—where a time-scale
separation is justified by the short length, and consequently fast dynamics, of the lines [27]. Nevertheless, as mentioned
in Remark 2.5, the model (II.19)-(II.20) is suitable for the description of a very general class of dc grids. By taking this
model as a point of departure, we thus introduce a reduced model that is particularly appropriate for the description of
a special class of dc grids, i.e. dc grids with short lines configurations. This class includes, among the others, the widely
popular case of dc microgrids [28] and the case of hvdc transmission systems with back-to-back configurations [29]. For
these configurations, we can then make the following assumption.
Assumption 3.1: The dynamics of the dc transmission lines evolve on a time-scale that is much faster than the time-scale
at which the dynamics of the voltage capacitors evolve.

Under Assumption 3.1, (II.17) reduces to:
iT ≡ i?T = GT vT ,

(III.1)

RT−1

where i?T is the steady-state vector of the line currents and GT :=
the conductance matrix of the transmission lines.
By replacing the expression (III.1) into (II.19) we finally obtain:



     
L +G
Lm
CP v̇P
vP
u
0
=− P > P
+ P +
,
(III.2)
CV v̇V
uV
ūV
Lm
LV + GV + GZ vV
together with the algebraic constraints (II.20) and where we defined
LP : = BP GL BP> ,

Lm := BP GL BV> ,

Remark 3.2: The matrix:


Lm
∈ Rc×c
LV



LP
L>
m

L :=

LV := BV GL BV> .

is the Laplacian matrix associated to the weighted undirected graph Ḡ w , obtained from the (unweighted directed) graph G ↑
that describes the hvdc transmission system by: 1) eliminating the reference node and all edges connected to it; 2) assigning
as weights of the edges corresponding to transmission lines the values of their conductances. Similar definitions are also
encountered in [3], [7].

IV. CONDITIONS FOR EXISTENCE OF AN EQUILIBRIUM POINT
From an electrical point of view, the reduced system (II.19)-(II.20) is a linear RLC circuit, where at each node a constant
power device is attached. It has been observed in experiments and simulations that the presence of constant power devices
may seriously affect the dynamics of these circuits hindering the achievement of a constant, stable behavior of the state
variables—the dc voltages in the present case [10], [30]–[32]. A first objective is thus to determine conditions on the free
control parameters of the system (II.19)-(II.20) for the existence of an equilibrium point. Before presenting the main result
of this section, we make an important observation: since the steady-state of the system (II.19)-(II.20) is equivalent to the
steady-state of the system (III.2)-(II.20), the analysis of existence of an equilibrium point follows verbatim. Based on this
consideration, in the present section we will only consider the system (III.2)-(II.20), bearing in mind the the same results
hold for the system (II.19)-(II.20). To simplify the notation, we define
ref
PPref : = col(PP,j
) ∈ Rp ,

RP := LP + GP ∈ Rp×p ,

ref
PVref : = col(PV,k
) ∈ Rv ,

RV := LV + GV + GZ ∈ Rv×v .

(IV.1)

Furthermore, we recall the following lemma, the proof of which can be found in [10].
Lemma 4.1: Consider m quadratic equations of the form fi : Rn → R,
fi (x) :=

1 >
x Ai x + x> Bi ,
2

i ∈ [1, m],

(IV.2)

n×n
where Ai = A>
, Bi ∈ Rn , ci ∈ R and define:
i ∈R

A(T ) : =

m
X

ti Ai ,

B(T ) :=

i=1

m
X

ti Bi ,

C(T ) :=

i=1

m
X

t i ci .

i=1

If the following LMI

Υ(T ) :=


A(T )
B(T )
> 0,
B > (T ) −2C(T )

is feasible, then the equations
fi (x) = ci ,

i ∈ [1, m],

(IV.3)

have no solution.
We are now ready to formulate the following proposition, that establishes necessary, control parameter-dependent, conditions for the existence of equilibria of the system (III.2)-(II.20).
Proposition 4.2: Consider the system (III.2)-(II.20), for some given PPref ∈ Rp , PVref ∈ Rv . Suppose that there exist two
diagonal matrices TP ∈ Rp×p and TV ∈ Rv×v such that:
Υ(TP , TV ) > 0,

(IV.4)

with


TP RP + RP TP
?
Υ := 
?

