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DC power transmission systems comprising multiple VSC-HVDC equipment
Electrical Power and Energy Systems 107 (2019) 140–148
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Real-power economic dispatch of AC/DC power transmission systems comprising multiple VSC-HVDC equipment
T
Luis M. Castroa, , J.H. Tovar-Hernándezb, N. González-Cabrerac, J.R. Rodríguez-Rodrígueza ⁎
a
National Autonomous University of Mexico (UNAM), Department of Electrical Energy, Mexico Instituto Tecnológico de Morelia (ITM), Research and Postgraduate Program in Electrical Engineering, Mexico c Instituto Tecnológico Superior de Irapuato (ITESI), Electromechanics Department, Mexico b
ARTICLE INFO
ABSTRACT
Keywords: AC/DC power systems Lagrange multipliers Incremental transmission loss Real-power economic dispatch VSC-HVDC equipment VSC stations
This paper presents a practical approach for the real-power economic dispatch of AC/DC power transmission systems formed by VSC-HVDC equipment. It uses the Lagrange multipliers method combined with a comprehensive representation of the AC/DC power system. The power losses of the AC systems, DC grid and power converters are considered in this novel formulation by using incremental transmission loss factors. For realistic solutions, the different control strategies of the power converters forming any VSC-based power grid are considered. Interestingly, this approach permits to conclude that the AC grids coupled to converters charged with DC voltage control define the nodal marginal prices in the DC network, and in turn, the energy prices in the VSCconnected passive grids are inherited from the DC network. Overall, this modelling approach yields great flexibility to model any arbitrary VSC-based HVDC power grid interconnecting various AC systems. The developed method has been validated using a two-terminal VSC-HVDC network interconnecting two otherwise independent AC grids. Its results are compared against those obtained by the sequential quadratic programming method available in the optimisation toolbox of Matlab©, with both solutions exhibiting differences inferior to 0.30% in the total generation costs, therefore concurring very well between each other. The practicality of this approach is also demonstrated by carrying out the real-power economic dispatch of an AC/DC transmission system comprising seven VSC units which give rise to a thirteen-node DC grid.
1. Introduction High Voltage Direct Current (HVDC) systems using VSC-based power stations are rapidly becoming the key platform of modern transmission grids with which higher power throughputs are being reached [1]. VSC-HVDC links enable the interconnection of neighbouring AC systems thus generating a variety of positive technical aspects [2]. It may be argued that one of the main motivations in using HVDC systems is the possibility to facilitate increased energy transactions aiming at reducing energy prices. This is especially true when there is high demand and null availability of low-cost power generation. It should be said that, however, the use of VSC-based HVDC grids per se may not necessarily lead to a more economic operation of the various HVDC-connected AC systems because there may be DC transmission paths being restricted by VSC stations [3]. This is one of the reasons why creditable efforts have been recently dedicated to study the optimal power flow (OPF) problem in HVDC grids, placing emphasis on the security-constrained OPF [4–6].
⁎
This is a very complex problem that may be solved using nonlinear optimisation techniques such as quadratic programming, interior-point methods, or heuristic methods. Generally, the straightforward option is to resort to commercial optimisation solvers such as Knitro, GAMS, AMPL, Matlab optimisation toolbox, etc. Unexpectedly, only a few works addressing this all-important economic aspect in hybrid AC/DC systems have been published [7–13]. It should be also said that the previously-mentioned solution approaches have failed to represent the various VSC types that may give rise to a generic AC/DC power grid having any number of VSC units. That is, they all lack the inclusion of VSC stations controlling their DC bus voltages, VSC units charged with fixed power transmission and VSC stations feeding into passive grids [14,15], in a single formulation as is the case of this paper. For the sake of generality and applicability, this is taken care of in this work, thus leading to very important conclusions regarding the economic operation of VSC-based HVDC power grids. Interestingly, the introduced method allows to conclude that the AC grids coupled to converters charged with DC voltage control define the nodal marginal prices in the
Corresponding author. E-mail address: [email protected] (L.M. Castro).
https://doi.org/10.1016/j.ijepes.2018.11.018 Received 16 March 2018; Received in revised form 6 September 2018; Accepted 16 November 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 107 (2019) 140–148
L.M. Castro et al.
Nomenclature
FPsch FPass FDC JAC/DC ΔΦSlack ΔΦPsch ΔΦPass ΔEDC Gdc θSlack θPsch θPass BSlack
θ nodal voltage phase angle V nodal voltage magnitude Edc DC voltage ma modulation ratio ϕ phase-shifting angle Rdc, Ldc, Cdc DC resistance, inductance and capacitance nac, ndc number of AC and DC nodes Ng number of generators Pgi generated power generator cost function ci(Pgi) λj, λk, λn Lagrange multipliers rph + jxph converter’s coupling impedance Ploss power losses of the system Psch scheduled power Ω nodal variable vector P power injection vector Pd demand power vector β vector of power loss sensitivity factors μ Lagrange multipliers associated with VSCPsch constraints χ Lagrange multipliers associated with power generation enforcements FAC mismatch vector of the AC grids FSlack mismatch vector of the VSCSlack converter model
BPsch BPass AC DC VSC VSCSlack VSCPsch VSCPass ITL HVDC
mismatch vector of the VSCPsch converter model mismatch vector of the VSCPass converter model mismatch vector of the DC grid model Jacobian matrix of the AC/DC grid model state variable vector of the VSCSlack converter model state variable vector of the VSCPsch converter state variable vector of the VSCPass converter state variables vector of the DC grid nodal conductance matrix of the DC grid phase angles of the AC grid coupled to VSCSlack phase angles of the AC grid coupled to VSCPsch phase angles of the AC grid coupled to VSCPass nodal susceptance matrix of the AC grid coupled to VSCSlack nodal susceptance matrix of the AC grid coupled to VSCPsch nodal susceptance matrix of the AC grid coupled to VSCPass Alternating Current Direct Current Voltage Source Converter slack VSC power-scheduled VSC passive VSC Incremental Transmission Loss High Voltage Direct Current
DC network, and in turn, the value of the nodal marginal prices in the VSC-connected passive systems are directly inherited from the DC network. The VSC units exerting power regulation, on the other hand, are found to raise the price of the energy in the whole hybrid AC/DC system, something that needs special attention if transmission system operators are to pursue an economic operation in these modern power grids. As opposed to previous works related to the optimal power flows in VSC-based transmission networks, this paper focuses on the real-power economic dispatch of HVDC grids fed by multiple VSC stations, one where each power converter is modelled with its corresponding control task. The Lagrange multipliers method, combined with a lossless representation of the AC/DC network, is employed. In turn, Incremental Transmission Loss (ITL) factors, derived for both the AC grids and DC grid, are used to incorporate the power losses of all, the AC systems, DC grid and power converters comprising the AC/DC grid. This comprehensive modelling approach has been validated using a two-terminal VSC-HVDC grid. Its solution is directly compared against the sequential quadratic programming method available in the optimisation toolbox of Matlab©, with both solutions agreeing quite well between each other.
stations exerting power regulation. In this context, the formulation for the real-power economic dispatch in AC/DC power grids comprising various VSC stations is derived next.
2. Economic operation of HVDC power transmission systems formed by VSC stations
A VSC station consists of a power electronic converter, AC filter and an on-load tap-changing transformer (OLTC) to connect to an AC subnetwork, as depicted in Fig. 1. The steady-state, fundamental frequency operation of AC/DC systems containing several VSC stations may be computed using the Newton-based power flow formulation summarised by (1). This equation implies a multi-terminal HVDC system with m VSCSlack units, n
2.1. Generalised AC/DC power grid modelling incorporating multiple VSC units System-wide studies aimed at HVDC networks formed by multiple VSC terminals may be carried out using a unified approach. The interactions between the interconnected AC sub-networks and the DC grid are well assessed by assigning the VSC units different control strategies to satisfy specific operating requirements. In this sense, the VSC stations may be classified as follows [14,15]: i. Slack converters VSCSlack - these stations provide voltage control at its DC terminal. ii. Power-scheduled converters VSCPsch - these stations control the power transmission to a constant value. iii. Passive converters VSCPass - these converters interface the DC grid with AC passive networks having no generation of their own.
The economic operation problem of electrical systems is associated with the minimisation of all generation cost functions. This objective is subjected to nodal power balances and to constraints related to (i) power transmission capacities, (ii) nodal voltage operating limits, and (iii) power generation limits of dispatchable generators. Fundamentally, the problem consists in finding the optimum generation dispatch that reduces the system power losses. This fact becomes more important in cases involving the transmission of electrical energy over very large distances, as may be the case of modern VSC-based power grids. It is also well-known that congested transmission paths significantly affect the generation dispatch, causing the local nodal energy prices to rise [16,17]. In general terms, any transmission constraint will lead to a pricier system operation. Admittedly, the same conclusions may be anticipated for AC/DC networks due to the presence of VSC
Fig. 1. VSC station with ancillary elements. 141
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L.M. Castro et al.
VSCPsch units, r VSCPass units and an arbitrary DC network. Notice that α AC sub-networks, α = m + n + r, are interlinked through the DC grid. The fundamental state variables of the DC grid and VSC stations, DC voltage Edc, modulation ratio ma, phase-shifting angle ϕ and OLTC’s tap Tv, are suitably accommodated in vectors ΔΦSlackm, ΔΦPschn, ΔΦPassr and ΔEDC, which are adjusted by iteration to satisfy the mismatch equations of the corresponding converters FSlackm, FPschn, FPassr and DC grid FDC [14,15]. In this sense, the vector FACα contains the power mismatch equations of the interconnected AC sub-networks whose associated state variables, nodal voltage magnitudes and phase angles, are placed in ΔΨACα. Clearly, the Jacobian matrix JAC/DC simultaneously accommodates the first-order partial derivatives of all mismatch equations emerging from the AC/DC network with respect to all state variables involved in the power flow problem.
FAC FSlackm FPschn FPassr FDC
i
AC
constant parameters whereas the phase-shifting angles ϕ of the converters VSCPsch enable the enforcement of their scheduled power flow Psch [14].
