1

Time scales tools a control solution for MTDC complex system with

Plug and Play requirements

Carmen Cardozo, Abdelkrim Benchaib

Confidential ALSTOM GRID

2

Time scales tools a control solution for MTDC complex system with

Plug and Play requirements

This work proposes a global control strategy for VSC-MTDC grid connected to

AC network, taking into account not only the DC grid intrinsic dynamics but also

interconnected systems and sub-systems time variations. To achieve this goal

different time scales are imposed to manage the DC grid, given that they are a

key element when moving towards system of systems approaches. Hence, both

steady-state and dynamic models for High Voltage Direct Current Multi-

Terminal systems based on Voltage Source Converters (VSC MTDC), suitable

for studying slow transients due to operating point changes, are introduced, in

order to represent the recovery of the energy balance in the DC grid after a

perturbation. It is beyond the scope of this paper to develop new efficient local

control law algorithms, but to consider the ones already used in power system

engineering. The major contribution of this work is to perform a cascade primary

and secondary control, with different time constants, that is able to manage and

secure the recovery of the VSC-MTDC grid after disturbances, similarly to what

has been developed for meshed AC power network. Thus, a DC voltage control

philosophy that ensures not only an adequate redistribution of the power

unbalance among the selected converters (AC systems) through a primary control

action, but that also considers a secondary control, where the DC voltages are

taken back to values around their nominal operating points is defined. At a

different stage of the proposed strategies, simulation results will be shown to

highlight the feasibility and effectiveness of such philosophy.

Keywords: MTDC grid, Control strategies, Complex system, Plug and Play

1. Nomenclature

1.1. Acronyms

AC Alternating Current

AGC Automatic Generation Control

CSC Current Source Converter

3

DC Direct Current

HVDC High Voltage Direct Current

IGBT Insulated Gate Bipolar Transistor

LRSP Load Reference Set Point

M/S Master/Slave DC voltage control

MTDC Direct Current Multi-Terminal systems

PF Power Flow

PLL Phase Locked Loop

PV Photo-Voltaic

RES Renewable Energy Sources

SC Secondary Control

TSO Transmission System Operators

VD DC Voltage Droop Control

VM Voltage Margin Control

VSC Voltage Source Converter

2.1. Variables Definition

Vs AC system voltage.

Is current injected by the AC system into VSC stations.

Ps active power injected by the AC system.

Qs reactive power injected by the AC system.

Vc voltage at the AC side of the converter.

Ic current injected by the converter into the station.

Pc active power injected by converters into the station.

Qc reactive power injected by converters into the station.

Vdc voltage at the DC side of the converter.

Idc current injected by the converters into the DC grid.

4

Pdc active power injected by converters into the DC grid.

Vsd d-axis component of the AC system voltage.

Vsq q-axis component of the AC system voltage.

Vdref d-axis component of converters’ reference voltage.

Vqref q-axis component of converters’ reference voltage.

I current injected into the AC system (-Is).

Id d-axis component of measured current I.

Iq q-axis component of measured current I.

Idref d-axis component of the reference current Iref.

Iqref q-axis component of the reference current Iref.

Pref active power set point at the AC system node.

Qref reactive power set point at the AC system node.

Kp Proportional gain of the PI controller.

Ki Integral gain of the PI controller.el one headings

2. Introduction

For more than one hundred years, the generation, transmission, distribution, and

utilization of electric energy has been based principally on Alternating Current (AC),

since Westinghouse proposal overcame Edison’s in the debate of the best technology

for power transmission, at the beginning of the electricity era. However, the increasing

world population, technology advances and environmental issues have driven changes

in the energy sector outlook, which have impact national energy policies across the

world, associated with the exploitation of Renewable Energy Sources (RES), and have

brought electricity sector reforms (deregulation).

Under this scenario, the widely deployed AC transmission technology presents

limitations, since higher efficiencies, power flow control capabilities and growing

5

interconnections with asynchronous features are required for a more economical and

reliable system operation [1]. Moreover, the long-distance bulk-power transmission and

the synchronous connections between AC networks are claimed, under certain

conditions, to deteriorate the overall dynamic behaviour of the AC systems; while the

submarine cable crossing length is limited by the management of the reactive power [1].

These applications among others, together with the development of power

semiconductors, microprocessors, and new insulation materials, have spurred the

advancement of the High Voltage Direct Current (HVDC) technology as an alternative

for power transmission [2], notwithstanding the technological challenges that remains,

such as the improvement of DC breakers and cables rating. Moreover, it is expected that

HVDC will contribute to improve the integration of economies across regional and

national borders, and the economy exploitation of renewable energy resources,

inasmuch as these sources are often located far away from the load centres,

necessitating long distance transmission.

More recently, the advent of the high power Insulated Gate Bipolar Transistor (IGBT)

and its turn-off capability has given rise to a new converter technology: the Voltage

Source Converter (VSC) or Forced-Commuted Converter (FCC) that is capable to

provide reactive power support and control the active power flow, reason why it is

considered to outmatch its predecessor, the well-known Current Source Converter

(CSC) or Line-Commuted Converter (LCC) [3].

Traditionally, when considering an HVDC link, only two converters are taken into

account, connected through a DC cable. In this scenario, significant research has been

carried out, such as [4]-[6]. A basic configuration of a VSC based HVDC transmission

system is presented in figure 1, where the converter is represented by a six-pulse bridge

6

equipped with self-commutating switches and anti-parallel connected diodes.

Figure 1. Two Terminal VSC system

It is worth mentioned that this point-to-point connection may be achieved using both

aforementioned technologies (CSC and VSC). Nevertheless, an outstanding difference,

from the control point of view, appears. When thyristor-based converters (CSC) are

used, only one degree of freedom is available at each terminal and its operating

principle is based on keeping a constant DC current (reason why is called current source

converter), while both magnitude and direction of the power flow are controlled by

changing the magnitude and direction of DC voltage as the control system acts through

firing angle adjustments of the valves and through tap changer adjustments on the

converter transformers [4]. Thus, the power reversal in CSC based HVDC links is

obtained by reversing the polarity of the direct voltages at both ends, which represents

an issue when considering multi-terminal operation [4].

In contrast, VSC converters offer two degree of freedom at each terminal, which allows

the independent control of the active and reactive power. Moreover, each converter can

be used to synthesize a balanced set of three phase voltages like an inertia-less

synchronous machine, provided that the DC voltage is free from oscillations during

disturbances and fault occurrences on the AC sides of the VSC-HVDC stations [7].

Furthermore, unlike conventional HVDC transmission (CSC), the transistor-based

7

converters (VSC) themselves have no reactive power demand and can actually control

their reactive power to regulate AC system voltage. The dynamic voltage support of the

AC voltage offered at each VSC terminal improves the voltage stability and can

increase the transfer capability of the sending and receiving AC systems without much

need for AC system reinforcement.

Finally, the power flow direction in the VSCs is controlled by changing the direction of

the current, which make them suitable for multi-terminal operation.

The concept of Multi-Terminal DC grids (MTDC), where DC cables connect more than

two AC/DC converter stations, as shown in figure 2., is nowadays spreading driven by

different criteria, all necessaries for large-capacity RES grid integration, such as larger

flexibility of grid operation, possibility of reversible power flows, increased

redundancy, and reduction of maximum power loss in case of a grid disturbance [7] [8].

Figure 2. VSC - Three Terminal DC Grid

On the other hand, the eagerness of Renewable Energy Sources (RES) exploitation has

forced Transmission System Operators (TSO) to deal with non dispatchable productions

8

centres, such as Photo-Voltaic (PV) and wind parks, that risk rendering the existing AC

system more exposed to collapse. As a solution to massive RES integration, without

engaging the sturdiness of the electrical system, or even enhancing it, a DC grid can be

superpose to the existing electrical network [9]. Therefore, suitable control techniques,

structures and strategies must be developed to guarantee the managing and operation of

both grids in unison, considering the global system behaviour in scheduling task, but

acting local and automatically at the interconnection points for fast control.