TP Lm + L>
m TV
TV RV + RV TV
?



0
−TV ūV
ref
>
−2(1>
T
P
P
p
P + 1v TV

,

PVref )

where PP? , PV? , RP and RV are defined in (IV.1). Then the system (III.2)-(II.20) does not admit an equilibrium point.
Proof: First of all, by setting the left-hand of the differential equations in (III.2) to zero and using (IV.1), we have:
0 = − RP vP? − Lm vV? + u?P ,
?
?
?
0 = − L>
m vP − RV vV + uV + ūV .
?
?
Left-multiplying the first and second set of equations by vP,j
and vV,k
respectively, with j ∼ EP , k ∼ EV , we get
ref
?
>
?
?
PP,j
= vP,j
RP,j
vP? + vP,j
L>
m,j vV ,
ref
?
?
>
?
PV,k
= vV,k
Lm,k vP? + vV,k
RV,k
vV? − vV,k
ūV,k ,

which, after some manipulations, gives
ci =

1 ? >
(v ) Ai v ? + (v ? )> Bi ,
2

with i ∼ EP ∪ EV , v ? := col(vP? , vV? ) ∈ Rc and

 
RP Lm
RP
Ai : = ei e>
+
i
L>
RV
Lm
m


L>
m
e e> ,
RV i i

Bi := ei e>
i

(IV.5)




0
,
ūV

ci := e>
i

 ref 
PP
.
PVref

Let consider the map f (v ? ) : Rc → Rc with components
1 ? >
(v ) Ai v ? ,
2
with i ∼ EP ∪ EV and denote by F the image of Rc under this map. The problem of solvability of such equations can be
formulated as in Lemma 4.1, i.e. if the LMI (IV.4) holds, then col(c?i ) is not in F , thus completing the proof.
fi (v ? ) =

Remark 4.3: Note that the feasibility of the LMI (IV.4) depends on the system topology reflected in the Laplacian matrix
L and on the system parameters, among which GZ , ūV and PVref are free (primary) control parameters. Since the feasibility
condition is only necessary for the existence of equilibria for (II.19), it is of interest to determine regions for these parameters
that imply non-existence of an equilibrium point.
V. CONDITIONS FOR POWER SHARING
As already discussed, another control objective of primary control is the achievement of power sharing among the voltagecontrolled units. This property consists in guaranteeing an appropriate (proportional) power distribution among these units
in steady-state. We next show that is possible to reformulate such a control objective as a set of quadratic constraints on the
equilibrium point, assuming that it exists. Since it is a steady-state property, the same observation done in Section IV applies,
which means that the results obtained for the system (III.2)-(II.20) also hold for the system (II.19)-(II.20). We introduce the
following definition.
?
Definition 5.1: Let be v ? := (vP? , vV? ) ∈ Rc and PDC,V (v ? ) := col(PDC,k (vC,k
)) ∈ Rv respectively an equilibrium point
for the system (III.2)-(II.20) and the collection of injected powers as defined by (II.15), and let be Γ := diag{γk } ∈ Rv×v ,
a positive definite matrix. Then v ? is said to possess the power sharing property with respect to Γ if:

ΓPDC,V (v ? ) = 1v .

(V.1)

Then we have the following lemma.
Lemma 5.2: Let v ? = (vP? , vV? ) ∈ Rc be an equilibrium point for (III.2)-(II.20) and Γ := diag{γk } ∈ Rv×v a positive
definite matrix. Then v ? possesses the power sharing property with respect to Γ if an only if the quadratic equations
1 ? > ps ?
(v ) Ak v + (Bkps )> v ? = pps
k ,
2

(V.2)

with k ∼ EV and where:
Aps
k



0
:=2
0


0
e e> ,
ΓGZ k k

Bkps




0
:=
e e> ,
ΓūV k k

admit a solution.
Proof: From (V.1) we have that by definition:
ref
γk PDC,k
(vC,k ) = 1,

pps
k

:=

e>
k



0
ΓPVref ,



with k ∼ EV , which by recalling (II.15), is equivalent to:
ref
2
γk (PV,k
+ µI,k vC,k + µZ,k vC,k
) = 1.