(a )
Pschn Passr
(1)
2.2. Problem formulation for HVDC power grids with multiple VSCconnected AC systems
f j = Pgjac
Pdjac
Pgimin
Pgi
|Pjm | s. t.
dc fk = Pdk fn = PVSCn
C=
Ng c (P ) i = 1 i gi
AC sub - networks
max P jm dc Pcalk =0 Pschn = 0
n VSCPsch & DC
(a )
max |Pkm | Pkm
(2)
k · fk
= =
j
=0
j
n · fn
i=j
j = 1, ...,nac;
j slack generators
=0
k = 1, ...,ndc;
fn
=0
n = 1, ...,n VSCPsch
k E k n
i = 1, ...,Ng ;
fk
n
Pkm = Gkm (Ek2
(c ) Pij =
k
VSCSlack
Ek Em)
Vi2 Gii
kma Ek Vi [Gph cos(
+ Vi Vj [Gij cos(
i
j)
+ Bij sin(
i)
+ Bph sin(
i
j )]
i )]
(5)
where i and j are generic nodes pertaining to any of the AC sub-netmax max works or DC grid; P jm and Pkm stand for the maximum power that can be transmitted through AC and DC transmission lines, respectively; PVSCn is the power transmitted through the n-th VSCPsch station with Pschn being its corresponding scheduled power. To find the constrained optimum of (2), the augmented Lagrange cost function (3) may be used. This equation suggests that the optimisation of C, subject to the equality constraints fj = 0, fk = 0 and fn = 0, is akin to searching for the unconstrained optimum of C*, where λ j, λk and λ n are the so-called Lagrange multipliers. The way inequality constraints are treated in the proposed formulation is explicitly detailed in Section 3. j ·f j
n
=
(b) Pki = k2ma2 Ek2 Gph
grid
C =C
(d )
C*
j
=0
Fig. 2 shows a VSC unit interfacing a DC line with an AC line. This is indeed the basic building block upon which the proposed method is derived. This section is dedicated to formulating the overall AC/DC grid using lossless power flow models of all, the AC and DC transmission elements along with the VSC units considering their corresponding control strategies, as addressed in Section 2.1. Since there may be n converters VSCPsch responsible for restricting their power flow, it becomes necessary to derive an AC/DC grid model where the transmission path models are available, as opposed to the case when all the generating units are assumed to be connected to a single node to supply the total system load. From the outset, it may be then predicted that these conditions will lead to different nodal marginal prices in various nodes of the VSC-based AC/DC grid, a fact that will be confirmed and duly analysed in the section of study cases. The steady-state power flow equations for DC transmission lines, VSC stations and AC transmission lines may be given by (5a), (5b) and (5c), respectively [14].
ac Pcalj =0
Pgimax
(c )
C* Ek
fj
2.3. Derivation of the lossless HVDC grid model fed by multiple VSC units
The optimum dispatch concerns with the determination of the individual real power settings of all Ng generation units of the α AC subnetworks interconnected through the various VSC units. To this end, the sum of the individual production cost of each dispatchable generator i, ci(Pgi) = ai + biPgi + di P2gi +∙∙∙, is the total cost function ($/h) to be minimised. This is subjected to constraint equations that ensure the nodal real-power balances in the AC/DC system, while simultaneously observing physical limitations in both power generation and power transmission, that is,
min
C*
fj j P gi
C Pgi
(4)
i
EDC
(b )
=
The real-power economic dispatch of AC systems, with transmission line constraints considered, bases on a network’s linear power flow model which stems from lossless transmission elements; following this philosophy, this is carried out in Section 2.3 for generic VSC-HVDC power grids. Subsequently, the system power losses may be considered by using Incremental Transmission Loss (ITL) factors. These may be regarded as sensitivities associated with changes in the system power losses with respect to nodal power injections; these sensitivities are derived, for AC/DC grids formed by VSC stations, in Section 2.4.
Slackm
(JAC / DC)i
=
C* Pgi
where Gij + jBij = (rij + jxij) ; Gph + jBph = (rph + jxph)−1 is the admittance of the converter’s coupling phase reactor relating to the filtering device (not shown in the figure), with rph accounting for its conduction losses; k = √(3/8) is a constant associated with the AC-to-DC voltage conversion in two-level, three-phase VSC stations. The following simplifying assumptions are adopted to obtain the aforesaid reduced AC/DC model: Gkm = rkm−1,
−1
(i) The power losses of the AC system and VSC are neglected by setting rij = rph = 0; (ii) Since the angular differences (ϕ-θi) and (θi-θj) are very small, then
(3)
The real-power economic dispatch is carried out with all AC voltage magnitudes fixed at 1 p.u. [16,17], therefore if the α AC sub-networks contain nac AC nodes and the DC grid comprises ndc DC nodes, the optimisation problem resides in solving the set of equations (4) which stems from the partial derivation of (3) with respect to all variables involved. As it turns out, the reference angles θ of the corresponding AC grids and the DC bus voltages E regulated by stations VSCSlack are all
Fig. 2. Generic VSC station interfacing AC with DC elements. 142
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L.M. Castro et al.
sin(ϕ-θi) ≈ (ϕ-θi) and sin(θi-θj) ≈ (θi-θj); (iii) A lossless DC transmission line model may be obtained by resorting to the alternative DC current expression Pkm ≈ Ikm.
C* C = Pgi Pgi
= Bij (
Em),
(b) Pki = Bph (
i
i ),
=
(c ) Pij
Using the linear power flow equations (6), the set of nodal power injections of the overall VSC-based transmission network may be straightforwardly obtained as a function of the AC phase angles and DC voltages. In a similar manner to Section 2.1, a multi-terminal arrangement, consisting of m VSCSlack converters, n VSCPsch converters and r VSCPass converters and a DC network, would bring about the linear power flow AC/DC model shown in (7), with i = 1…m, j = 1…n and k = 1…r. The derivation of this lossless AC/DC network model is exemplified in the section of case studies.
C / Pgi 1
Nac / dc
Ng
C* =
Nac / dc
ci (Pgi ) i=1
( i=1
i=1
Ploss )
j +1
Ek Ek + 1
nac
(14)
Endc ]T
dP net = dP jnet + dPknet +
Nac / dc
dP net ;
j
,
i
=1
(15) The total derivatives of the net power injections dPγ are explicitly given by (16). For ease of representation, the use of the superscript ‘net’ has been dropped in these equations.
dP1 =
P1 j
d j+
dPnac =
Pnac
dP1dc =
P1dc
dPndc =
Pndc
j j
j
+
P1 nac
d
d j+
+
Pnac
d j+
+
P1dc
d j+
+
Pndc
nac nac
nac
+
nac
P1 dEk + Ek
P1 dEndc Endc
+
d
nac
+
Pnac dEk + Ek
+
Pnac dEndc Endc
d
nac
+
P1dc dEk + Ek
+
P1dc dEndc Endc
d
nac
+
Pndc dEk + Ek
+
Pndc dEndc Endc
(16)
Using (14) and (16), the vectors (17) and (18) may be defined, all of dimension (ndc + nac-2 m-n)x1.
dP = [ dPj dPj + 1 = [d
d
d
j
j +1
dPnac dPk dPk + 1
dPndc ]T
(17)
d
dEndc ]T
(18)
nac
dEk dEk + 1
The power-injection vectors (19) and (20) may be also defined for those nodes associated with slack AC buses, j′, and DC buses, k′, respectively. The Jacobian matrix (21) encompasses the derivatives of the power injections with respect to all unknown variables involved in (16).
Pj
Nac / dc
Pdi
(13)
=1
net
(9)
Pgi
P net
=1
Nac / dc
Ploss = 0
j
Nac / dc
One way to consider the power losses in the real-power economic dispatch of AC networks is by means of Incremental Transmission Loss (ITL) factors [16,17], an enterprise not yet addressed in the context of VSC-based DC transmission networks. If the power losses of the overall system are Ploss, then the nodal power balances of the whole AC/DC network may be defined by (9), with Nac/dc = nac + ndc. Consequently, the constrained optimum problem may be defined as in (10).
Pdi
=1
dPloss =
(8)
i=1
Nac / dc
Pd =
Eqs. (13) and (14) suggest that power loss is a function of both the AC bus voltage phase angles and the DC nodal voltages, i.e., Ploss = f (Ω). From (13), equation (15) is obtained by differentiation, where j′ and k′ stand for the slack AC buses and DC buses where voltage regulation is exerted by VSCSlack units, respectively.
2.4. Derivation of the incremental transmission loss factors for AC/DC grids formed by VSC units
Pgi
(12)
Since there is a slack generator for each AC network coupled to the m VSCSlack units and n VSCPsch units whilst there are also m regulated DC bus voltages by VSCSlack units, it follows that the system’s set of nodal variables may be defined by the (ndc + nac-2 m-n)x1-dimensional vector Ω, shown in (14).
Eq. (7) may be written in compact form as in (8), where BSlacki, BPschj, and BPassk are diagonal matrices which accommodate the lossless models of the AC grids connected to converters VSCSlack, VSCPsch and VSCPass, respectively. Matrices BS and BPa provide the link between the AC terminals of the VSCSlack and VSCPass with their corresponding DC nodes. The phase reactor susceptances of converters VSCSlack are placed in BSdc whereas GSdc contains the DC network conductances connecting with the DC buses of VSCSlack stations. Lastly, matrix Gdc accommodates the lossless DC grid model, excluding the row and column relating to the DC bus of the slack converters.
i=1
Nac / dc
Pg =1
(7)
Nac / dc
i = 1, ...,Ng
Ploss / Pgi
Ploss =
=[
PAC / DC = C· ,
(11)
where the term ∂Ploss/∂Pgi is usually referred to as the ITL associated with each generating unit i. From (12), it may be stated that ITLi = ∂Ploss/∂Pgi for i = 1,…,Ng, where ITLi = 0 for the bus relating to the slack generator. As described in Section 2.1, various AC networks are interconnected through the HVDC power grid, signifying that the overall AC/DC network model will be featured by various slack AC buses, a fact that needs special attention to successfully obtain the ITL in these new power grids. Reformulating (9), the total power loss Ploss of the AC/DC network may be expressed as
(6)
j)
i = 1, ...,Ng
Solving (11) for λ, the ensuing relationship is obtained:
Due to these suppositions, Eqs. (5) are reduced to those shown in (6). where the power flows, in the opposite direction, may be found by simply exchanging the corresponding subscripts. It should be said that these suppositions may imply for the proposed method, at most, more global iterations to converge. Also, the AC/DC grid’s optimum dispatch is expected to slightly differ with respect to more complex approaches such as OPF type solutions due to the fundamental differences in the optimisation tools. This is discussed in the section of case studies.
(a) Pkm = Gkm (Ek
Ploss =0 Pgi
+
(10)
Pk
The following expression must be satisfied for an optimum power dispatch: 143
=( =(
Pj j
Pk j
Pj
Pj
Pj
Pj
Pj
j +1
nac
Ek
Ek + 1
Endc
Pk
Pk
j+ 1
nac
Pk Ek
Pk Ek + 1
)T
Pk T Endc )
(19) (20)
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L.M. Castro et al.
P
=(
P1
P1
j
nac
Pnac
Pnac
j
nac
P1dc
P1dc
j
j
Pndc
Pndc
j
nac
P1 Ek
P1 Endc
Pnac Ek
Pnac Endc
P1dc Ek
P 1dc Endc
Pndc Ek
Pndc Endc
zero-padded matrices of suitable orders. A is a diagonal matrix accommodating the terms 2di for each generating plant i resulting from the derivatives ∂ci/∂Pgi. The connectivity matrix B contains the term −1 in locations where there is a link between row i and column j, i.e., B (i,j) = −1, recalling that i = 1,…,Ng and j = 1,…,nac. Matrix C results from the lossless AC/DC grid model (7) and (8). Matrix D, on the other hand, ensures the fixed power transmission in power-scheduled converters, i.e., this matrix suitably accommodates the expressions bph(ϕθi) = −Psch, according to (6b), for the n VSCPsch units. Matrices A to D are given in detail, in the section of study cases, whereas the various vector terms of (27) are explicitly shown in (28).