Concerning MTDC grids, a few DC voltage control strategies have been proposed in

literature, starting with a centralized control: Master/Slave (M/S). This strategy would

be an extrapolation of the one used in a point-to-point DC connection (two terminals),

since just one converter controls the DC voltage acting like a “battery” or “slack bus”.

This means that it will provide or absorb sufficient active power to achieve the power

balance in the DC grid and to hold its DC voltage at the reference value [10]; therefore

the active power injected by this converter mathematically compensates all the losses in

the DC circuit. Meanwhile, all the other converters operate in a constant active power

injection mode.

Consequently, the slack converter must be connected to a strong node in the AC system

and it has to be oversized to react fast to DC grid transient such as a loss of converters

or DC lines [11]. Hence, the geographic location of the slack converter becomes an

issue since the selected TSO is requested to cope with all the problems in the DC grid.

Finally, an outage of this converter could not be tolerated because this would mean the

loss of the DC voltage control.

A feasible solution would be to control the DC voltage at different nodes of the DC grid

simultaneously. Despite such duplication could increase the overall reliability of the

9

system, it gives rise to suboptimal operating points (static), and to voltage and power

oscillations [11]. Therefore, different solutions have been proposed to distribute the

control of the DC voltage among a certain number of converters [10] [11] [12], to

increase reliability and to enhance transient recovery.

Afterwards, at least four other strategies for regaining the power balance of the DC grid

after a disturbed condition can be quoted: the Voltage Droop control (VD), the Voltage

Margin (VM), the Ratio Control and the Priority Control. Nevertheless, some of them

may entail a deviation in the DC grid voltage profile. This behaviour is already

acknowledged from the AC system, where the Automatic Generation Control (AGC) is

responsible for regulating frequency to the specified nominal value and maintaining the

interchange power between control areas at the scheduled values by adjusting the output

of selected generators through a primary speed control action that will result in a steady

state frequency deviation depending on the governor droop characteristic and frequency

sensibility of the load [13]. Then, a supplementary control action adjusts the Load

Reference Set Point (LRSP) and restores the system frequency.

Therefore, a natural analogy appears between the AC frequency and the DC voltage

since they both reflex the power balance in the grid. However, attention must given to

the fact that different frequencies at different points in an AC system indicate transient

processes, while the DC voltage at each node is not the same in steady state due to the

power flow.

In the same way, an analogy can be made between the aforementioned DC voltage

control strategies, especially the DC voltage droop, and the primary speed control action

used in AC generators, since they aim to restore the power balance of the network by

acting locally flowing an given characteristic, even if the “global” variable

10

(frequency/DC voltage) strays from its specified value.

In this work, both steady-state and dynamic models for High Voltage Direct Current

systems based on Voltage Source Converters (VSC HVDC) are presented, comparing

different DC voltage control strategies, to finally include a secondary control (SC), now

needed to reduce the DC voltage deviation caused by the primary control. Furthermore,

a cascade distributed and centralized DC voltage control, in different time scales, is

proposed to improve the DC grid management.

Subsequently, in section 3 the steady state model for a VSC MTDC system is developed

to obtain a first approach of both steady-state conditions, before and after a disturbance

is introduced, and to initialize the dynamic simulation. In section 4, the adopted control

system structure and the above-mentioned DC voltage control strategies are explained.

In section 5, the dynamic model is finally assemble to estimate the transient response,

within the approximations made, of the VSC MTDC system. In section 6, the results

obtained for the simplest MTDC grid, a three-terminal VSC system, are shown, while

their discussion is presented in Section 7. Finally some conclusions are drawn out in

Section 8.

3. Steady State VSC-MTDC Model

The steady state of a system refers to an equilibrium condition, where the effect of

transients is no longer relevant. In power systems, any specific steady state operating

point can be determined through an analytical technique called Power Flow analysis

(PF, also known as load-flow), where the loads are modelled as bulk power delivery

points and the voltages of the transmission network can be determined [13].

An analogous study for a VSC-MTDC system was conducted in [14]. The DC power

11

flow calculation can be used to find the starting point for dynamic analysis, considering

that the VSC-MTDC system is operating under a centralized DC voltage control such as

M/S, where only one converter (known as slack converter) is designated to keep the DC

voltages in a close band around its rated value through active power compensation.

Once more, an outstanding difference between the AC and DC systems must be kept on

sight. In an AC transmission system, where the line resistances are neglected given the

inductance higher values, the active power flow is dominated by the angle differences

between different buses, while the reactive power flow is mainly associated with the

voltage magnitude deviation of different nodes [13].

When considering a DC grid, there is not reactive power flow; therefore, the power flow

calculation is solely concerned about the active power, which, contrary to the AC case,

is dictated by the differences in the voltage magnitudes between the different nodes.

Another specification of the DC grid is the absence of the frequency; hence, only

resistances are introduced in the nodal admittance matrix (Ydc).

In the following, the selected sign convention is defined. Afterwards, the DC side model

is built, followed by the DC power flow calculation. Then, the steady state AC side

model is presented, where the filter and other station equipment are not considered, and

the VSC is represented by its average model, approximation consistent with the

phenomena of interest. Finally, a numerical example is used to illustrate the developed

concepts (results achieved using MATLAB).

3.1. Convention

The DC power and current are assumed to be positive when they are flowing away from

the VSC, as shown in figure 3. Therefore the DC powers (and currents) are considered

12

as positive when they are being injected into the DC grid.

Figure 3. Power and Current directions

3.2. DC side model

In steady-state the DC network is modelled by the link resistances (Rij). It is common

practice to add a shunt resistance to represent the converter losses as shown in figure 4.

Figure. 4. Monopolar symmetrically grounded VSC-MTDC system

In a MTDC system of n DC nodes, the current injected in a node i is calculated as the

sum of all the currents flowing to the other (n-1) interconnected nodes and the current

13

going into the shunt resistance (Rdc,ii), as written in equation (1).

iidc

idcjdcidc

n

ijj ijdc

idcR

VVV

RI

,

,,,

1 ,, )(

1++++

−−−−⋅⋅⋅⋅==== ∑∑∑∑≠≠≠≠====

(1)

3.3. DC Power Flow calculation (DC PF)

The active power in each node is the product of its DC voltage and its DC current. For a

monopolar symmetrically grounded converter, as the one shown in figure 4, and for

bipolar converters the power injection is given by equation (2).

idcidcidc IVP ,,, 2==== (2)

Substituting equation (1) in (2), it is possible to calculate the power injection into each

DC node, when the DC voltage values are known beforehand as shown in equation (3).

++++

−−−−⋅⋅⋅⋅⋅⋅⋅⋅==== ∑∑∑∑≠≠≠≠==== iidc

idcjdcidc

n

ijj ijdc

idcidcR

VVV

RVP

,

,,,

1 ,,, )(

12 (3)

However, in a VSC-MTDC system under a centralized DC voltage control (M/S), the

DC voltage is only known for the slack converter, while it remains as an unknown in the

other nodes where the VSCs are operating in a constant active power injection mode

(Pdc,i known); giving rise to a nonlinear equations system that should be rewritten as

equation (4).

02

V,

,dc ====

−−−−

idc

idcdc

V

PY (4)

Where Ydc is the DC nodal admittance matrix, since

14

dcdc VI dcY==== (5)

The solution of the equation system given in equation (4), gives the power injection at

the slack converter and the DC voltages of all the VSCs operating in a constant active

power injection mode. Therefore, after performing the DC PF, the voltage and active

power injection of all the DC nodes in the MTDC grid will be perfectly specified.

3.4. AC side model

VSC converters are considered to be connected to the AC system through a phase

reactor as shown in figure 5. It is common practice to include a transformer, but for the

purposes of this study (where filters and others station equipment are not model), its

representation is omitted.