After some straightforward manipulations, the above equalities can be rewritten as (V.2), completing the proof.
An immediate implication of this lemma is given in the following proposition, which establishes necessary conditions for
the existence of an equilibrium point that verifies the power sharing property.
Proposition 5.3: Consider the system (III.2)-(II.20), for some given PPref , PVref and Γ. Suppose that there exist three
diagonal matrices TP ∈ Rp×p , TV ∈ Rv×v , TVps ∈ Rv×v , such that:
Υ(TP , TV ) + Υps (TVps ) > 0,
with
Υps


0
:= ?
?

0
2TVps ΓGZ
?

(V.3)


0
.
TVps ΓūV
ps
−2TV (1v − ΓPVref )

Then the system (III.2)-(II.20) does not admit an equilibrium point that verifies the power sharing property.
Proof: The proof is similar to the proof of Proposition 4.2. By using Lemma 5.2 the power sharing constraints can be
indeed rewritten as quadratic equations, similarly to (IV.5). Hence, it suffices to apply Lemma 4.1 to the quadratic equations
(IV.5), (V.2) to complete the proof.
VI. CONDITIONS FOR LOCAL ASYMPTOTIC STABILITY
We now present a result on stability of a given equilibrium point for the system (II.19)-(II.20). The result is obtained by
applying Lyapunov’s first method.
Proposition 6.1: Consider the system (II.19)-(II.20) and assume that v ? = (vP? , vV? , i?T ) ∈ Rm is an equilibrium point. Let
)
(
(
)
ref
ref
PV,k
PP,j
p×p
?
?
∈ R , GV := diag
GP : = diag
∈ Rv×v ,
(VI.1)
? )2
? )2
(vP,j
(vV,k
and



−CP−1 (GP + G?P )
0
−CP−1 BP
J(v ? ) := − 
0
−CV−1 (GV + G?V ) −CV−1 BV  .
−1 >
>
LT BP
L−1
−L−1
T BV
T RT

Then if:
- all eigenvalues λi of J are such that
Re{λi [J(v ? )]} < 0,
the equilibrium point v ? is locally asymptotically stable;
- there exists at least one eigenvalue λi of J such that
Re{λi [J(v ? )]} > 0,
the equilibrium point v ? is unstable.
Proof: The first-order approximation of the system (II.19)-(II.20) around v ? is given by:

 
    ∂iP
 
0
0
CP v̇P
−GP
0
−BP
vP
vP
∂vP v ?
∂iV
CV v̇V  =  0



−GV −BV  vV  +  0
v
0
V
∂vV v ?
iT
iT
BP>
BV> −RT
LT i̇T
0
0
0

(VI.2)

Differentiating (II.20) with respect to vP , vV , yields:
0p×p =

∂iP
· diag{vP,j } + diag{iP,j },
∂vP

0v×v =

∂iV
· diag{vV,k } + diag{iV,k }.
∂vV

By using (VI.1), it follows that
∂iP
∂vP

= −G?P ,
v?

∂iV
∂vV

= −G?V .
v?

The proof is completed by replacing into (VI.2) and invoking Lyapunov’s first method.
VII. AN ILLUSTRATIVE EXAMPLE
In order to validate the results on existence of equilibria and power sharing for the system (II.19)-(II.20) we next provide
an illustrative example. Namely, we consider the four-terminal hvdc transmission system depicted in Fig. 3, the parameters
of which are given in Table I.

TABLE I: System parameters.
Gi
Ci

Value
0 Ω−1
20 µF

?
PV,1

G12

Value
30 M W
0.1 Ω−1

?
PP,2

G14

Value
−20 M W
0.15 Ω−1

?
PV,3

G23

Value
9 MW
0.11 Ω−1

1

?
PP,4

G24

Value
−24 M W
0.08 Ω−1

4

2

3

Fig. 3: Four-terminal hvdc transmission system.