)
(21)
Using (14)–(21), the set of expressions (16) may be rearranged as (22), in compact form, with the power injections (19) and (20) redefined as shown in (23).
dP = (
dPj = (
P
)d
Pj
d
)T d
P
=(
dPk = (
Pk
) 1d P
(22)
)T d
(23)
(27)
Substituting dΩ of (22) into (23) and (15) yields (24). Nac / dc
dPloss =
·d P +
dP net
Pg1
j
,
i
(24)
=1
Pj
Pg =
where = [( )T + ( k )T ]( ) 1 are sensitivity factors of power losses with respect to power injections. Eq. (24) may be used to compute the incremental changes in power loss following an incremental change in power injections at any bus of the various AC networks or DC network. That is, knowing from (17) that dP = [dPj, dPj+1,…, dPnac, dPk, dPk+1,…,dPndc]T and that according to (24), β = [βj, βj+1,…, βnac, βk, βk+1,…,βndc]T, it is clear that the element-by-element multiplication β∙dP will lead to a vector of the same order as ΣdPγ net = [dPj, dPj+1,…, dPnac, dPk, dPk+1,…,dPndc]T shown in (24). Therefore, equation (24) takes the form of (25), after factoring the corresponding terms. It should be remarked that there will be as many sensitivity factors β as the number of AC/DC nodes Nac/dc, excluding those related to the slack AC buses and DC buses with voltage regulation, nodes j′ and k′, respectively. This is so because the power loss function Ploss is independent of the state variables associated with j′ and k′, i.e., the reference AC voltage phase angles and voltages of voltage-controlled DC nodes are known a priori.
dPloss = (1 + +
1) dP1
P
+
+ (1 +
+ (1 +
nac ) dPnac
+ (1 +
dPloss = (1 + dPgi
i)
=
i
j
µ=
Nac / dc
µ1 µn
b1 ,
b=
, bNg
Pd
P1dc
Pdnac , Ek Gk 1dc
Pndc
Ek Gk ndc
Psch =
Psch1 Pschn (28)
To further clarify the procedure involved in the real-power economic dispatch of AC/DC power grids, Fig. 3 shows a flow diagram summarising its most important steps. It should be said that this proposed approach suggests that each AC network and DC grid comprising the AC/DC system must take care for their corresponding power losses. This means that the power losses ought to be included into the otherwise independent AC networks and DC network as demanded power by properly modifying Pd in (28). A suggested practice is to add the AC and DC power losses, for instance, to those nodes related to slack AC and DC nodes, j′ and k′, respectively. 3.1. Handling of inequality constraints
(25)
i = 1, ...,Ng ;
,
Pd1
1dc ) dP1dc
ndc ) dPndc
=
PgNg
P
It should be mentioned that in the case that a generator has exceeded one of its limits, Pgmax or Pgmin, then the augmented Lagrange cost function (3) must be properly amended to include its corresponding inequality constraint as an equality function. This is shown in (29) where fm = Pgi –Pgmax = 0 or fm = Pgi – Pgmin = 0 should be included into the optimisation problem, depending on the exceeded power limit. In explicit manner, this may be treated in the proposed formulation as shown in (30). As expected, the matrix and vectors of (27) are suitably expanded to conform them according to the number of equality constraints plus the m violated power generation limits.
If all generators’ output powers are kept constant, except the one connected to node i, therefore, Eq. (26) holds.
ITLi =
1
,
(26)
This last expression facilitates the inclusion of the power losses, in the real-power economic dispatch of HVDC power grids comprising various VSC-connected AC systems, by applying (12). Notice that to obtain the ITL, the Jacobian matrix ∂P/∂Ω (21), the vectors ∂Pj′/∂Ω (19) and ∂Pk′/∂Ω (20), must be available; all these terms are suitably calculated with the information corresponding to the last iteration when solving (1), for a given load/generation pattern.
C =C
j ·f j
k · fk
n · fn
m ·fm
(29)
3. Economic dispatch for AC/DC power networks comprising various VSC-connected AC grids Assuming quadratic generation cost curves, ci(Pgi) = ai + biPgi + diP2gi, then the resulting economic dispatch problem is a linear one, from the mathematical standpoint. This fact combines very well with the derivations carried out for AC/DC power grids, (2)–(4), and with the generalised HVDC grid model fed by various VSC-connected AC systems derived in Section 2.3. In this context, the proposed approach aimed at the economic dispatch problem may be summarised by (27), where the 0 entries are
(30) In Eq. (30), E is a matrix of order Ngxm containing −1 in suitable locations corresponding to the generators whose limits are being enforced; χ is an mx1 vector that contains the Lagrange multipliers 144
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L.M. Castro et al.
Fig. 3. General procedure for carrying out the economic dispatch in AC/DC grids containing various VSC-connected AC systems.
associated with each of the m restrictions and the vector Pglim accommodates the power limits of the corresponding generators. All these vectors are explicitly shown in (31).
E=
0 0 0
1 0 0 0
0 0 0
Pglim 1
1
1
,
=
HVDC power grids with numerous VSC-connected AC grids enabling the simultaneous computation of the nodal marginal prices at the AC and DC systems. In this regard, this proposed approach contributes new knowledge to this all-important field since:
,
Plim g
=
m
lim Pgm
(i) It clearly shows the application of the Lagrange multipliers theory to solve the real-power economic dispatch problem of practical AC/DC grids, which is the hallmark of the introduced real-power dispatch scheme aimed at VSC-based HVDC systems. (ii) Contrary to existing approaches, the introduced method attains a complete representation of the AC/DC grids. That is, it makes due provisions for multiple VSC stations whose control strategies are suitably considered and reflected on the manner they are modelled. These are: slack converters VSCSlack, power-scheduled converters VSCPsch, and passive converters VSCPass. (iii) The proposed formulation also considers the power losses of the AC and DC transmission systems by using the Incremental Transmission Loss factors, thus taking a relatively few iterations to find the optimum solution whilst attaining practical results, as verified in Section 4.
(31)
It should be said that a similar procedure to the one previously described may be carried out for cases when the power flows in certain transmission paths need to be enforced in any of the AC or DC grids. 3.2. Kuhn-Tucker’s optimality conditions: As illustrated in Fig. 3, to find the optimum operation of the generators comprising the AC/DC grids, Kuhn-Tucker’s optimality conditions need to be assessed when the limits of one or more generators have been enforced; the application of such criterion is shown in (32). The theoretical foundation of the Kuhn-Tucker’s optimality conditions can be found in [16,17]. dci (Pgi) d Pgi dci (Pgi ) d Pgi dci (Pgi) d Pgi
=
Pgimin < Pgi < Pgimax
4. Cases of study
Pgi = Pgimax Pgi = Pgimin
4.1. Validation test - proposed approach vs sequential quadratic programming
(32)
For ease of reproduction, the VSC-based transmission network shown in Fig. 4 is used to validate the real-power economic dispatch solution approach developed in previous sections. This is an AC/DC grid that consists of two-terminal VSC-HVDC link interconnecting two otherwise independent
3.3. Summary of advantages of the proposed approach: All in all, this generalised formulation enables the simulation of
Fig. 4. AC/DC grid with a two-terminal VSCHVDC link.
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AC systems. In turn, each of these grids comprise an equivalent dispatchable generator whose cost curve are c1(Pg1) = 78 + 8Pg1 + 0.005P2g1 [$/hr] and c2(Pg2) = 310 + 7.8Pg2 + 0.002P2g2 [$/hr]. Both AC systems are equally loaded with S2 = S5 = 1.5 + j0.5p.u (100-MVA base). The converter VSCPsch, which acts as a rectifier, is set to restrict its power flow to Psch = 100 MW. The rest of the parameters are provided in the Appendix. Since G1 and G2 are the slack generators for their corresponding AC systems, thus θ1 = θ4 = 0, whilst the DC voltage E1 is kept constant by VSCSlack, i.e., E1 = E1nom. From Fig. 4, it may be also established that nac = 6, ndc = 2, Nac/dc = 8, and hence, the matrices and vectors of (27) may be straightforwardly obtained for this AC/DC network, as in (33). As previously detailed, expressions (33) imply a lossless AC/DC grid model. To include the power losses of the AC systems and HVDC network, by means of the ITL factors (26), into the optimisation problem, the general procedure provided in Fig. 3 must be followed. To validate the correctness of this novel approach, its solution is compared against that obtained by using the sequential quadratic programming (SQP) method available in the optimisation toolbox of Matlab© where a full representation of the AC/DC grid model, defined by (1), is employed.
VSC units, which is a topic that lies outside the scope of this work but it is the subject of a forthcoming research paper. On the other hand, from the results presented in Table 2, it may be concluded that the energy costs are superior in AC 2 since the HVDC link is limiting the power flow to Psch = 1 pu via VSCPsch. It should be recalled that a similar situation also occurs when the power flow through a certain AC transmission line is restricted so as not to exceed its power rating. 4.2. Economic dispatch of an AC/DC power grid formed by seven VSC stations The power grid shown in Fig. 5 is used to carry out the real-power economic dispatch to demonstrate the applicability of the proposed method to larger networks. This AC/DC system contains seven otherwise independent AC grids each coupled to a VSC station, which are connected to a thirteen-node DC system. Notice that there are six generators taking part in the economic dispatch, contained in AC 1 to AC 3, and that there are four 50-MW passive grids, AC 4 to AC 7, which are all fed by passive converters, VSCPass. Additionally, the two powerscheduled converters, VSCPsch1 and VSCPsch2, keep their power transmission constant to 120 MW and 100 MW, respectively, as shown in Fig. 5. The converter coupled to AC 1 is selected to play the role of the slack converter, VSCSlack. The generators’ cost curves of G1 to G6 are shown in Table 3; the rest of the parameters of this AC/DC grid are shown in the Appendix. The VSC types are also specified in Fig. 5, according to Section 2.1; for simplicity of representation, their AC filters are not represented in this figure, but they are considered as depicted in Fig. 1. In this study case, special emphasis is placed on the interactions between the AC and DC grids in terms of nodal marginal prices, this being the reason why the employed AC networks are rather simplified. However, this fact implies no loss of generality as for the solution approach since large-scale AC networks may be straightforwardly implemented by suitably expanding matrices B in (7) and A to C in (27). Table 4 shows the nodal marginal prices, as calculated by the proposed approach defined by (27). These results permit to obtain very important conclusions regarding the energy prices throughout an AC/ DC grid, such as the one being analysed in this study case. It is observed that nodal prices in networks AC 1, AC 2 and AC 3 differ among them, with AC 1 incurring higher nodal energy prices, λAC1 = 10.2598 [$/MWhr]. Not only does the DC grid have the same nodal prices as AC 1, but also, these prices are the same for the passive grids AC 4 to AC 7. In summary, this fact permits to conclude that the AC networks coupled to converters responsible for controlling their corresponding DC bus voltages, VSCSlack, will define the nodal prices in the whole DC network. In addition, it is seen that the energy prices in the AC passive networks are inherited from the DC grid.