Figure 5. Single phase VSC representation

The phase reactor is represented by a resistance (Rpr) and an inductance (Lpr). In the

following, the per unit system is adopted as defined in Appendix A. The basic equation

of this circuit, following the Kirchhoff's Voltage Law (KVL), is:

dt

dILIRVV c

prcprsc ++++====−−−− (6)

15

Hypothesis:

• If a balanced three-phase system is considered, the equation (6) can be

transformed to a rotating dq0 reference frame, using the Park-Clarke

transformation (P) defined as:

(((( ))))

(((( ))))

++++−−−−

−−−−−−−−−−−−

++++

−−−−

====

2

1

2

1

2

1

3

2sin

3

2sinsin

3

2cos

3

2coscos

3

2 ππππθθθθ

ππππθθθθθθθθ

ππππθθθθ

ππππθθθθθθθθ

P

Such that, in steady-state,

qprdprsdcd IXIRVV −−−−====−−−− (7)

dprqprsqcq IXIRVV ++++====−−−− (8)

• Then, if the converters losses are not considered (Rii=∞),

dcc PP −−−−==== (9)

• And the phase reactor resistance and losses are neglected,

sc PP −−−−==== (10)

The active power at the system node and both side of the VSC is the same.

• Moreover, if the AC system voltages (Vs) are considered as known (from the AC

PF) and,

• The Voltage Oriented Control (VOC) method [15] is applied (Vsq=0):

16

dsdss IVP2

3==== (11)

sqdss IVQ2

3−−−−==== (12)

If the active and reactive powers injected by any AC system are defined as set points by

the associated TSO; then, the converters AC side currents (Id and Iq) can be determined

by equations (11) and (12), since no filter is considered (thus, Is = - Ic). Finally, the

converters AC side voltages (Vcd and Vcq) are given by equations (7) and (8).

3.5. Study Case: three-terminal VSC under M/S

In order to determine a realistic initial condition of any particular VSC-MTDC system,

a DC PF must be calculated when considering a Master/Slave (M/S) DC voltage control

strategy. Figure 2 shows a representative topology of a three terminal VSC system. In

Table 1, the system bases used in further calculation (using MATLAB) are presented.

Parameter Units Values

AC system three phase Voltage (Vac) kV (rms) 115

Bases Power (S) MVA 140 (2x70)

DC voltage (Vdc) kV 80

AC frequency (fo) Hz 50

Table 1. System Bases

The per unit set points at each terminal are presented in Table 2 and represented in bold

at figure 6. The calculated values, after performing a DC Power Flow (in real values)

17

are presented in Table 3 and depicted in italic on figure 6 (in per unit). In Table 4 the

links parameters are presented.

Node Type of Node Pref Qref Vdcred

1 P controlled 1 0 ---

2 P controlled -0,25 -0,2 ---

3 Slack --- 0,1 1

Table 2. Reference System Set-Points in Per Unit

Figure 6. 3T VSC system. Set points and DC PF results. Initial steady state condition.

18

Node Type of Node Pdc [MW] Vdc [kV]

1 P controlled 140,0000 81,0068

2 P controlled -35,0000 80,1169

3 Slack -103,3111 80,0000

Table 3. DC Power Flow Calculation

Leaving

Node

Arriving

Node

Rdc

[Ω/km]

Length

[km]

Pdc i,j

[MW]

1 2 0,0366 200 42,9074

1 3 0,0366 100 97,0926

2 3 0,0366 150 7,4360

Table 4. DC Links Parameters (500mm2 copper) and Power Flows

Finally, as explained in section D, the steady state values of the converters AC side

voltage and current in the rotating dq0 reference frame can also be determined from the

power reference set points, assuming as known the AC system voltages (Vs) and the

phase reactor impedance (Lpr=50mH, resistance neglected). The results are presented in

Table 5.

19

Node Vsd Vcd Vcq Id Iq

1 1,0000 1,0000 0,1663 -1,0000 0,0000

2 1,0000 1,0333 0,0416 0,2500 -0,2000

3 1,0000 0,9834 0,1227 0,7379 0,1000

Table 5. AC side Calculation in Per Unit

Then, a disturbed condition could be considered; in this case, AC system 1 loses half of

its power injection. The steady state condition after the perturbation, considering that

the system was able to cope with this contingency, can be found by repeating the

calculation presented above, but changing the initial condition of VSC 1, such that

Pref,1=0,5pu.

Figure 7. 3T VSC system. Set points and DC PF results. New steady state condition,

considering M/S DC voltage control.

20

In this subsection, it has been shown that the steady state model and the DC PF

calculation allow determining different operating points of the VSC-MTDC system. The

transition between them will be represented through the dynamic model, including the

controllers’ response (Section IV). In following, the considered control system is

discussed.

4. Control System of VSC based on VOC

In response to the forthcoming massive renewable energy integration, which will be

connected to the existing AC electrical network through power converters, great efforts

have been devoted to the development of more performing control system that, amongst

others, will contribute to mitigate the inherent problem of intermittent but clean RES.

In the literature, special attention has been given to the controllers design, including

classical Proportional Integral (PI), hysteresis regulators, and the development of

customized adaptive algorithms [16] [17]. Also, robust nonlinear control techniques

based Lyapunov and Backstepping methodologies has been proposed, in which the

system decoupling is obtained via cascade structure and using the inherent time scales

of the studied system [18]. Finally, the Dead-Beat (DB) control including current and

voltage limitations has been suggested when high dynamic response is mandatory [19].

Additionally, available control methods have been evaluated for VSC-HVDC

transmission systems applications, such Direct Power Control (DPC) and vector control

in different reference frames such as the Voltage Oriented Control (VOC) in dq0 or

alpha-beta [20]. However, in this study the well-known closed-loop vector control

method in conjunction with PI controllers in cascade structure is adopted according to

21

[15], since this work aims to offer a contribution in the understanding of the overall

VSC-MTDC system behaviour in order to establish a DC voltage control philosophy,

when further contributions regarding stability analysis as well as robustness properties

to obtain effective control scheme can be subsequently included.

Therefore, in this section only the control structure is presented, considering their

integral and proportional gains as known. In practice, the inner (current) loop was tuned

according to “modulus optimum” technique while the outer (power/voltage) loop was

tuned following the “symmetrical optimum” criteria as suggested in [21].

The active and reactive power control structure, based on a closed-loop structure under

synchronous frame Voltage Oriented Control (VOC) is shown in figure 8, as presented

in [15], where PI-based controllers decide the reference current d and q components

from the error between the actual injected powers and their reference values, resulting

in a four loops control structure.

Figure 8. PQ closed-loop VOC implemented on the dq synchronous frame

22

4.1. Inner Current Controller

The converter’s AC side current can be controlled by two parallel inner loops using PI

controllers that transform the current errors into voltage signals as depicts in figure 9.

Figure 9. Current Controller Structure - VOC based on dq synchronous frame

The representative equation of the PI regulator is:

++++====++++====

S

SK

s

KKsG

i

ip

ipPI

ττττ

ττττ1)( (13)

Where the proportional gain Kp and integral time constant τi or the integral gain Ki are

the design parameters calculated according to [21].

Using separate current controller loops for Id and Iq, the converter reference voltage

signals for the two axes (Vdref and Vqref in figure 9), are generated from the current error

and the PI regulator transfer function.

(((( ))))ddrefi

pcdref IIs

KKV −−−−

++++====

11

(14)

23

(((( ))))qqrefi

pcqref IIs

KKV −−−−

++++====

11

(15)

In equations (16) and (17) the grid voltage has been feed-forwarded and the cross-

coupling terms between the two control loops are added to compensate the ones

introduced by the transformation to the rotating reference frame [15].

(((( ))))ddrefi

pqprsdcdref IIs

KKILVV −−−−

++++++++−−−−====

11ωωωω

(16)

(((( ))))qqrefi

pdprsqcqref IIs

KKILVV −−−−

++++++++++++====

11ωωωω (17)

4.2. Outer Power Controller

An active and reactive power closed-loop control structure is employed; hence, the

reference currents for the inner loop (Idref and Iqref) are calculated in an outer power loop

where the active and reactive powers injected are estimated using measurements (of

three-phase voltages and currents) at the Point of Common Coupling (PCC) and their

values are compared to the set points.