Since c = t = 4, the graph associated to the hvdc system has n = 4 + 1 = 5 nodes and m = 4 + 4 = 8 edges. We then
make the following assumptions.
- Terminal 1 and Terminal 3 are equipped with primary control, from which it follows that there are p = 2 PQ units and
v = 2 voltage-controlled units. More precisely we take
δk (vC,k ) = −

dk
nom
? (vC,k − vC ),
Vd,k

k = {1, 3}.

nom
is the nominal voltage
This is the well-known voltage droop control, where dk is a free control parameter, while vC
of the hvdc system, see also Remark 2.4.
- The power has to be shared equally among terminal 1 and terminal 3, from which it follows that Γ = I2 in Definition
5.1.
The next results are obtained by investigating the feasibility of the LMIs (IV.4), (V.3) as a function of the free control
parameters d1 and d3 . For this purpose, CVX, a package for specifying and solving convex programs, has been used to
solve the semidefinite programming feasibility problem [33]. By using a gridding approach, the regions of the (positive)
parameters that guarantee feasibility (yellow) and unfeasibility (blue) of the LMI (IV.4) are shown in Fig. 4, while in Fig. 5
the same is done with respect to the LMI (V.3). We deduce that a necessary condition for the existence of an equilibrium
point is that the control parameters are chosen inside the blue region of Fig. 4. Similarly, a necessary condition for the
existence of an equilibrium point that further possesses the power sharing property is that the control parameters are chosen
inside the blue region of Fig. 5.

VIII. CONCLUSIONS AND FUTURE WORKS
In this paper, a new nonlinear model for primary control analysis and design has been derived. Primary control laws
are described by equivalent ZIP models, which include the standard voltage droop control as a special case. A necessary
condition for the existence of equilibria in the form of an LMI—which depends on the parameters of the controllers—is
established, thus showing that an inappropriate choice of the latter may lead to non-existence of equilibria for the closed-loop
system. The same approach is extended to the problem of existence of equilibria that verify a pre-specified power sharing
property. The obtained necessary conditions can be helpful to system operators to tune their controllers such that regions

7000

6000

d 3 ·V d,3

5000

4000

3000

2000

1000

0
0

2000

4000

6000

8000

10000

12000

d 1 ·V d,1

Fig. 4: Feasibility regions of the LMI (IV.4) on the plane (d1 , d3 ) of droop control parameters. Regions are yellow-coloured
if the LMI is feasible and blue-coloured if the LMI is unfeasible.

7000

6000

d 3 ·V d,3

5000

4000

3000

2000

1000

0
0

2000

4000

6000

8000

10000

12000

d 1 ·V d,1

Fig. 5: Feasibility regions of the LMI (V.3) on the plane (d1 , d3 ) of droop control parameters. Regions are yellow-coloured
if the LMI is feasible and blue-coloured if the LMI is unfeasible.

where the closed-loop system will definitely not admit a stationary operating point are excluded. In that regard, the present
paper is a first, fundamental stepping stone towards the development of a better understanding of how existence of stationary
solutions of hvdc systems are affected by the system parameters, in particular the network impedances and controller gains.
A final contribution consists in the establishment of conditions of local asymptotic stability of a given equilibrium point.
The obtained results are illustrated on a four-terminal example.
Starting from the obtained model, future research will concern various aspects. First of all, a better understanding of how
the feasibility of the LMIs are affected by the parameters is necessary. A first consideration is that the established conditions,
besides on the controllers parameters, also depends on the network topology and the dissipation via the Laplacian matrix
induced by the electrical network. This suggests that the location of the voltage-controlled units, as well as the network
impedances, play an important role on the existence of equilibria for the system. Similarly, it is of interest to understand in
which measure the values of Z, I and P components of the equivalent ZIP mode affect the LMIs, in order to provide guidelines
for the design of primary controllers. Furthermore, the possibility to combine the obtained necessary conditions with related
(sufficient) conditions from the literature, e.g. in [34], is very interesting and timely. Other possible developments will focus
on the establishment of necessary (possibly sufficient) conditions for the existence of equilibria in different scenarios: small
deviations from the nominal voltage [4], [9]; power unit outages [4]; linear three-phase, ac circuit, investigating the role
played by reactive power [32].

IX. ACKNOWLEDGMENTS
The authors acknowledge the support of: the Future Renewable Electric Energy Distribution Management Center (FREEDM),
a National Science Foundation supported Engineering Research Center, under grant NSF EEC-0812121; the Ministry of
Education and Science of Russian Federation (Project14.Z50.31.0031); the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No. 734832.
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