(33) The comparison is reported in Table 1 which also shows the number of iterations and the computed power losses whose difference is barely 0.0105 p.u. As expected, there are small differences between the two quite distinct solution methods and grid models. This may be explained by the fact that when the full AC/DC network model is solved by the SQP method, the nodal voltages of the two AC systems tend to be higher than 1p.u thus leading to a reduction of the active power losses in both AC systems. Admittedly, this phenomenon cannot be captured by the developed method because, from the outset, it is assumed that all nodal AC voltage magnitudes are kept at 1p.u, as inferred from Sections 2.3 and 2.4. Nevertheless, it is observed that the computed errors for the generated powers stand at 0.23% and 0.36% for G1 and G2, respectively. Table 2 shows the Lagrange multipliers furnished by the two methods, including the total generation costs which only differ by 0.30%. Certainly, the solution obtained by the proposed method could be used, for instance, as a suitable starting point for Newton-based optimal power flow (OPF) algorithms dealing with HVDC grids fed by Table 1 Comparison of power flow results [p.u] for the validation test. Values
Method SQP
Values Proposed
Method SQP
Proposed
Pg1 Pg4 V1 V2 V3 V4
0.5147 2.5738 1.045/0o 1.018/−4.06o 1.05/−0.14o 1.05/0o
0.5159 2.5831 1.0/0o 1.0/−4.34o 1.0/−0.16o 1.0/0o
V5 V6 E1 E2 ϕSlack ϕPsch
0.992/−21.32 1.05/−25.59o 2.0000 2.0104 3.712o −29.411o
1.0/−22.20o 1.0/−26.49o 2.0000 2.0285 4.144o −26.497o
P12 P23 P3Slack Pdc12
0.5147 −0.9880 −0.9906 −0.9948
0.5159 −0.9948 −0.9948 −1.0
P45 P56 P6Psch
2.5738 1.0092 1.0052
2.5831 1.0 1.0
Power losses
0.08849
0.09898
Iterations
24
3
146
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Table 2 Nodal marginal prices [$/MWhr] for the AC systems and DC grid. Method
AC 1
SQP Proposed
AC 2
λ1
λ2
λ3
λ4
λ5
λ6
8.5147 8.5159
8.5987 8.5159
8.5573 8.5159
8.8295 8.8332
9.3501 8.8332
9.4049 8.8332
DC grid SQP Proposed
λ1dc = 8.4849 λ1dc = 8.5159
AC 1
G1
AC 6
G2 12 11
3
13
VSCPass4
10
11
1
AC grids DC grid
AC 4 10
5
G5
7
9
Generation [MW] Power flows [MW]
AC 3
8 6
4
G6 Power flows [MW]
7 8
2
VSCPsch1
VSCPsch2 6
Psch=1.2p.u
4
G4
9
3
G3
5
AC 2
2953.125 [$/hr] 2962.277 [$/hr]
Table 5 Power flow solution for the real-power economic dispatch.
13
VSCPass1
VSCPass2
AC 5
VSCSlack
VSCPass3
λ2dc = 9.4878 λ2dc = 8.5159
AC 7
12
1 2
Total generation costs
Psch= 1.0p.u
AC 1
AC 2
AC 3
Pg1 = 225.98 Pg2 = 225.98 P12 = 3.83 P13 = 227.90 P23 = 224.07
Pg3 = 166.41 Pg4 = 166.41 P45 = 8.14 P46 = 170.49 P56 = 162.36
Pg5 = 25.56 Pg6 = 25.56 P78 = −7.66 P79 = 21.74 P89 = 29.41
P12 = 88.73 P15 = 92.12 P23 = 90.09 P612 = 60.46 P89 = 28.62 P813 = 46.27
DC network P24 = 61.43 P25 = 57.20 P34 = −9.90 P910 = 28.62 P1112 = −6.31 P1213 = 3.93
P45 = −23.37 P610 = 21.59 P611 = 43.89 P47 = 24.96 P48 = 49.93 P56 = 125.95
Fig. 5. AC/DC power network with seven AC grids and a thirteen-node DC grid. Table 3 Generator cost curves, ci(Pgi) = ai + biPgi + diP2gi [$/hr], and AC loads. Generator
ai
bi
di
Pmin, Pmax [MW]
AC loads [MW]
G1 - G2 G3 - G4 G5 - G6
78 310 561
8.0 7.8 7.92
0.005 0.002 0.00154
20, 300 20, 300 20, 300
Pd3 = 250 Pd6 = 200
Pd9 = 150 Pd10 → Pd13 = 50
Table 4 Nodal marginal prices for the AC systems and DC grid. Network
Nodal prices [$/MWhr]
Network
Nodal prices [$/MWhr]
AC AC AC AC
λ1 → λ 3: λ4 → λ 6: λ7 → λ 9: λ10 :
AC AC AC DC
λ11: λ12: λ13: λ1 → λ 13:
1 2 3 4
10.2598 8.4657 7.9987 10.2598
5 6 7 grid
Fig. 6. Convergence characteristics of the developed real-power economic dispatch formulation.
10.2598 10.2598 10.2598 10.2598
Table 6 Nodal prices when Psch of converters VSCPsch are optimised.
Total generation costs: 9138.214 [$/hr]
Networks
Nodal prices [$/MWhr]
Network
Nodal prices [$/MWhr]
AC 1 → AC 7
λ:
DC grid
λ:
8.5217
8.5217
Total generation costs: 8917.353 [$/hr]
It is also noted that AC 3 is the network incurring minimum nodal prices, λAC3 = 7.9987 [$/MWhr]. This may be well explained by the fact that this network imports 100 MW from the DC grid via VSCPsch2, thus leading to a generation reduction in AC 3 and, consequently, fewer generation costs. It may be argued that a portion of this imported power is indirectly extracted from the grid coupled to VSCSlack, i.e., AC 1. This may be deduced since VSCPsch1 injects 120 MW to the DC grid but the passive grids AC 4 to AC 7 are consuming 200 MW together. Therefore, the remaining 80 MW plus 100 MW demanded by VSCPsch1 must be provided by AC 1. This can be corroborated from Table 5 which reports the power flow solution for the real-power economic dispatch of the HVDC power grid. In summary, the network AC 1, via VSCSlack, is balancing the powers in the whole AC/DC power grid, which also
explains why the nodal prices in both AC 1 and DC grids are the same in this case study. The results reported in Tables 4 and 5 were obtained considering a very stringent tolerance of 1e−4 to check the numerical performance of the developed real-power economic dispatch formulation, taking 12 iterations to converge in this case. Fig. 6 illustrates the convergence characteristics (evolution of the generation power mismatches), through the iterative loops detailed in Fig. 3, for each of the six generators comprised in the AC/DC system. It is corroborated in this manner the efficiency and prowess of the solution method to find optimum solutions even for quite stringent tolerances. 147
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On the other hand, since the power settings of the converters VSCPsch1 and VSCPsch2 were randomly selected, Psch1 = 120 MW and Psch2 = −100 MW, this led the overall AC/DC grid to incur higher generation costs. One way to find the optimal power settings, Psch, for VSCPsch1 and VSCPsch2 is by omitting its corresponding power restrictions, fn = PVSCn-Pschn in (2). Table 6 shows the nodal marginal prices when the power flows through the converters VSCPsch are optimised. As expected, the nodal marginal prices in the AC and DC grids are the same because all dispatchable generators operate at equal incremental production costs. In this case, the total generation costs are reduced to 8917.35 [$/hr] with the power settings of VSCPsch1 and VSCPsch2 standing at Psch1 = 145.33 MW and Psch2 = 211.69 MW, respectively.
4.1: z12 = z45 = 0.01 + j0.15 p.u.; rdc12 = 0.02135 p.u., Study case 4.2: z12 = z13 = z23 = 0.01 + j0.10 p.u., bsh12 = bsh13 = bsh23 = 0.025; z45 = z46 = z56 = 0.02 + j0.15 p.u., bsh45 = bsh46 = bsh56 = 0.05; z78 = z79 = z89 = 0.03 + j0.25 p.u., bsh78 = bsh79 = bsh89 = 0.075; rdc12 = 0.01938 p.u., rdc15 = 0.05403 p.u., rdc23 = 0.04699 p.u., rdc24 = 0.05811 p.u., rdc25 = 0.05695 p.u., rdc34 = 0.06701 p.u., rdc45 = 0.01335 p.u., rdc47 = rdc48 = rdc56 = rdc78 = 0.02 p.u. rdc610 = 0.09498 p.u., rdc611 = 0.12291 p.u., rdc612 = 0.06615 p.u., rdc89 = 0.03181 p.u., rdc813 = 0.12711 p.u., rdc910 = 0.08205 p.u., rdc1112 = 0.22092 p.u., rdc1213 = 0.17093 p.u.
5. Conclusions
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijepes.2018.11.018.
Appendix A. Supplementary material
The real-power economic dispatch for practical AC/DC power grids containing multiple VSC stations has been addressed in this paper. The Lagrange multipliers method is employed to solve the constrained optimisation problem to find the optimum economic dispatch of the various interconnected AC systems and the HVDC power grid, in a simultaneous fashion. The different control strategies of the converters comprising any VSC-based transmission system are considered in the modelling approach for practical solutions. In this sense, a lossless representation of AC/DC networks formed by multiple VSC units and a DC network have been initially derived. The power losses of the AC systems, DC grid and power converters are subsequently considered by using incremental transmission loss factors, an approach which contrasts with papers found in the open literature. Admittedly, a straightforward extension of this research work should be the development of an OPF for AC/DC power grids incorporating the three VSC converter types considered in the present paper. This modelling approach yields great flexibility to model any HVDC grid with multiple VSC units coupled to AC systems possessing various dispatchable generators. It has been validated using a two-terminal HVDC network interconnecting two otherwise independent AC grids its results concurred quite well with those obtained by the sequential quadratic programming method available in the optimisation toolbox of Matlab©. An AC/DC power grid formed by seven AC systems, seven VSC units and a thirteen-node DC grid has been also used to carry out the real-power economic dispatch. Interestingly, it is established that the nodal marginal prices in the DC grid are imposed by the AC system coupled to the VSC unit responsible for exerting DC voltage regulation, i.e., slack type converter VSCSlack. It is also concluded that the nodal marginal prices in VSC-connected passive grids are inherited from the DC grid. Admittedly, this novel approach will be very useful for power system analysts dealing with the economic operation of practical HVDC power grids incorporating multiple VSC stations.
References [1] ABB review. Special Report – 60 years of HVDC. ABB Group, Editorial Council, ISSN:1013-3119, July 2014. [2] Hertem DV, Ghandhari M. Multi-terminal VSC-HVDC for the European supergrid: obstacles. Renew Sustain Energy Rev 2010;14(Dec.):3156–63. [3] Zhao Q, Garcia-Gonzalez J, Gomis-Bellmunt O, Prieto-Araujo E, Echavarren FM. Impact of converter losses on the optimal power flow solution of hybrid networks based on VSC-MTDC. Electr Power Syst Res 2017;151(Oct.):395–403. [4] Yi L, Zhang-Sui L, Liu-Qing Q, Ming-Ye Z. An improved approach on static security constrained OPF for AC/DC meshed grids with MMC-HVDC. In: IEEE PES Asia Pacific power and energy conf.; Oct. 2016. p. 1–7. [5] Saplamidis V, Wiget R, Andersson G. Security constrained optimal power flow for mixed AC and multi-terminal HVDC grids. In: IEEE Eindhoven PowerTech; July 2015. p. 1–6. [6] Chatzivasileiadis S, Andersson G. Security constrained OPF incorporating corrective control of HVDC. In: Power Syst. Comp. Conf. (PSCC), Wroclaw, Poland; Aug. 2014. p. 1–8. [7] Yang Z, Zhong H, Bose A, Xia Q, Kang C. Optimal power flow in AC-DC grids with discrete control devices. IEEE Trans Power Syst 2017(June):1–10. [8] Ayan K, Kilic U. Optimal power flow of two-terminal systems using backtracking search algorithm. Electr Power Syst Res 2016;78(June):326–35. [9] Nanou SI, Tzortzopoulos OD, Papathanassiou A. Evaluation of an enhanced power dispatch control scheme for multi-terminal HVDC grids using Monte-Carlo simulation. Electr Power Syst Res 2016;140(Nov.):925–32. [10] Aragues-Peñalba M, Beerten J, Rimez J, Hertem DV, Gomis-Bellmunt O. Optimal power flow tool for hybrid AC/DC systems. In: IET Int. Conf. AC/DC Power Trans.; Feb. 2015. p. 1–7. [11] Aragues-Penalba M, Alvarez AE, Galceran-Arellano S, Gomis-Bellmunt O. Optimal power flow tool for mixed high-voltage alternating current and high-voltage direct current systems for grid integration of large wind power plants. IET Renew Power Gen 2015;9:876–81. [12] Gonzalez-Longatt FM. Optimal power flow in multi-terminal HVDC networks for dcsystem operator: constant current operation. In: Intl. Universities Power Eng. Conf. (UPEC); Sept. 2015. p. 1–6. [13] Baradar M, Hesamzadeh MR, Ghandhari M. Modelling of Multi-terminal HVDC Systems in optimal power flow Formulation. In: IEEE Electrical Power and Energy Conference (EPEC); Oct. 2012. [14] Castro LM, Acha E. A unified modeling approach of multi-terminal VSC-HVDC links for dynamic simulations of large-scale power systems. IEEE Trans Power Syst 2016(Feb.). [15] Acha E, Castro LM. A generalized frame of reference for the incorporation of multiterminal VSC-HVDC systems in power flow solutions. Electr Power Syst Res 2016;136(July):415–24. [16] Wood AJ, Wollenberg BF. Power generation, operation, and control. 2nd ed. John Wiley & Sons; 1996. [17] Elgerd OI. Electric energy systems: an introduction. 2nd ed. McGraw-Hill; 1982.