Special attention must be given to the selected convention. If the power set points

(Pset_point and Qset_point) are defined by the TSO as injected by the AC system into the

VSC station, while the controlled current (Id and Iq), has been defined as injected by the

VSC into the AC grid, the power reference value given to the control system (Pref and

Qref in figure 8) must be defined in the same direction of the current, such that,

int_ posetref PP −−−−==== (18)

int_ posetref QQ −−−−==== (19)

24

Active Power Controller

When the M/S DC voltage control strategy is implemented in a VSC MTDC system

with an n DC nodes, n − 1 converters control the active power, and one controls the DC

voltage. Therefore, the variable Idref is calculated from the active power set-point of all

converters, except for the slack converter, through a combination of an open loop and a

PI controller as expressed in equation (20)

(((( ))))PPs

KK

V

PI ref

ip

ppsd

ref

dref −−−−

++++++++==== (20)

A feed-forward term representing the d-axis current (Id) that should be generated when

the injected power is equal to its reference is added to minimize disadvantage of slow

dynamic response of cascade control. Additionally, this contributes to improve stability

since the load variation can be greatly reduced and, with it, the gain of voltage

controller [4].

According to equation (11), it should be a 3/2 factor in the feed-forward term, but it has

disappeared due to the chosen per unit system: the power and voltage bases have been

set as the Rated three-phase power of the converter [VA] and the Peak value of rated

line-to-neutral voltage [V] (see Appendix A).

Reactive Power Controller

Analogously, the q-axis current reference (Iqref) is calculated from the reactive power

reference value (Qref), where a combination of an open loop and a PI controller is used

to drive the reactive power to its desired value, leading to:

(((( ))))QQs

KK

V

QI ref

iq

pqsd

ref

qref −−−−

++++++++−−−−====

(21)

25

Alternatively to the reactive power, the magnitude of the AC grid voltage could be

controlled; however, an extra control loop is needed to provide the reactive power

reference to the reactive power control loop. In computer models, the reactive power

injection necessary to keep the AC system’s voltage in a given value can be calculated

by an AC Power Flow, where the converter is considered as a PV node [13]. A

sequential AC/DC power flow has been proposed in [22]-[23].

In [24] it is proposed to calculate the q-axis current reference directly from the voltage

reference using a PI controller as in equation (22), but this scenario is not considered in

this work since none further models of the AC grid has been so far included; Thus, the

evolution of the grid voltage (Vsd) in response to the control q-axis current is unknown.

(((( ))))sdacrefiv

pvqref VVs

KKI −−−−

++++==== (22)

4.3. DC voltage Controller

Master/Slave DC voltage control (M/S)

When the Master/Slave DC voltage control strategy (M/S) is implemented, one

converter of the VSC MTDC system controls the DC voltage. The control of the DC

voltage could act directly on the d-axis reference current (idref) as shown in figure 10.

The design parameters of the controller are found through a linearization of the power

equation [15]. Otherwise, in this work, the control of the DC voltage is considered to act

on the DC current, here referred as idcref, and the d-axis reference current (idref) is found

directly from the power balance equation, as represented in figure 11. Therefore, a PI

controller is used to keep the DC voltage of the slack converter at its reference value

and the steady-state DC current (as a function of the actual DC power injected by the

slack converter into the DC grid, Pdc,slack, and the reference DC voltage) is feed-

26

forwarded. Subsequently, the reference DC current of the slack converter can be

expressed as:

(((( ))))slackdcdcrefidc

pdcdcref

slackdcslackdcref VV

s

KK

V

PI ,

,,

2−−−−

++++++++====

(23)

Where Pdc,slack could be calculated using measurement at the DC bus (24), or, if the

converter losses have been neglected, as a function of the calculated AC side converters

voltage and current (25).

slackdcslackdcslackdc VIP ,,, 2==== (24)

cqqcddslackcslackdc VIVIPP −−−−−−−−====−−−−==== ,, (25)

Finally, the d-axis reference current (Idref,slack) is defined as:

cd

cqqdcslackdcref

slackdrefV

VIVII

++++−−−−====

,

,

2 (26)

27

Figure 10. Vdc-Q closed-loop VOC implemented on the dq synchronous frame

Figure 11. DC Voltage Control structure

DC Voltage – Active Power Droop (VD)

When the control of the DC voltage is considered to be distributed implementing a

control strategy such as the DC Voltage Droop (VD), a group of converters may have a

third control loop where the active power reference is generated as a function of a active

power set-point (Pdc0) and the error between the actual DC voltage and its set-point

(Vdc0) by a proportional controller, as shown in figure 12, such that:

(((( ))))dcdcpddcdroopref VVKPP −−−−++++==== 00_ (27)

Figure 12. DC voltage droop Proportional Controller

28

In MTDC networks, this distributed control employs the droop mechanism to regulate

the DC voltages by adjusting the converter power (or current) injections as shown in

figure 13.

dcP

dcV

0,dcP

0,dcV

k

INV REC

Figure 13. Voltage droop characteristics: DC Voltage / Active Power

There is an outstanding particularity in the DC VD with respect to the aforementioned

centralized DC voltage control strategy (M/S). In this kind of distributed control, the

concept of “set points” of active power and DC voltage is changed by the idea of

“nominal operating points” (NOP).

The difference lies in the new steady-state condition after a disturbance. In the droop

control, although the NOP (Vdc,0 and Pdc,0) are equally given by the TSO as a result of

an Optimal Power Flow (OPF) in a nominal condition; it is not expected that neither the

power, nor the voltage, return to their exactly values after this primary control has faced

a contingency.

29

In other words, in a non-disturbed condition, identical to the one established by the TSO

as the nominal, the controllers will follow the NOPs as their reference values and the

desired power flow will be achieved.

However, when the active power balance is disturbed, the DC voltages deviate and the

converters under droop control (as those operating in DC voltage control mode, if any)

will change permanently their active power injection to regain the power balance. In

this new condition, the DC voltages of droop and active power controlled converters

have also moved away from their NOP.

This strategy is suitable for really large networks with high reliability standards since a

minimum of communication is required inasmuch as only the NOP are transmitted and

the real references for the control system are determined locally.

In the implementation, the set of DC voltage Vdc,o and active power Pdc,o nominal

operating points for the converters under voltage droop control might be obtained from

the system working in steady-state conditions (DC PF), without a droop control being

active [11]. Concerning the VD proportional gain Kpd, the higher its value the more the

converter adapts its output power. When Kpd 0 the converter works in constant active

power mode (Pdc = Pdc,o), and when Kpd ∞ the converter controls the DC voltage (Vdc

= Vdc,o).

In principle, normal and disturbed operations are treated identically by the DC VD and

differentiated optimization is not possible.

To face this drawback, different solutions were proposed in [25] as a function of the

droop constant (k), inversely proportional to the proportional gain of the controller

(Kpd):

30

Dead-band Droop control

Inside of the band of normal operation just one converter controls the DC voltage

(figure 14) while the control action of the others reminds inactive (figure 15), and in

failure condition all converters take their share to compensate the disturbance via their

k-value according to their capabilities and limitations.

dcP

dcV

0,dcP

0,dcV

k

min,dcV

max,dcV

INV REC

Figure 14. Voltage droop characteristics with Voltage Dead-band

dcP

dcV

0,dcP

0,dcV

k

min,dcV

max,dcV

INV REC

Figure 15. Voltage droop characteristics with Power Dead-band

31

Figure 15 shows the behaviour of the n-1 converters that operate under constant power

injection in normal conditions, i.e. for a specific band of DC voltage deviation;

however, once these limits are exceeded, they will contribute to regain the power

balance in the DC link.

Undead-band Droop control

The control activity of the converters within the band is not fully dead, but only

reduced, since in normal operation several converters could have non-zero k-value and

contribute to the DC voltage control, even if only a few of them could carry the main

load by having higher gains, while the other converters could have smaller values.