Appendix Parameters of each VSC unit, AC filter and OLTC (on a 100 MVA base): Snom = 2 p.u., Enom = 2.0 p.u, G0 = 4e−3 p.u, rph = 7.5e−4 p.u, xph = 0.075 p.u., yft = 0.4 p.u. zoltc = 0.0025 + j0.075 p.u. Study case
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Real-power economic dispatch of AC/DC power transmission systems comprising multiple VSC-HVDC equipment
T
Luis M. Castroa, , J.H. Tovar-Hernándezb, N. González-Cabrerac, J.R. Rodríguez-Rodrígueza ⁎
a
National Autonomous University of Mexico (UNAM), Department of Electrical Energy, Mexico Instituto Tecnológico de Morelia (ITM), Research and Postgraduate Program in Electrical Engineering, Mexico c Instituto Tecnológico Superior de Irapuato (ITESI), Electromechanics Department, Mexico b
ARTICLE INFO
ABSTRACT
Keywords: AC/DC power systems Lagrange multipliers Incremental transmission loss Real-power economic dispatch VSC-HVDC equipment VSC stations
This paper presents a practical approach for the real-power economic dispatch of AC/DC power transmission systems formed by VSC-HVDC equipment. It uses the Lagrange multipliers method combined with a comprehensive representation of the AC/DC power system. The power losses of the AC systems, DC grid and power converters are considered in this novel formulation by using incremental transmission loss factors. For realistic solutions, the different control strategies of the power converters forming any VSC-based power grid are considered. Interestingly, this approach permits to conclude that the AC grids coupled to converters charged with DC voltage control define the nodal marginal prices in the DC network, and in turn, the energy prices in the VSCconnected passive grids are inherited from the DC network. Overall, this modelling approach yields great flexibility to model any arbitrary VSC-based HVDC power grid interconnecting various AC systems. The developed method has been validated using a two-terminal VSC-HVDC network interconnecting two otherwise independent AC grids. Its results are compared against those obtained by the sequential quadratic programming method available in the optimisation toolbox of Matlab©, with both solutions exhibiting differences inferior to 0.30% in the total generation costs, therefore concurring very well between each other. The practicality of this approach is also demonstrated by carrying out the real-power economic dispatch of an AC/DC transmission system comprising seven VSC units which give rise to a thirteen-node DC grid.
1. Introduction High Voltage Direct Current (HVDC) systems using VSC-based power stations are rapidly becoming the key platform of modern transmission grids with which higher power throughputs are being reached [1]. VSC-HVDC links enable the interconnection of neighbouring AC systems thus generating a variety of positive technical aspects [2]. It may be argued that one of the main motivations in using HVDC systems is the possibility to facilitate increased energy transactions aiming at reducing energy prices. This is especially true when there is high demand and null availability of low-cost power generation. It should be said that, however, the use of VSC-based HVDC grids per se may not necessarily lead to a more economic operation of the various HVDC-connected AC systems because there may be DC transmission paths being restricted by VSC stations [3]. This is one of the reasons why creditable efforts have been recently dedicated to study the optimal power flow (OPF) problem in HVDC grids, placing emphasis on the security-constrained OPF [4–6].
⁎
This is a very complex problem that may be solved using nonlinear optimisation techniques such as quadratic programming, interior-point methods, or heuristic methods. Generally, the straightforward option is to resort to commercial optimisation solvers such as Knitro, GAMS, AMPL, Matlab optimisation toolbox, etc. Unexpectedly, only a few works addressing this all-important economic aspect in hybrid AC/DC systems have been published [7–13]. It should be also said that the previously-mentioned solution approaches have failed to represent the various VSC types that may give rise to a generic AC/DC power grid having any number of VSC units. That is, they all lack the inclusion of VSC stations controlling their DC bus voltages, VSC units charged with fixed power transmission and VSC stations feeding into passive grids [14,15], in a single formulation as is the case of this paper. For the sake of generality and applicability, this is taken care of in this work, thus leading to very important conclusions regarding the economic operation of VSC-based HVDC power grids. Interestingly, the introduced method allows to conclude that the AC grids coupled to converters charged with DC voltage control define the nodal marginal prices in the
Corresponding author. E-mail address: [email protected] (L.M. Castro).
https://doi.org/10.1016/j.ijepes.2018.11.018 Received 16 March 2018; Received in revised form 6 September 2018; Accepted 16 November 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.
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Nomenclature
FPsch FPass FDC JAC/DC ΔΦSlack ΔΦPsch ΔΦPass ΔEDC Gdc θSlack θPsch θPass BSlack
θ nodal voltage phase angle V nodal voltage magnitude Edc DC voltage ma modulation ratio ϕ phase-shifting angle Rdc, Ldc, Cdc DC resistance, inductance and capacitance nac, ndc number of AC and DC nodes Ng number of generators Pgi generated power generator cost function ci(Pgi) λj, λk, λn Lagrange multipliers rph + jxph converter’s coupling impedance Ploss power losses of the system Psch scheduled power Ω nodal variable vector P power injection vector Pd demand power vector β vector of power loss sensitivity factors μ Lagrange multipliers associated with VSCPsch constraints χ Lagrange multipliers associated with power generation enforcements FAC mismatch vector of the AC grids FSlack mismatch vector of the VSCSlack converter model
BPsch BPass AC DC VSC VSCSlack VSCPsch VSCPass ITL HVDC
mismatch vector of the VSCPsch converter model mismatch vector of the VSCPass converter model mismatch vector of the DC grid model Jacobian matrix of the AC/DC grid model state variable vector of the VSCSlack converter model state variable vector of the VSCPsch converter state variable vector of the VSCPass converter state variables vector of the DC grid nodal conductance matrix of the DC grid phase angles of the AC grid coupled to VSCSlack phase angles of the AC grid coupled to VSCPsch phase angles of the AC grid coupled to VSCPass nodal susceptance matrix of the AC grid coupled to VSCSlack nodal susceptance matrix of the AC grid coupled to VSCPsch nodal susceptance matrix of the AC grid coupled to VSCPass Alternating Current Direct Current Voltage Source Converter slack VSC power-scheduled VSC passive VSC Incremental Transmission Loss High Voltage Direct Current
DC network, and in turn, the value of the nodal marginal prices in the VSC-connected passive systems are directly inherited from the DC network. The VSC units exerting power regulation, on the other hand, are found to raise the price of the energy in the whole hybrid AC/DC system, something that needs special attention if transmission system operators are to pursue an economic operation in these modern power grids. As opposed to previous works related to the optimal power flows in VSC-based transmission networks, this paper focuses on the real-power economic dispatch of HVDC grids fed by multiple VSC stations, one where each power converter is modelled with its corresponding control task. The Lagrange multipliers method, combined with a lossless representation of the AC/DC network, is employed. In turn, Incremental Transmission Loss (ITL) factors, derived for both the AC grids and DC grid, are used to incorporate the power losses of all, the AC systems, DC grid and power converters comprising the AC/DC grid. This comprehensive modelling approach has been validated using a two-terminal VSC-HVDC grid. Its solution is directly compared against the sequential quadratic programming method available in the optimisation toolbox of Matlab©, with both solutions agreeing quite well between each other.
stations exerting power regulation. In this context, the formulation for the real-power economic dispatch in AC/DC power grids comprising various VSC stations is derived next.
2. Economic operation of HVDC power transmission systems formed by VSC stations
A VSC station consists of a power electronic converter, AC filter and an on-load tap-changing transformer (OLTC) to connect to an AC subnetwork, as depicted in Fig. 1. The steady-state, fundamental frequency operation of AC/DC systems containing several VSC stations may be computed using the Newton-based power flow formulation summarised by (1). This equation implies a multi-terminal HVDC system with m VSCSlack units, n
2.1. Generalised AC/DC power grid modelling incorporating multiple VSC units System-wide studies aimed at HVDC networks formed by multiple VSC terminals may be carried out using a unified approach. The interactions between the interconnected AC sub-networks and the DC grid are well assessed by assigning the VSC units different control strategies to satisfy specific operating requirements. In this sense, the VSC stations may be classified as follows [14,15]: i. Slack converters VSCSlack - these stations provide voltage control at its DC terminal. ii. Power-scheduled converters VSCPsch - these stations control the power transmission to a constant value. iii. Passive converters VSCPass - these converters interface the DC grid with AC passive networks having no generation of their own.
The economic operation problem of electrical systems is associated with the minimisation of all generation cost functions. This objective is subjected to nodal power balances and to constraints related to (i) power transmission capacities, (ii) nodal voltage operating limits, and (iii) power generation limits of dispatchable generators. Fundamentally, the problem consists in finding the optimum generation dispatch that reduces the system power losses. This fact becomes more important in cases involving the transmission of electrical energy over very large distances, as may be the case of modern VSC-based power grids. It is also well-known that congested transmission paths significantly affect the generation dispatch, causing the local nodal energy prices to rise [16,17]. In general terms, any transmission constraint will lead to a pricier system operation. Admittedly, the same conclusions may be anticipated for AC/DC networks due to the presence of VSC
Fig. 1. VSC station with ancillary elements. 141
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VSCPsch units, r VSCPass units and an arbitrary DC network. Notice that α AC sub-networks, α = m + n + r, are interlinked through the DC grid. The fundamental state variables of the DC grid and VSC stations, DC voltage Edc, modulation ratio ma, phase-shifting angle ϕ and OLTC’s tap Tv, are suitably accommodated in vectors ΔΦSlackm, ΔΦPschn, ΔΦPassr and ΔEDC, which are adjusted by iteration to satisfy the mismatch equations of the corresponding converters FSlackm, FPschn, FPassr and DC grid FDC [14,15]. In this sense, the vector FACα contains the power mismatch equations of the interconnected AC sub-networks whose associated state variables, nodal voltage magnitudes and phase angles, are placed in ΔΨACα. Clearly, the Jacobian matrix JAC/DC simultaneously accommodates the first-order partial derivatives of all mismatch equations emerging from the AC/DC network with respect to all state variables involved in the power flow problem.