Figures 16 and 17 depict this strategy for both kinds of controlled converters, those

operating in Voltage Undead-band control and those controlled with a Power Undead-

band respectively.

dcP

dcV

0,dcP

0,dcV

kD

min,dcV

max,dcV

INV REC

kN

Figure 16. Voltage droop characteristics with Voltage Undead-band

32

dcP

dcV

0,dcP

0,dcV

kD

min,dcV

max,dcV

INV REC

kN

min,dcP max,dcP

Figure 17. Voltage droop characteristics with Power Undead-band

DC Voltage Margin

Another option proposed in literature to cope with the main drawbacks of the M/S

control without been a distributed control (without entailing an “uncontrollable” DC

voltage deviation) is the DC Voltage Margin control strategy (VM).

Within this strategy one converter acts as DC voltage controller (slack) as long as it

does not reach the upper or lower limits of its power injection: Pmax - Pmin [26].

dcV

refdcV ,

AB PP >

INV REC

AB PP =

AB PP <

APmaxPminP

Figure 18. Voltage margin strategy

33

In figure 18 Terminal A is considered to control the DC voltage. However, as soon as

the DC grid power demand exceeds the upper limit (Pmax), the DC voltage decreases,

and when it gets smaller than the lower limit (Pmin) the DC voltage increases. Then,

another terminal (B) will take the slack role.

This means, when the DC voltage rises, another terminal (with next higher DC voltage

reference setting) takes over the duty of DC voltage regulation. This is illustrated in

figure 19.

dcV

INV REC

Operatingpoint

Terminal A

Terminal B

AP

Voltage margin

dcV∆

maxPminP

Figure 19. Operating point in voltage margin strategy

dcV

INV REC

Operating

point

Terminal ATerminal C

AP

Voltage margin

dcV∆

maxPminP

Figure 20. Operating point in voltage margin strategy

When the DC voltage level reduced, the terminal with the next lower DC voltage setting

34

(Terminal C) will start to act as DC slack bus, as shown in figure 20.

The downside of the voltage margin is that only one converter is capable of controlling

the system DC voltage at a given time. Otherwise, the DC voltage is restricted to ± 10%

of the nominal voltage, limiting the amount of converters that could participate in this

distributed DC voltage control strategy in a MTDC system with a high number of

nodes.

Ratio control

The stations controlling the DC voltage share the power according to a certain ratio.

Using this control strategy, the TSO could vary the power ratio between the voltage

controlling terminals by changing the slope of the droop characteristic [27]. The

expression of the ratio (r) between two terminals is:

2,

1,

2,

1,

dc

dc

dc

dc

I

I

P

Pr ≈≈≈≈====

(28)

This approximation is due to the fact that the DC voltage is considered to be the “same”

in both converters; thus, the power shared is approximately equal to the DC current

ratio. The drawback of this control strategy is that communication is necessary if there

are more than two DC voltage controlling stations.

Priority control

The priority control is a personalised strategy where DC voltage droop controllers are

used to perform priority power sharing. The terminal with the highest priority controls

the DC voltage, taking all the power injected into the DC grid, until its rated capacity or

a maximum power set point is reached. The second converter will start transmitting

power following a droop characteristic only when the first has attained its limits, and so

35

on [27].

If a three terminal VSC-HVDC system, integrated by a wind farm and two AC grids (A

and B), is taken as an example, and the priority control is implemented to define the

sharing of the power produced by the wind farm such that System A is prioritised, it

will obtain all the power produced by the wind farm, until a particular limit is achieved.

At this point System B will take the power excess following a DC voltage / Power

droop. In other words, as long as the power delivered to the first prioritised system does

not reach the limit, the second prioritised system will not receive any power from the

generation system.

Comparison of DC voltage control strategies

The comparison of the five (5) first control strategies previously discussed is

summarised in Table 6 for a three terminal HVDC system through the introduction of

droop constants for both normal (kN) and disturbed (kD) operation. The last converter

(3) has the role of the slack bus when considering M/S control strategy and therefore its

droop constant value is 0.

Within the VM control strategy, converter 3 is the slack bus in normal condition, until it

achieves its limit for a given disturbed condition and converter 2 becomes the slack bus

as converter 3 changes its operating mode into constant active power injection.

As a general principle, concerning the VD control strategy, the same droop constant is

applied to the three converters in normal operation and disturbed operation. However,

when a dead-band is considered, the droop constant is set to 0 for the slack bus

converter and to ∞ for a P controller in normal operation, while they have values

different from 0 or ∞ in disturbed operation.

36

Finally, when an undead-band droop control strategy is implemented, the droop

constants are set to values different from to zero and ∞ even for normal operation.

VSC 1 VSC 2 VSC 3

Control strategy kN kD kN kD kN kD

∞ ∞ ∞ ∞ 0 0 Master/slave

∞ ∞ ∞ 0 0 ∞ Voltage Margin

K1 K1 K2 K2 K3 K3 Droop

∞ K1 ∞ K2 0 K3 Dead-band droop

K1N K1D K2N K2D K3N K3D Undead-band droop

Table 6. Comparison of the control strategies

4.3. Primary and Secondary Control of VSC-MTDC systems

Within the steady state analysis, a modification of the above-developed DC PF was

proposed in [11] to include the DC voltage droop as a distributed DC voltage control

strategy. Fig 21 presents the results found for the system presented in figure 6, when

considering the same contingency presented in Fig 7 (where VSC 1 has change its

power injection from 1 pu to 0,5 pu), but this time, the DC VD, instead of the M/S, is

considered as the primary control. The droop characteristic parameters are presented in

Table 7, and identified in figure 21 as the Nominal Operating Points (NOP). The NOP

are taken from the DC PF under M/S (initial condition figure 6) and the droop constants

are arbitrarily defined. The new steady state, after the action of the primary control (DC

VD) is defined as Operating Point (OP2 and OP3).

37

Figure 21. 3T VSC system. DC PF results, considering distributed DC voltage control

(Droop)

Node Type of Node Pdc,0 Vdc,o k_droop

1 P controlled --- --- ---

2 Droop -0,25 1,0015 0,25

3 Droop -0,7379 1 0,10

Table 7. Nominal Operating Points in Per Unit (From initial DC PF under M/S)

The DC Voltage Droop has proved to guarantee the desired distribution of the power

unbalance among the selected converters by calculating locally the new reference set-

points (OP), as a function of their droop characteristic, where the TSO has define the

NOP and the droop constant.

However, as aforementioned, the distribution of the DC voltage control entails an

38

“uncontrollable” DC voltage deviation that, even if it could be bounded through the

droop constant, leaves the system exposed in case of a consecutive contingency.

Therefore, a Secondary Control (SC) is needed to take back the DC voltage profile

around its nominal value.

The idea is to implement a sort of centralized control (M/S), once the Droop has acted

as primary control. In this case, the new redistribution of power given by the DC VD

must be held in n-1 converters, which will operate in constant power injection mode,

while one converter (VSC 3 in the study case) will impose its DC voltage, as a “slack”

bus.

It is worth noticing that in this scenario the “slack” bus is not request to cope with the

DC grid contingencies, neither it is responsible for the DC voltage control at every

instant, which represent the main drawbacks of the M/S strategy. Within the secondary

control, the selected VSC is only request to return the DC voltage profile around its

nominal value (transitorily charge or discharge the DC capacitors).

Figures 22 and 23 show four (4) different operating points for VSC 2 and VSC 3

(responsible for the SC) that include, the initial condition (NOP), the action of the

primary control (OP) and action of the secondary control (SCOP).

As shown in figures 22 and 23, this secondary control can be interpreted as a

displacement of the droop characteristic, since the operating points will be replaced

closer of the nominal DC voltage (by the action of the SC), and the droop characteristic

of each VSC must be rebuilt based on the new operating points physically defined

(SCOP). In this case the droop constants were kept unchanged, (under discretion of the

TSO).

39

Figure 22. Operation Points of VSC 3, considering primary and secondary DC voltage

control. Specified DC voltage during SC (1 pu).

Figure 23. Operation Points of VSC 2, considering primary and secondary DC voltage

control. Constant Active Power injection during SC.

A subtlety must be highlighted regarding the way that the secondary control redefines

40

the reference points to locally build the new droop characteristic. In VSC 2 (and any n-1

converters not contributing to the secondary control) the transition between OP2 and

SCOP2 is made in constant active power mode as shown in figure 23, responding to the

control structure given in figure 8, where the active power reference value (Pref) is the

output of the DC VD (primary control).

Meanwhile, VSC 3, responsible for the SC, will respond to the control structure given in

figure 11, which means that it will impose the DC voltage, while moving from OP3 to

SCOP3, through transitory active power compensation (not reflected in figure 22). In

the final steady-state condition (after de SC) a slightly different active power is found

(Pdc,3_sc in figure 22), due to the changes DC grid losses at the new power flow

condition. In other words, the considered disturbed condition was a power income loss

for the DC grid that entailed a drop in the DC voltage profile, to overcome this the

secondary control has acted, raising the voltage at the n-1 node working in a constant

power injection mode, resulting in a reduction in the DC links current transits, and with

it the DC grid losses.

The state of the system obtained after the contingency has been overcome, is also

presented as OP’, considering that the “displaced” droop has acted as primary control.

Once more, the DC voltage profile has strayed from its nominal value and the SC is

responsible to take it back, as closer as possible to 1 pu. Finally, if the system returns to

its initial condition, the initial DC PF is achieved (NOPs).

Figure 24 shows de sequence of event abode described, representing only the steady-

state conditions (the dynamic transitions are not included). After the fault insertion at 50

s the DC voltages drop and the SC restores the DC voltage profile at 100 s with an

imperceptible active power variation at VSC 3. Then, the initial power distribution is

41

imposed at 150 s, and the SC establishes the initial PF (NOPs).

Figure 24. primary and secondary DC voltage control

Fault

appearance

Fault clearance

Primary

VD control Primary

VD control Secondary

control

Secondary

control

42

5. VSC-MTDC Dynamic Model

A dynamic model for VSC-MTDC systems, valid for every DC grid configuration and

regardless of the converters’ topology was developed in [24], starting with the basic

differential equations of the AC and DC circuits to finally add the controller and

coupling equations. In this section the time domain expressions are obtained to

assemble a Differential-Algebraic Equations (DAE) system that allows describing the

dynamic behaviour of VSC-MTDC systems, where:

• The model is only valid for small deviations from the fundamental frequency,

since quasi-stationary phasor representation is used. A full electromagnetic

model, valid for a broad frequency range, would require representing the AC

and DC filter, valves, etc.

• The AC grid voltage is considered as perfectly constant, balanced and stable

since the dynamic models of generators, exciters, governors, AGC, automatic

voltage regulator (AVR), power system stabilizer (PSS) among others are not

integrated, even though it is acknowledged that the electrical network has its

own dynamic performance, which can be influenced by faults, resonances,

overload, etc.

5.1. AC side dynamic equations

The dynamic of the converter AC side is given by the phase reactor current, in this work

in the dq-frame is used,

(((( ))))(((( ))))sdcd

prqd

pr

prd VVL

IIL

R

dt

Id−−−−++++++++−−−−====

1ωωωω

(29)

43

(((( ))))(((( ))))sqcq

prdq

pr

prqVV

LII

L

R

dt

Id−−−−++++−−−−−−−−====

1ωωωω

(30)

It is worth noticing that no assumption has been made about the relation between AC

and DC side voltages.

5.2. DC side dynamic equations

The DC circuit of a VSC HVDC system consists of a large capacitor at the converter

station and a DC cable. Figure 25 shows the DC circuit representation of a two-terminal

VSC HVDC system with two cables of opposite voltage.

Figure 25. VSC - DC side representation

The basic equations of a two terminal system are:

(((( ))))(((( ))))(((( )))) (((( ))))2,11

1)1( dcdc

dcdc II

dt

VdC −−−−==== (31)

(((( ))))(((( ))))(((( )))) (((( )))) (((( )))) (((( ))))2,121,22

2)2( dcdcdcdc

dcdc IIII

dt

VdC ++++====−−−−==== 32)

(((( )))))2,1()2()1(

)2,1(

)2,1( dcdcdcdc

dc

dc IRVVdt

IdL −−−−−−−−==== (33)

In a MTDC system several converters can be simultaneously connected, as shown in

figure 26.

Idc,(i,j)

Cdc,i Cdc,j

44

Figure 26. Multi-terminal VSC - DC side [24]

The current directions in the DC lines are fixed such that the current flowing from

converter i to converter j is defined positive if the subscript i is smaller than j.

Finally, the generalised dynamic equations of a MTDC circuit can then be written as

follows:

(((( ))))

−−−−==== ∑∑∑∑

====

dcN

j

jidcidcidc

idcII

Cdt

Vd

1

),()()(

)( 1 (34)

(((( ))))(((( ))))),(),()()(

),(

),( 1jidcjidcjdcidc

jidc

jidcIRVV

Ldt

Id−−−−−−−−====

(35)

5.3. Converter dynamic equations

The actual converter voltages are equal to their references only in steady-state. During

transients, the fast current controller provides a reference value for the AC side

converter voltage but its actual value lags the reference due to the time-delay introduced

45

by the converter’s power electronics and digital control circuits.

The relation between the actual and the reference value has been approximated by a

time constant Tσ [24]:

1

1

++++====

sTV

V

cdref

cd

σσσσ

(((( )))) (((( ))))cdcdrefcd

VVTdt

Vd−−−−====

σσσσ

1 (36)

1

1

++++====

sTV

V

cqref

cq

σσσσ

(((( ))))

(((( ))))cqcqrefcq

VVTdt

Vd−−−−====

σσσσ

1 (37)

Another delay might be considered at the slack converter for delivering the DC current,

approximated by a time constant Tdc as suggested in [28].

1

1

,

,

++++====

sTI

I

dcslackdcref

slackdc (((( )))) (((( ))))slackdcslackdcrefdc

dcII

Tdt

Id,,

1−−−−==== (38)

5.4. Inner Controller Dynamic Equation

Two additional variables, Md and Mq, as represented in figure 27, are introduced to

rewrite equations (16) and (17) algebraically and to represent the controller responses

through a set of first order differential equations given by (39) and (40)

Figure 27. PI Controller decomposition

(((( )))) (((( ))))ddrefid IIK

dt

Md−−−−==== 1

(39)

qprsd ILV ωωωω−−−−

dI

cdrefV

dM

drefI

46

(((( ))))(((( ))))qqrefi

qIIK

dt

Md−−−−==== 1

(40)

(((( )))) dddrefpqprsdcdref MIIKILVV ++++−−−−++++−−−−==== 1ωωωω (41)

(((( )))) qqqrefpdprsqcqref MIIKILVV ++++−−−−++++++++==== 1ωωωω (42)

The expressions for the reference voltages (38) and (39) can be substituted in equations

(33) and (34) to represent the AC side converter voltage dynamic as the following first

order differential equation:

(((( )))) (((( )))) (((( )))) dsdcdqpr

ddrefpcd M

TVV

TI

T

LII

T

K

dt

Vd

σσσσσσσσσσσσσσσσ

ωωωω111

++++−−−−−−−−−−−−−−−−==== (43)

(((( ))))(((( )))) (((( )))) qsqcqd

pr

qqref

pcqM

TVV

TI

T

LII

T

K

dt

Vd

σσσσσσσσσσσσσσσσ

ωωωω111

++++−−−−−−−−++++−−−−==== (44)

5.5. Outer Controller Dynamic Equation

Active Power Controller

In an analogous way, a new variable, Nd, is introduced to represent the active power

controller response as depicted in figure 28 that leads to an extra differential equation

(45).