FAC FSlackm FPschn FPassr FDC
i
AC
constant parameters whereas the phase-shifting angles ϕ of the converters VSCPsch enable the enforcement of their scheduled power flow Psch [14].
(a )
Pschn Passr
(1)
2.2. Problem formulation for HVDC power grids with multiple VSCconnected AC systems
f j = Pgjac
Pdjac
Pgimin
Pgi
|Pjm | s. t.
dc fk = Pdk fn = PVSCn
C=
Ng c (P ) i = 1 i gi
AC sub - networks
max P jm dc Pcalk =0 Pschn = 0
n VSCPsch & DC
(a )
max |Pkm | Pkm
(2)
k · fk
= =
j
=0
j
n · fn
i=j
j = 1, ...,nac;
j slack generators
=0
k = 1, ...,ndc;
fn
=0
n = 1, ...,n VSCPsch
k E k n
i = 1, ...,Ng ;
fk
n
Pkm = Gkm (Ek2
(c ) Pij =
k
VSCSlack
Ek Em)
Vi2 Gii
kma Ek Vi [Gph cos(
+ Vi Vj [Gij cos(
i
j)
+ Bij sin(
i)
+ Bph sin(
i
j )]
i )]
(5)
where i and j are generic nodes pertaining to any of the AC sub-netmax max works or DC grid; P jm and Pkm stand for the maximum power that can be transmitted through AC and DC transmission lines, respectively; PVSCn is the power transmitted through the n-th VSCPsch station with Pschn being its corresponding scheduled power. To find the constrained optimum of (2), the augmented Lagrange cost function (3) may be used. This equation suggests that the optimisation of C, subject to the equality constraints fj = 0, fk = 0 and fn = 0, is akin to searching for the unconstrained optimum of C*, where λ j, λk and λ n are the so-called Lagrange multipliers. The way inequality constraints are treated in the proposed formulation is explicitly detailed in Section 3. j ·f j
n
=
(b) Pki = k2ma2 Ek2 Gph
grid
C =C
(d )
C*
j
=0
Fig. 2 shows a VSC unit interfacing a DC line with an AC line. This is indeed the basic building block upon which the proposed method is derived. This section is dedicated to formulating the overall AC/DC grid using lossless power flow models of all, the AC and DC transmission elements along with the VSC units considering their corresponding control strategies, as addressed in Section 2.1. Since there may be n converters VSCPsch responsible for restricting their power flow, it becomes necessary to derive an AC/DC grid model where the transmission path models are available, as opposed to the case when all the generating units are assumed to be connected to a single node to supply the total system load. From the outset, it may be then predicted that these conditions will lead to different nodal marginal prices in various nodes of the VSC-based AC/DC grid, a fact that will be confirmed and duly analysed in the section of study cases. The steady-state power flow equations for DC transmission lines, VSC stations and AC transmission lines may be given by (5a), (5b) and (5c), respectively [14].
ac Pcalj =0
Pgimax
(c )
C* Ek
fj
2.3. Derivation of the lossless HVDC grid model fed by multiple VSC units
The optimum dispatch concerns with the determination of the individual real power settings of all Ng generation units of the α AC subnetworks interconnected through the various VSC units. To this end, the sum of the individual production cost of each dispatchable generator i, ci(Pgi) = ai + biPgi + di P2gi +∙∙∙, is the total cost function ($/h) to be minimised. This is subjected to constraint equations that ensure the nodal real-power balances in the AC/DC system, while simultaneously observing physical limitations in both power generation and power transmission, that is,
min
C*
fj j P gi
C Pgi
(4)
i
EDC
(b )
=
The real-power economic dispatch of AC systems, with transmission line constraints considered, bases on a network’s linear power flow model which stems from lossless transmission elements; following this philosophy, this is carried out in Section 2.3 for generic VSC-HVDC power grids. Subsequently, the system power losses may be considered by using Incremental Transmission Loss (ITL) factors. These may be regarded as sensitivities associated with changes in the system power losses with respect to nodal power injections; these sensitivities are derived, for AC/DC grids formed by VSC stations, in Section 2.4.
Slackm
(JAC / DC)i
=
C* Pgi
where Gij + jBij = (rij + jxij) ; Gph + jBph = (rph + jxph)−1 is the admittance of the converter’s coupling phase reactor relating to the filtering device (not shown in the figure), with rph accounting for its conduction losses; k = √(3/8) is a constant associated with the AC-to-DC voltage conversion in two-level, three-phase VSC stations. The following simplifying assumptions are adopted to obtain the aforesaid reduced AC/DC model: Gkm = rkm−1,
−1
(i) The power losses of the AC system and VSC are neglected by setting rij = rph = 0; (ii) Since the angular differences (ϕ-θi) and (θi-θj) are very small, then
(3)
The real-power economic dispatch is carried out with all AC voltage magnitudes fixed at 1 p.u. [16,17], therefore if the α AC sub-networks contain nac AC nodes and the DC grid comprises ndc DC nodes, the optimisation problem resides in solving the set of equations (4) which stems from the partial derivation of (3) with respect to all variables involved. As it turns out, the reference angles θ of the corresponding AC grids and the DC bus voltages E regulated by stations VSCSlack are all
Fig. 2. Generic VSC station interfacing AC with DC elements. 142
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sin(ϕ-θi) ≈ (ϕ-θi) and sin(θi-θj) ≈ (θi-θj); (iii) A lossless DC transmission line model may be obtained by resorting to the alternative DC current expression Pkm ≈ Ikm.
C* C = Pgi Pgi
= Bij (
Em),
(b) Pki = Bph (
i
i ),
=
(c ) Pij
Using the linear power flow equations (6), the set of nodal power injections of the overall VSC-based transmission network may be straightforwardly obtained as a function of the AC phase angles and DC voltages. In a similar manner to Section 2.1, a multi-terminal arrangement, consisting of m VSCSlack converters, n VSCPsch converters and r VSCPass converters and a DC network, would bring about the linear power flow AC/DC model shown in (7), with i = 1…m, j = 1…n and k = 1…r. The derivation of this lossless AC/DC network model is exemplified in the section of case studies.
C / Pgi 1
Nac / dc
Ng
C* =
Nac / dc
ci (Pgi ) i=1
( i=1
i=1
Ploss )
j +1
Ek Ek + 1
nac
(14)
Endc ]T
dP net = dP jnet + dPknet +
Nac / dc
dP net ;
j
,
i
=1
(15) The total derivatives of the net power injections dPγ are explicitly given by (16). For ease of representation, the use of the superscript ‘net’ has been dropped in these equations.
dP1 =
P1 j
d j+
dPnac =
Pnac
dP1dc =
P1dc
dPndc =
Pndc
j j
j
+
P1 nac
d
d j+
+
Pnac
d j+
+
P1dc
d j+
+
Pndc
nac nac
nac
+
nac
P1 dEk + Ek
P1 dEndc Endc
+
d
nac
+
Pnac dEk + Ek
+
Pnac dEndc Endc
d
nac
+
P1dc dEk + Ek
+
P1dc dEndc Endc
d
nac
+
Pndc dEk + Ek
+
Pndc dEndc Endc
(16)
Using (14) and (16), the vectors (17) and (18) may be defined, all of dimension (ndc + nac-2 m-n)x1.
dP = [ dPj dPj + 1 = [d
d
d
j
j +1
dPnac dPk dPk + 1
dPndc ]T
(17)
d
dEndc ]T
(18)
nac
dEk dEk + 1
The power-injection vectors (19) and (20) may be also defined for those nodes associated with slack AC buses, j′, and DC buses, k′, respectively. The Jacobian matrix (21) encompasses the derivatives of the power injections with respect to all unknown variables involved in (16).
Pj
Nac / dc
Pdi
(13)
=1
net
(9)
Pgi
P net
=1
Nac / dc
Ploss = 0
j
Nac / dc
One way to consider the power losses in the real-power economic dispatch of AC networks is by means of Incremental Transmission Loss (ITL) factors [16,17], an enterprise not yet addressed in the context of VSC-based DC transmission networks. If the power losses of the overall system are Ploss, then the nodal power balances of the whole AC/DC network may be defined by (9), with Nac/dc = nac + ndc. Consequently, the constrained optimum problem may be defined as in (10).
Pdi
=1
dPloss =
(8)
i=1
Nac / dc
Pd =
Eqs. (13) and (14) suggest that power loss is a function of both the AC bus voltage phase angles and the DC nodal voltages, i.e., Ploss = f (Ω). From (13), equation (15) is obtained by differentiation, where j′ and k′ stand for the slack AC buses and DC buses where voltage regulation is exerted by VSCSlack units, respectively.
2.4. Derivation of the incremental transmission loss factors for AC/DC grids formed by VSC units
Pgi
(12)
Since there is a slack generator for each AC network coupled to the m VSCSlack units and n VSCPsch units whilst there are also m regulated DC bus voltages by VSCSlack units, it follows that the system’s set of nodal variables may be defined by the (ndc + nac-2 m-n)x1-dimensional vector Ω, shown in (14).
Eq. (7) may be written in compact form as in (8), where BSlacki, BPschj, and BPassk are diagonal matrices which accommodate the lossless models of the AC grids connected to converters VSCSlack, VSCPsch and VSCPass, respectively. Matrices BS and BPa provide the link between the AC terminals of the VSCSlack and VSCPass with their corresponding DC nodes. The phase reactor susceptances of converters VSCSlack are placed in BSdc whereas GSdc contains the DC network conductances connecting with the DC buses of VSCSlack stations. Lastly, matrix Gdc accommodates the lossless DC grid model, excluding the row and column relating to the DC bus of the slack converters.
i=1
Nac / dc
Pg =1
(7)
Nac / dc
i = 1, ...,Ng
Ploss / Pgi
Ploss =
=[
PAC / DC = C· ,
(11)
where the term ∂Ploss/∂Pgi is usually referred to as the ITL associated with each generating unit i. From (12), it may be stated that ITLi = ∂Ploss/∂Pgi for i = 1,…,Ng, where ITLi = 0 for the bus relating to the slack generator. As described in Section 2.1, various AC networks are interconnected through the HVDC power grid, signifying that the overall AC/DC network model will be featured by various slack AC buses, a fact that needs special attention to successfully obtain the ITL in these new power grids. Reformulating (9), the total power loss Ploss of the AC/DC network may be expressed as
(6)
j)
i = 1, ...,Ng
Solving (11) for λ, the ensuing relationship is obtained:
Due to these suppositions, Eqs. (5) are reduced to those shown in (6). where the power flows, in the opposite direction, may be found by simply exchanging the corresponding subscripts. It should be said that these suppositions may imply for the proposed method, at most, more global iterations to converge. Also, the AC/DC grid’s optimum dispatch is expected to slightly differ with respect to more complex approaches such as OPF type solutions due to the fundamental differences in the optimisation tools. This is discussed in the section of case studies.
(a) Pkm = Gkm (Ek
Ploss =0 Pgi
+
(10)
Pk
The following expression must be satisfied for an optimum power dispatch: 143
=( =(
Pj j
Pk j
Pj
Pj
Pj
Pj
Pj
j +1
nac
Ek
Ek + 1
Endc
Pk
Pk
j+ 1
nac
Pk Ek
Pk Ek + 1
)T
Pk T Endc )
(19) (20)
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P
=(
P1
P1
j
nac
Pnac
Pnac
j
nac
P1dc
P1dc
j
j
Pndc
Pndc
j
nac
P1 Ek
P1 Endc
Pnac Ek
Pnac Endc
P1dc Ek
P 1dc Endc
Pndc Ek
Pndc Endc
zero-padded matrices of suitable orders. A is a diagonal matrix accommodating the terms 2di for each generating plant i resulting from the derivatives ∂ci/∂Pgi. The connectivity matrix B contains the term −1 in locations where there is a link between row i and column j, i.e., B (i,j) = −1, recalling that i = 1,…,Ng and j = 1,…,nac. Matrix C results from the lossless AC/DC grid model (7) and (8). Matrix D, on the other hand, ensures the fixed power transmission in power-scheduled converters, i.e., this matrix suitably accommodates the expressions bph(ϕθi) = −Psch, according to (6b), for the n VSCPsch units. Matrices A to D are given in detail, in the section of study cases, whereas the various vector terms of (27) are explicitly shown in (28).
)
(21)
Using (14)–(21), the set of expressions (16) may be rearranged as (22), in compact form, with the power injections (19) and (20) redefined as shown in (23).
dP = (
dPj = (
P
)d
Pj
d
)T d
P
=(
dPk = (
Pk
) 1d P
(22)
)T d
(23)
(27)
Substituting dΩ of (22) into (23) and (15) yields (24). Nac / dc
dPloss =
·d P +
dP net
Pg1
j
,
i
(24)
=1
Pj
Pg =
where = [( )T + ( k )T ]( ) 1 are sensitivity factors of power losses with respect to power injections. Eq. (24) may be used to compute the incremental changes in power loss following an incremental change in power injections at any bus of the various AC networks or DC network. That is, knowing from (17) that dP = [dPj, dPj+1,…, dPnac, dPk, dPk+1,…,dPndc]T and that according to (24), β = [βj, βj+1,…, βnac, βk, βk+1,…,βndc]T, it is clear that the element-by-element multiplication β∙dP will lead to a vector of the same order as ΣdPγ net = [dPj, dPj+1,…, dPnac, dPk, dPk+1,…,dPndc]T shown in (24). Therefore, equation (24) takes the form of (25), after factoring the corresponding terms. It should be remarked that there will be as many sensitivity factors β as the number of AC/DC nodes Nac/dc, excluding those related to the slack AC buses and DC buses with voltage regulation, nodes j′ and k′, respectively. This is so because the power loss function Ploss is independent of the state variables associated with j′ and k′, i.e., the reference AC voltage phase angles and voltages of voltage-controlled DC nodes are known a priori.
dPloss = (1 + +
1) dP1
P
+
+ (1 +
+ (1 +
nac ) dPnac
+ (1 +
dPloss = (1 + dPgi
i)
=
i
j
µ=
Nac / dc
µ1 µn
b1 ,
b=
, bNg
Pd
P1dc
Pdnac , Ek Gk 1dc
Pndc
Ek Gk ndc
Psch =
Psch1 Pschn (28)
To further clarify the procedure involved in the real-power economic dispatch of AC/DC power grids, Fig. 3 shows a flow diagram summarising its most important steps. It should be said that this proposed approach suggests that each AC network and DC grid comprising the AC/DC system must take care for their corresponding power losses. This means that the power losses ought to be included into the otherwise independent AC networks and DC network as demanded power by properly modifying Pd in (28). A suggested practice is to add the AC and DC power losses, for instance, to those nodes related to slack AC and DC nodes, j′ and k′, respectively. 3.1. Handling of inequality constraints
(25)
i = 1, ...,Ng ;
,
Pd1
1dc ) dP1dc
ndc ) dPndc
=
PgNg
P
It should be mentioned that in the case that a generator has exceeded one of its limits, Pgmax or Pgmin, then the augmented Lagrange cost function (3) must be properly amended to include its corresponding inequality constraint as an equality function. This is shown in (29) where fm = Pgi –Pgmax = 0 or fm = Pgi – Pgmin = 0 should be included into the optimisation problem, depending on the exceeded power limit. In explicit manner, this may be treated in the proposed formulation as shown in (30). As expected, the matrix and vectors of (27) are suitably expanded to conform them according to the number of equality constraints plus the m violated power generation limits.
If all generators’ output powers are kept constant, except the one connected to node i, therefore, Eq. (26) holds.
ITLi =
1
,
(26)
This last expression facilitates the inclusion of the power losses, in the real-power economic dispatch of HVDC power grids comprising various VSC-connected AC systems, by applying (12). Notice that to obtain the ITL, the Jacobian matrix ∂P/∂Ω (21), the vectors ∂Pj′/∂Ω (19) and ∂Pk′/∂Ω (20), must be available; all these terms are suitably calculated with the information corresponding to the last iteration when solving (1), for a given load/generation pattern.
C =C
j ·f j
k · fk
n · fn
m ·fm
(29)
3. Economic dispatch for AC/DC power networks comprising various VSC-connected AC grids Assuming quadratic generation cost curves, ci(Pgi) = ai + biPgi + diP2gi, then the resulting economic dispatch problem is a linear one, from the mathematical standpoint. This fact combines very well with the derivations carried out for AC/DC power grids, (2)–(4), and with the generalised HVDC grid model fed by various VSC-connected AC systems derived in Section 2.3. In this context, the proposed approach aimed at the economic dispatch problem may be summarised by (27), where the 0 entries are
(30) In Eq. (30), E is a matrix of order Ngxm containing −1 in suitable locations corresponding to the generators whose limits are being enforced; χ is an mx1 vector that contains the Lagrange multipliers 144
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Fig. 3. General procedure for carrying out the economic dispatch in AC/DC grids containing various VSC-connected AC systems.
associated with each of the m restrictions and the vector Pglim accommodates the power limits of the corresponding generators. All these vectors are explicitly shown in (31).
E=
0 0 0
1 0 0 0
0 0 0
Pglim 1
1
1
,
=
HVDC power grids with numerous VSC-connected AC grids enabling the simultaneous computation of the nodal marginal prices at the AC and DC systems. In this regard, this proposed approach contributes new knowledge to this all-important field since:
,
Plim g
=
m
lim Pgm
(i) It clearly shows the application of the Lagrange multipliers theory to solve the real-power economic dispatch problem of practical AC/DC grids, which is the hallmark of the introduced real-power dispatch scheme aimed at VSC-based HVDC systems. (ii) Contrary to existing approaches, the introduced method attains a complete representation of the AC/DC grids. That is, it makes due provisions for multiple VSC stations whose control strategies are suitably considered and reflected on the manner they are modelled. These are: slack converters VSCSlack, power-scheduled converters VSCPsch, and passive converters VSCPass. (iii) The proposed formulation also considers the power losses of the AC and DC transmission systems by using the Incremental Transmission Loss factors, thus taking a relatively few iterations to find the optimum solution whilst attaining practical results, as verified in Section 4.
(31)
It should be said that a similar procedure to the one previously described may be carried out for cases when the power flows in certain transmission paths need to be enforced in any of the AC or DC grids. 3.2. Kuhn-Tucker’s optimality conditions: As illustrated in Fig. 3, to find the optimum operation of the generators comprising the AC/DC grids, Kuhn-Tucker’s optimality conditions need to be assessed when the limits of one or more generators have been enforced; the application of such criterion is shown in (32). The theoretical foundation of the Kuhn-Tucker’s optimality conditions can be found in [16,17]. dci (Pgi) d Pgi dci (Pgi ) d Pgi dci (Pgi) d Pgi
=
Pgimin < Pgi < Pgimax
4. Cases of study
Pgi = Pgimax Pgi = Pgimin
4.1. Validation test - proposed approach vs sequential quadratic programming
(32)
For ease of reproduction, the VSC-based transmission network shown in Fig. 4 is used to validate the real-power economic dispatch solution approach developed in previous sections. This is an AC/DC grid that consists of two-terminal VSC-HVDC link interconnecting two otherwise independent
3.3. Summary of advantages of the proposed approach: All in all, this generalised formulation enables the simulation of
Fig. 4. AC/DC grid with a two-terminal VSCHVDC link.
145
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AC systems. In turn, each of these grids comprise an equivalent dispatchable generator whose cost curve are c1(Pg1) = 78 + 8Pg1 + 0.005P2g1 [$/hr] and c2(Pg2) = 310 + 7.8Pg2 + 0.002P2g2 [$/hr]. Both AC systems are equally loaded with S2 = S5 = 1.5 + j0.5p.u (100-MVA base). The converter VSCPsch, which acts as a rectifier, is set to restrict its power flow to Psch = 100 MW. The rest of the parameters are provided in the Appendix. Since G1 and G2 are the slack generators for their corresponding AC systems, thus θ1 = θ4 = 0, whilst the DC voltage E1 is kept constant by VSCSlack, i.e., E1 = E1nom. From Fig. 4, it may be also established that nac = 6, ndc = 2, Nac/dc = 8, and hence, the matrices and vectors of (27) may be straightforwardly obtained for this AC/DC network, as in (33). As previously detailed, expressions (33) imply a lossless AC/DC grid model. To include the power losses of the AC systems and HVDC network, by means of the ITL factors (26), into the optimisation problem, the general procedure provided in Fig. 3 must be followed. To validate the correctness of this novel approach, its solution is compared against that obtained by using the sequential quadratic programming (SQP) method available in the optimisation toolbox of Matlab© where a full representation of the AC/DC grid model, defined by (1), is employed.
VSC units, which is a topic that lies outside the scope of this work but it is the subject of a forthcoming research paper. On the other hand, from the results presented in Table 2, it may be concluded that the energy costs are superior in AC 2 since the HVDC link is limiting the power flow to Psch = 1 pu via VSCPsch. It should be recalled that a similar situation also occurs when the power flow through a certain AC transmission line is restricted so as not to exceed its power rating. 4.2. Economic dispatch of an AC/DC power grid formed by seven VSC stations The power grid shown in Fig. 5 is used to carry out the real-power economic dispatch to demonstrate the applicability of the proposed method to larger networks. This AC/DC system contains seven otherwise independent AC grids each coupled to a VSC station, which are connected to a thirteen-node DC system. Notice that there are six generators taking part in the economic dispatch, contained in AC 1 to AC 3, and that there are four 50-MW passive grids, AC 4 to AC 7, which are all fed by passive converters, VSCPass. Additionally, the two powerscheduled converters, VSCPsch1 and VSCPsch2, keep their power transmission constant to 120 MW and 100 MW, respectively, as shown in Fig. 5. The converter coupled to AC 1 is selected to play the role of the slack converter, VSCSlack. The generators’ cost curves of G1 to G6 are shown in Table 3; the rest of the parameters of this AC/DC grid are shown in the Appendix. The VSC types are also specified in Fig. 5, according to Section 2.1; for simplicity of representation, their AC filters are not represented in this figure, but they are considered as depicted in Fig. 1. In this study case, special emphasis is placed on the interactions between the AC and DC grids in terms of nodal marginal prices, this being the reason why the employed AC networks are rather simplified. However, this fact implies no loss of generality as for the solution approach since large-scale AC networks may be straightforwardly implemented by suitably expanding matrices B in (7) and A to C in (27). Table 4 shows the nodal marginal prices, as calculated by the proposed approach defined by (27). These results permit to obtain very important conclusions regarding the energy prices throughout an AC/ DC grid, such as the one being analysed in this study case. It is observed that nodal prices in networks AC 1, AC 2 and AC 3 differ among them, with AC 1 incurring higher nodal energy prices, λAC1 = 10.2598 [$/MWhr]. Not only does the DC grid have the same nodal prices as AC 1, but also, these prices are the same for the passive grids AC 4 to AC 7. In summary, this fact permits to conclude that the AC networks coupled to converters responsible for controlling their corresponding DC bus voltages, VSCSlack, will define the nodal prices in the whole DC network. In addition, it is seen that the energy prices in the AC passive networks are inherited from the DC grid.