Figure 28. Active Power Control block diagram

sd

ref

V

P

refP

dsd IVP ====

drefI

dN

47

( ) ( )PPKdt

Ndrefip

d −= (45)

Finally, the d-axis reference current of the active power controlled converters is found

through an algebraic equation:

(((( ))))dsdrefppdsd

refdref IVPKN

V

PI −−−−++++++++==== (46)

Reactive Power Controller

Nq is now introduced to represent the dynamic of the PI controller associated with the

reactive power control loop, resulting in an additional differential equation:

(((( ))))(((( ))))(((( ))))qsdrefiq

qIVQK

dt

Nd−−−−−−−−==== (47)

Finally, substitution leads to an algebraic expression for the q-axis reference current:

(((( ))))qsdrefpqqsd

refqref IVQKN

V

QI ++++++++++++−−−−==== (48)

DC Voltage Controller

Introducing:

(((( )))) (((( ))))dcdcrefidcdc VVK

dt

Md−−−−==== (49)

Equation (23) can be rewritten as an algebraic equation:

(((( )))) dcslackdcdcrefpdcdcref

slackdcslackdcslackdcref MVVK

V

IVI ++++−−−−++++==== ,

,,,

(50)

It is worth noticing that for each converter either Nd or Mdc is defined.

48

Then, equation (50) can be replaced in equation (38) to write the first order differential

equation for the slack converter DC current:

(((( ))))(((( ))))

++++−−−−++++++++−−−−==== dcdcdcrefpdc

dcref

slackdcdcslackdc

dc

slackdcMVVK

V

IVI

Tdt

Id ,,

, 1 (51)

5.6. Power Balance equations

The AC and DC circuit equations have to be coupled. The coupling equations follow

from the active power balance between AC and DC side of the converter.

0====++++++++ clossdc PPP (52)

If the converters’ losses are neglected, it is possible to calculate the injected DC currents

of all converters as follows:

cd

cqqdcslackdcslackdref

V

VIVII

++++−−−−====

,,

2 Slack converter (53)

dc

cqqcdddc

V

VIVII

2

++++−−−−==== All others converters (54)

5.7. Reduced Order Dynamic Model (RODM)

From the equations presented above a non-linear Differential Algebraic Equation

(DAE) system was built as developed in [24], where the electrical parameters and

controllers’ gains are assumed as known [18]. Finally the vector of unknown variables

X is assembled as shown below, starting with the DC and AC current and voltage, and

the controllers’ variables of each converter to finish with adding all the DC link currents

as defined in Eq. (35). For the slack converter Nd is replaced for Mdc.

49

====

dc

dc

q

d

cq

cd

q

d

q

d

V

I

N

N

V

V

M

M

I

I

X

+

−−−− ),1(

)2,1(

...

NdcNdcdc

dc

I

I

In a first place, for the purpose of this study, the delay of converters can be neglected to

work with a reduced model [24], so that the AC side converters’ voltage is equal to their

reference, given by algebraic equations (41) and (42). In the same way, the DC injected

current of the slack converter is given by equation (50). Figure 29 shows a summary of

the equations that define each variable of interest.

Figure 29. Summary of Equation in the DAE system.

50

6. Simulation Results

The system under concern is the same as presented in figure 6. In this section the

dynamic evolution between the two aforementioned steady states, before and after a

contingency are presented, considering both DC voltage control strategies, the

centralized M/S and the distributed VD as primary control. To achieve numerical

solutions some additional parameters must be defined as the DC capacitance (250 µF),

and the phase reactor (50 mH) and DC line inductances (0,2 mH/km).

At t = 0,3s the 0,5 pu active power step change is considered in VSC 1. Then, at t = 0,7s

a reactive power step change is considered in VSC 2.

6.1. Centralized DC voltage control (M/S)

Figure 30. Active Power Step change under M/S.

51

Figure 31. DC voltage profile under M/S.

Figure 32. DC currents under M/S.

Figure 33. D-axis currents under M/S.

52

6.2. Distributed DC voltage control (VD)

Figure 34. Active Power Step change under DC VD.

Figure 35. DC voltage profile under DC VD.

Figure 36. DC currents under DC VD.

53

Figure 37. D-axis currents under DC VD.

6.3. Reactive Power control

Figure 38. Reactive Power Step change.

Figure 39. Q-axis currents.

54

Figure 40. D-axis AC side voltages.

Figure 41. Q-axis AC side voltages.

7. Results Analysis

Regarding the active power injection of the VSCs into the DC grid, under the M/S DC

voltage control strategy, VSC 3 (the slack) is forced to assume all the power change

imposed to VSC 1, while VSC 2 remains under constant active power injection mode

(figure 30). On the other hand, when DC VD is considered (figure 34) the power step is

shared between VSC 2 and 3 as expected. Since the proportional gain of the droop

controller in VSC 3 has been set to a higher value than the one of VSC 2 (10 vs. 4 pu.),

VSC 3 takes a greater portion of the power unbalance.

55

Concerning the DC voltage, if a centralised DC voltage control strategy and the RODM

(converter’s delay neglected) are considered, figure 31 shows that the DC voltage

regulator is capable of restoring the DC voltage of the slack converter to its reference

value with a well amortised dynamic. On the other hand, figure 35 shows that under the

DC VD the DC voltage profile deviate from its nominal value.

The DC current and the d-axis AC current (figures 32 and 33, 36 and 37) follow the

same behaviour of the active power as could be predicted.

With respect to the q-axis AC current, there is not different between both DC voltage

control strategies since a decoupled active and reactive power has been achieved (figure

39), and it will respond to the reactive power steps changes (figure 38).

Following equation (7), when the phase reactor resistances is neglected, the overall

behaviour of the d-axis voltage (figure 40) is linked with the q-axis AC current, with a

transient response after a power step change. In the same way, the q-axis voltage

behaviour is associated to the d-axis current, and therefore to the active power step

changes, as described in equation (8)

8. Conclusions

The main feature of the proposed model is its suitability for studying slow transients in

any DC grid configuration regardless the converters topology. However, the filters,

switching components, PLL, fast acting protection, and firing control are not modelled,

which was necessary to confine the range of time constants to a practical wide,

appropriated to the phenomena of interest (around the fundamental frequency);

otherwise an enormous modelling effort and a huge computational power would be

required. Nevertheless these elements could be included in further works, as the AC

56

grid generators and their control.

The starting points for the dynamic simulation were found by writing the steady-state

DC equations in a matrix form and by intelligently building the admittance matrix,

where the slack converter was defined as the last node and its link admittance

information was placed in the last row (and column) of the DC nodal admittance matrix,

to finally solve a DC Power Flow, considering a centralised DC voltage control strategy

(M/S).

It must be kept in mind that the phasor modelling approach and an implicit integration

method (based on linear interpolation) were used, therefore the implemented model and

the results obtained are submitted to all their inherent limitations.

Later, the dynamic study began with a Reduced Order Dynamic model (RODM) that

was derived by neglecting the smallest time constants, given by the converters delay.

Even if the Full Order Dynamic Model (FODM) results are not presented, it was found

that the RODM provided the same overall behaviour as the FODM, with some

advantages that include less data requirements, less equations and, larger step sizes

could be used [24].

It is important to note that when the RODM is used, any power step change can be

imposed, and the controllers will be able to build and follow their references set-points.

However, when the converters’ delay is introduced, more moderated disturbance must

be considered to achieve numerical solution. On the other hand, as expected, the phase

reactor inductance represents a key parameter in the control of the current, an above all,

the reactive power, while the DC capacitor will determine the DC voltage dynamic

response, and therefore, it plays an important role in the active power control.

57

Thereafter, a primary control strategy accomplished by a Distributed DC voltage control

strategy was implemented based on a DC voltage / power droop characteristic. The

dynamic of the DC voltage, and with it the power, the DC current and all the d-axis

variables, is enhanced and more restricted power step can be tolerated with respect to

the Master/Slave strategy (FODM), since the power unbalance will be shared between

different nodes.

However, in this scenario, permanent DC voltage deviations remain in all VSC DC

nodes, reason why a secondary control strategy at a different and a slower time scale

has been proposed. Within this SC one converter will force its DC voltage, transitorily

changing its power (to end with an imperceptible power deviation due to new DC grid

losses in the succeeding PF condition), while the others work in a constant active power

mode, generating a vertical displacement of the droop characteristic.