(33) The comparison is reported in Table 1 which also shows the number of iterations and the computed power losses whose difference is barely 0.0105 p.u. As expected, there are small differences between the two quite distinct solution methods and grid models. This may be explained by the fact that when the full AC/DC network model is solved by the SQP method, the nodal voltages of the two AC systems tend to be higher than 1p.u thus leading to a reduction of the active power losses in both AC systems. Admittedly, this phenomenon cannot be captured by the developed method because, from the outset, it is assumed that all nodal AC voltage magnitudes are kept at 1p.u, as inferred from Sections 2.3 and 2.4. Nevertheless, it is observed that the computed errors for the generated powers stand at 0.23% and 0.36% for G1 and G2, respectively. Table 2 shows the Lagrange multipliers furnished by the two methods, including the total generation costs which only differ by 0.30%. Certainly, the solution obtained by the proposed method could be used, for instance, as a suitable starting point for Newton-based optimal power flow (OPF) algorithms dealing with HVDC grids fed by Table 1 Comparison of power flow results [p.u] for the validation test. Values
Method SQP
Values Proposed
Method SQP
Proposed
Pg1 Pg4 V1 V2 V3 V4
0.5147 2.5738 1.045/0o 1.018/−4.06o 1.05/−0.14o 1.05/0o
0.5159 2.5831 1.0/0o 1.0/−4.34o 1.0/−0.16o 1.0/0o
V5 V6 E1 E2 ϕSlack ϕPsch
0.992/−21.32 1.05/−25.59o 2.0000 2.0104 3.712o −29.411o
1.0/−22.20o 1.0/−26.49o 2.0000 2.0285 4.144o −26.497o
P12 P23 P3Slack Pdc12
0.5147 −0.9880 −0.9906 −0.9948
0.5159 −0.9948 −0.9948 −1.0
P45 P56 P6Psch
2.5738 1.0092 1.0052
2.5831 1.0 1.0
Power losses
0.08849
0.09898
Iterations
24
3
146
o
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L.M. Castro et al.
Table 2 Nodal marginal prices [$/MWhr] for the AC systems and DC grid. Method
AC 1
SQP Proposed
AC 2
λ1
λ2
λ3
λ4
λ5
λ6
8.5147 8.5159
8.5987 8.5159
8.5573 8.5159
8.8295 8.8332
9.3501 8.8332
9.4049 8.8332
DC grid SQP Proposed
λ1dc = 8.4849 λ1dc = 8.5159
AC 1
G1
AC 6
G2 12 11
3
13
VSCPass4
10
11
1
AC grids DC grid
AC 4 10
5
G5
7
9
Generation [MW] Power flows [MW]
AC 3
8 6
4
G6 Power flows [MW]
7 8
2
VSCPsch1
VSCPsch2 6
Psch=1.2p.u
4
G4
9
3
G3
5
AC 2
2953.125 [$/hr] 2962.277 [$/hr]
Table 5 Power flow solution for the real-power economic dispatch.
13
VSCPass1
VSCPass2
AC 5
VSCSlack
VSCPass3
λ2dc = 9.4878 λ2dc = 8.5159
AC 7
12
1 2
Total generation costs
Psch= 1.0p.u
AC 1
AC 2
AC 3
Pg1 = 225.98 Pg2 = 225.98 P12 = 3.83 P13 = 227.90 P23 = 224.07
Pg3 = 166.41 Pg4 = 166.41 P45 = 8.14 P46 = 170.49 P56 = 162.36
Pg5 = 25.56 Pg6 = 25.56 P78 = −7.66 P79 = 21.74 P89 = 29.41
P12 = 88.73 P15 = 92.12 P23 = 90.09 P612 = 60.46 P89 = 28.62 P813 = 46.27
DC network P24 = 61.43 P25 = 57.20 P34 = −9.90 P910 = 28.62 P1112 = −6.31 P1213 = 3.93
P45 = −23.37 P610 = 21.59 P611 = 43.89 P47 = 24.96 P48 = 49.93 P56 = 125.95
Fig. 5. AC/DC power network with seven AC grids and a thirteen-node DC grid. Table 3 Generator cost curves, ci(Pgi) = ai + biPgi + diP2gi [$/hr], and AC loads. Generator
ai
bi
di
Pmin, Pmax [MW]
AC loads [MW]
G1 - G2 G3 - G4 G5 - G6
78 310 561
8.0 7.8 7.92
0.005 0.002 0.00154
20, 300 20, 300 20, 300
Pd3 = 250 Pd6 = 200
Pd9 = 150 Pd10 → Pd13 = 50
Table 4 Nodal marginal prices for the AC systems and DC grid. Network
Nodal prices [$/MWhr]
Network
Nodal prices [$/MWhr]
AC AC AC AC
λ1 → λ 3: λ4 → λ 6: λ7 → λ 9: λ10 :
AC AC AC DC
λ11: λ12: λ13: λ1 → λ 13:
1 2 3 4
10.2598 8.4657 7.9987 10.2598
5 6 7 grid
Fig. 6. Convergence characteristics of the developed real-power economic dispatch formulation.
10.2598 10.2598 10.2598 10.2598
Table 6 Nodal prices when Psch of converters VSCPsch are optimised.
Total generation costs: 9138.214 [$/hr]
Networks
Nodal prices [$/MWhr]
Network
Nodal prices [$/MWhr]
AC 1 → AC 7
λ:
DC grid
λ:
8.5217
8.5217
Total generation costs: 8917.353 [$/hr]
It is also noted that AC 3 is the network incurring minimum nodal prices, λAC3 = 7.9987 [$/MWhr]. This may be well explained by the fact that this network imports 100 MW from the DC grid via VSCPsch2, thus leading to a generation reduction in AC 3 and, consequently, fewer generation costs. It may be argued that a portion of this imported power is indirectly extracted from the grid coupled to VSCSlack, i.e., AC 1. This may be deduced since VSCPsch1 injects 120 MW to the DC grid but the passive grids AC 4 to AC 7 are consuming 200 MW together. Therefore, the remaining 80 MW plus 100 MW demanded by VSCPsch1 must be provided by AC 1. This can be corroborated from Table 5 which reports the power flow solution for the real-power economic dispatch of the HVDC power grid. In summary, the network AC 1, via VSCSlack, is balancing the powers in the whole AC/DC power grid, which also
explains why the nodal prices in both AC 1 and DC grids are the same in this case study. The results reported in Tables 4 and 5 were obtained considering a very stringent tolerance of 1e−4 to check the numerical performance of the developed real-power economic dispatch formulation, taking 12 iterations to converge in this case. Fig. 6 illustrates the convergence characteristics (evolution of the generation power mismatches), through the iterative loops detailed in Fig. 3, for each of the six generators comprised in the AC/DC system. It is corroborated in this manner the efficiency and prowess of the solution method to find optimum solutions even for quite stringent tolerances. 147
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On the other hand, since the power settings of the converters VSCPsch1 and VSCPsch2 were randomly selected, Psch1 = 120 MW and Psch2 = −100 MW, this led the overall AC/DC grid to incur higher generation costs. One way to find the optimal power settings, Psch, for VSCPsch1 and VSCPsch2 is by omitting its corresponding power restrictions, fn = PVSCn-Pschn in (2). Table 6 shows the nodal marginal prices when the power flows through the converters VSCPsch are optimised. As expected, the nodal marginal prices in the AC and DC grids are the same because all dispatchable generators operate at equal incremental production costs. In this case, the total generation costs are reduced to 8917.35 [$/hr] with the power settings of VSCPsch1 and VSCPsch2 standing at Psch1 = 145.33 MW and Psch2 = 211.69 MW, respectively.
4.1: z12 = z45 = 0.01 + j0.15 p.u.; rdc12 = 0.02135 p.u., Study case 4.2: z12 = z13 = z23 = 0.01 + j0.10 p.u., bsh12 = bsh13 = bsh23 = 0.025; z45 = z46 = z56 = 0.02 + j0.15 p.u., bsh45 = bsh46 = bsh56 = 0.05; z78 = z79 = z89 = 0.03 + j0.25 p.u., bsh78 = bsh79 = bsh89 = 0.075; rdc12 = 0.01938 p.u., rdc15 = 0.05403 p.u., rdc23 = 0.04699 p.u., rdc24 = 0.05811 p.u., rdc25 = 0.05695 p.u., rdc34 = 0.06701 p.u., rdc45 = 0.01335 p.u., rdc47 = rdc48 = rdc56 = rdc78 = 0.02 p.u. rdc610 = 0.09498 p.u., rdc611 = 0.12291 p.u., rdc612 = 0.06615 p.u., rdc89 = 0.03181 p.u., rdc813 = 0.12711 p.u., rdc910 = 0.08205 p.u., rdc1112 = 0.22092 p.u., rdc1213 = 0.17093 p.u.
5. Conclusions
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijepes.2018.11.018.
Appendix A. Supplementary material
The real-power economic dispatch for practical AC/DC power grids containing multiple VSC stations has been addressed in this paper. The Lagrange multipliers method is employed to solve the constrained optimisation problem to find the optimum economic dispatch of the various interconnected AC systems and the HVDC power grid, in a simultaneous fashion. The different control strategies of the converters comprising any VSC-based transmission system are considered in the modelling approach for practical solutions. In this sense, a lossless representation of AC/DC networks formed by multiple VSC units and a DC network have been initially derived. The power losses of the AC systems, DC grid and power converters are subsequently considered by using incremental transmission loss factors, an approach which contrasts with papers found in the open literature. Admittedly, a straightforward extension of this research work should be the development of an OPF for AC/DC power grids incorporating the three VSC converter types considered in the present paper. This modelling approach yields great flexibility to model any HVDC grid with multiple VSC units coupled to AC systems possessing various dispatchable generators. It has been validated using a two-terminal HVDC network interconnecting two otherwise independent AC grids its results concurred quite well with those obtained by the sequential quadratic programming method available in the optimisation toolbox of Matlab©. An AC/DC power grid formed by seven AC systems, seven VSC units and a thirteen-node DC grid has been also used to carry out the real-power economic dispatch. Interestingly, it is established that the nodal marginal prices in the DC grid are imposed by the AC system coupled to the VSC unit responsible for exerting DC voltage regulation, i.e., slack type converter VSCSlack. It is also concluded that the nodal marginal prices in VSC-connected passive grids are inherited from the DC grid. Admittedly, this novel approach will be very useful for power system analysts dealing with the economic operation of practical HVDC power grids incorporating multiple VSC stations.
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Appendix Parameters of each VSC unit, AC filter and OLTC (on a 100 MVA base): Snom = 2 p.u., Enom = 2.0 p.u, G0 = 4e−3 p.u, rph = 7.5e−4 p.u, xph = 0.075 p.u., yft = 0.4 p.u. zoltc = 0.0025 + j0.075 p.u. Study case
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