Finally, a complex system represented by the MTDC grid is controlled using the

“system of systems” approach, a solution with different timescales, which is mandatory

to cope with “Plug and Play” requirements needed for networked systems. Similarly to

AC grids and using the concept of “thinking globally and acting locally”, a primary

control strategy which is a global but distributed control has been proposed. At a

different and slower timescale, the secondary control regulates the voltage deviation

across the grid – or the next system up in the system of systems approach. The proposed

work could usefully be applied to other fields if a formalizing theory that puts the basis

for global stability and without taking into account all the mathematical equations

representing the interconnected systems to proof the global stability. This will be the

subject of future work.

58

Appendix

A. Per unit System

The term 3/2 of power equations (11) and (12) disappears in the per unit

representation if the power base is defined as the three phase power and the voltage base

is the peak value of rated line-to-neutral voltage, such that:

basebasermsrmsbase IVIVS2

33 ========

Then, the power equation in the rotating frame can be simplified as follows:

(((( ))))(((( ))))

qpuqpudpudpu

basebase

qqdd

baseqqddpu IVIV

IV

IVIV

SIVIVP ++++====

++++

====++++====

2

32

3

1

2

3

qpudpudpuqpupu IVIVQ −−−−====

When dealing with converters, close attention should be given to the connection

between the bases of the AC and DC sides, so the per unit power is conserved.

basebasebasebasedcbasedcbasedc IVSIVP2

32 ___ ============

References

[1] T. J. Hammons, D. Woodford, J. Loughtan, M. Chamia, J. Donahoe, D. Povh, B. Bisewski, and W. Long, “Role of HVDC Transmissions in Future Energy Development,” IEEE Power Engineering Review, pp. 10-25, Feb. 2000.

[2] M. P. Bahrman and B. K. Johnson, “The ABCs of HVDC Transmission Technologies,” IEEE Power & Energy Magazine, pp.

32-44, March/April 2007. [3] N. Flourentzou, V. Agelidis, and G. Demetriades, “VSC-Based HVDC Power Transmission Systems: An Overview,” IEEE

Transactions on Power Electronics, vol. 24, No. 3, pp. 592-602, March 2009.

[4] C. Bajracharya, “Control of VSC-HVDC for wind power”. Norwegian University of Science and Technology, June 2008.

[5] M. Oliveira, J. Gazoli, A. Sguarezi, and E. Ruppert, “A Control Method for Voltage Source Inverter Without Dc Link

Capacitor”. IEEE 2008.

[6] G. Bon-G Gu and N. Kwanghee, “A DC-Link Capacitor Minimization Method Through Direct Capacitor Current Control”.

IEEE Transactions on Industry Applications, vol. 42, No. 2, pp. 573-581, March/April 2006.

[7] T. Haileselaissie, “Control of Multi-Terminal VSC-HVDC Systems”. Norwegian University of Science and Technology,

June 2008.

[8] N. Chaudhuri, R. Majumder, B. Chaudhuri, J.Pan, and R. Nuqui, “Modeling and Stability Analysis of MTDC Grids for

Offshore Wind Farms: A Case Study on the North Sea Benchmark System”. IEEE, 2011.

[9] Ahmed, N.; Haider, A.; Van Hertem, D.; Lidong Zhang; Nee, H.-P., "Prospects and challenges of future HVDC SuperGrids

with modular multilevel converters," Power Electronics and Applications (EPE 2011), Proceedings of the 2011-14th

European Conference on , vol., no., pp.1,10, Aug. 30 2011-Sept. 1 2011. [10] C. Barker and R. Whitehouse, “Autonomous converter control in a multiterminal HVDC system”. Proc. IET ACDC ’10,

London, UK, October 20–21, 2010.

[11] J. Beerten, D. Van Hertem, and R. Belmans, “VSC MTDC systems with a distributed DC voltage control – A power flow approach”. Proc. IEEE PowerTech ’11, Trondheim, Norway, June 19–23, 2011.

[12] L. Xu, L. Yao, and M. Bazargan, “DC Grid Management of a Multi-Terminal HVDC Transmission System for Large

Offshore Wind Farms”. International Conference on Sustainable Power Generation and Supply, SUPERGEN ’09, pp. 1-7, 2009.

[13] P. Kundur, “Power System Stability and Control”. McGraw-Hill, Inc, 1994.

[14] J. Beerten, S. Cole, and R. Belmans, “Generalized steady-state VSC MTDC model for sequential AC/DC power flow

algorithms”. IEEE Trans. Power Syst., 2012. [15] R. Teodorescu, M. Liserre, and P. Rodríguez, “Grid Converters for Photovoltaic and Wind Power Systems”. John Wiley &

Sons, Ltd., 2011.

59

[16] Milasi, R.M.; Lynch, A.F.; Li, Y.W., "Adaptive Control of a Voltage Source Converter for Power Factor Correction," Power

Electronics, IEEE Transactions on , vol.28, no.10, pp.4767,4779, Oct. 2013.

[17] Chaudhuri, N.R.; Chaudhuri, B., "Adaptive Droop Control for Effective Power Sharing in Multi-Terminal DC (MTDC)

Grids," Power Systems, IEEE Transactions on , vol.28, no.1, pp.21,29, Feb. 2013.

[18] J. Eloy-Garcıa, S. Poullain and A. Benchaib, Discrete-time Dead-Beat Control of a VSC Transmission Scheme including

Voltage and Current limitations for Wind Farm Connection, EPE-PEMC’2006 (12th International Power Electronics and

Motion Control Conference), Portoroz, Slovenia, August 30 - September 1, 2006.

[19] A. Benchaib, S. Poullain and J. L. Thomas, Robust DC bus voltage control based Backstepping and Lyapunov methods using

Kalman filter and adaptive control for long distance VSC transmission scheme “Les techniques avancées et les stratégies

innovantes en modélisation et commande robuste des procèdes industriels, ISA (The instrumental Systems & Automation

Society) Symposium, Martigues, France, 21st – 22nd September, 2004.

[20] Sood, V.; Patel, H., "Comparison between direct and vector control strategy for VSC-HVDC system in EMTP-RV," Power

Electronics, Drives and Energy Systems (PEDES) & 2010 Power India, 2010 Joint International Conference.

[21] C. Bajracharya, M. Molinas J. Suul, and T. Undeland, “Understanding of tuning techniques of converters controllers for VSC-HVDC”. Nordic Workshop on Power and Industrial Electronics, June 2008.

[22] J. Beerten, S. Cole, and R. Belmans, “Implementation aspects of a sequential AC/DC power flow computation algorithm for

multi-terminal VSC HVDC systems,” in Proc. IET ACDC ’10, London, U.K., October 20–21, 2010. [23] J. Beerten, S. Cole, and R. Belmans, “A sequential AC/DC power flow algorithm for networks containing multi-terminal

VSC HVDC systems,” in Proc. IEEE PESGM’10,Minneapolis, MN, July 25–29, 2010.

[24] S. Cole, “Steady-State and Dynamic Modelling of VSC HVDC Systems for Power Systems Simulation”. Katholieke Universiteit Leuven, September 2010.

[25] T. Vrana, L. Zeni, and O. Fosso, “Active Power Control with Undead-band Voltage & Frequency Droop for HVDC

Converters in Large Meshed DC Grids”. NTU, Trindheim and DTU, Denmark, 2012.

[26] T. Nakajima and S.Irokawa, “A Control System for HVDC Transmission by Voltage Sourced Converters”. Power Engineering Society Summer Meeting, IEEE vol. 2, pp. 1113–1119, 1999.

[27] R. Teixeira Pinto, “Dynamics and Control of VSC-based HVDC Systems". Master Thesis, Politecnico di Torino, Torino,

Italy, 2008. [28] S. Cole and J.Beerten, “Generalized Dynamic VSC MTDC Model for Power System Stability Studies”. IEEE Trans. Power

Sys., vol. 25, no. 3, pp. 1655–1662, August 2010.