WO2015042654A1 - Alternating current (ac) power flow analysis in an electrical power network - Google Patents

Abstract

This disclosure generally concerns electrical power networks, and more particularly, a computer-implemented method for alternating current (AC) power flow analysis in an electrical power network. The method comprises (a) determining a relaxation of AC power flows in the electrical power network and (b) optimising an objective function to determine a relaxed optimum. The relaxation comprises a first constraint and a second constraint for each of multiple cosine terms. The first constraint is a quadratic function and the second constraint defines a feasibility space with the first constraint such that values of the cosine term are between the first constraint and the second constraint. When optimising the objective function, the function variables are constraint to the feasibility space and as a result, it is guaranteed that there is no physically feasible solution that is better than the determined solution because the value of the cosine term is between the two constraints.

Description

"Alternating current (AC) power flow analysis in an electrical power network'^* Cross- Reference to Related Applications

The present application claims priori ty from Australian Provisional Patent Application No 2013 Q3769 filed on .30 September 2013, the content of which is incorporated herein by reference.

Technical Field

This disclosure generally concerns electrical power networks, and more particularly, a computer-implemented method for alternating current (AC) power flow analysis in ail electrical power network. This disclosure also concerns a computer system, a computer program and an electrical powe network employing the method.

Background Art

Optimisation technology is used in modern power systems to produce cost savings. However, the increasing role of demand response, the integration of renewable sources of energy, and the desire for more automation in fault detection and recovery pose new challenges for the planning and control of electrical power systems. Power grid now need to operate in more stochastic environments and under varying operating conditions, while still ensuring system reliability and security, Summary

A computer-implemented method for alternating current (AC) power flow analysis in an electrical, power network comprises;

(a) based on information relating to buses and transmission lines connecting the buses in the electrical power network, determining a relaxation of AC power flows in the electrical power network,

wherein the relaxation of AC power flows comprises a first constraint and a second constraint for each of multiple cosine terms associated with active power components and. reactive power components of the AC power flows, the first constraint fo that cosine term being mdtcative of a quadratic function and the second constramt for that cosine term defining a feasibility space with the first constraint for that cosine term, such that values of that cosine term are between the first constraint for that cosine term and the second constraint for that cosine term; and

(b) optimisin an objective function associated with the electrical power network to determine a relaxed optimum by constraining function variables of the objective function to the feasibility space.

Since the relaxation comprises a quadratic function;, the non-linear problem of AC power flow analysis is transformed int a quadratic problem. The quadratic problem can be solved by robust and computationally efficient industry standard solvers, which is an advantage of other methods that require less robust and computationally inefficient non-linear- solvers.

While it is not guaranteed that the result of the optimisation is physically feasible, it is guaranteed that there is no physically feasible solution that is better than the detennined solution because the value of the cosine term is always between the two constraints. n other words, all physically feasible solutions (the values of the cosine terms) are part of the feasibility space and therefore, if a better solution exi ts, it would be found in the optimisation, step. This i an advantage over other methods that may be able to determine a solution but cannot guarantee that this is the best possible solution.

The feasibility space may be a convex feasibility space. A conve problem can be solved by robust and. computationally efficient industry standard solvers, which is an advantage of other methods that require less robust and computationally inefficient non-convex solvers.

The relaxation of AC power flows may comprise a first constraint and a second constraint for each of multiple sine terms associated with active power components and reactive power components of the AC power flows, the first constraint for that sine term being indicative of a first linear function and the second constraint for that sine term being indicative of a second linear function and defining a feasibility space with the first constraint for that sine term, such that values of that sine term over a range of interest are between the first constraint for that sine term and the second constraint for that sine term.

The feasibility space may be further defined by a range of interest in relation to a voltage angle.

The method may further comprise determining a maximum error value of the relaxation of the AC power flows. Determining the relaxation of the AC power flows may comprise determining one or more redundancies and adding the one or more redundancies into the relaxation. It is an advantage that adding redundancies leads to a tighter relaxation and therefore, the solution is closer to the physically feasible solution and the error is reduced. The one or more redundancies may comprise linear combinations of power flow equations and each of the one or more redundancies may be representative of a power los on a transmission line. It is an advantage that power los equations allow an efficient factoring and therefore a refined relaxation. The method further may further comprise causing a real-time modification to the power network based on the relaxed optimum or determining a power network configuration.

The objective function may be associated with a cost of one or more of:

generating electricity in the electrical power network,

selecting one or more transmission lines to be included in the electrical power network, and

voltage control in the electrical power network.

Optimising the objective function ma comprise determining a minimum cost of generating electricity in the electrical power network and controlling a topology of the electrical power network. Controlling the topology of the electrical power network may comprise selecting one or more transmission lines to be included in the electrical power network, The relaxation of AC power flows may comprise a binary variable for each transmission line indicative of whether that transmission line is included in the electrical power network.

Optimising the objective function may comprise determining an optimum number of reactive, power compensation devices in the electrical power network.

A computer program is also provided comprising computer-executable instructions to cause a computer to perform the method for alternating current (AC) power flow analysis in an electrical power network as described above.

There is also provided a computer system, for alternating current (AC) power flow analysis in an electrical power network, the system comprising a processing device to perform the method described above.

Further, there is provided an electrical power network in which alternating current (AC J power flow analysis is performed using the method described above. Brief Description of Drawings

Examples will now be described with reference to the accompanying drawings, in which:

Fig. 1 is a schematic di agram of an example system for AC power flow analysis;

Fig. 2 i an illustration of solution space for optimisatio problems;

Fig. 3 is a flowchart of an example method for AC power flow analysis;

Fig. 4{a). Fig, 4(b) and Fig. 4(c) are plots of the (a) cosine function relaxation, (b) square function convex envelope and (c) sine function relaxation, respectively;

Fig. 5(a) and Fig. 5(b) are plots of the (a) general MacCorroick relaxation and (b) MacCoi'mick relaxation in power flows, respectively for bilinear terms;

Fig. 6 is a table on the sizes of power systems, benchmarks;

Fig. 7 is a plot of the relative quality loss when converting a DC-OPF solution into an AC -OFF solution; Fig. 8(a), Fig. 8(b) and Fig. 8(c) are plots of the (a) optimality gap between QC_l and £ C , (b) optimality gap between SD and, QC , and (c) runtime increase factor between SDPmdQC , respectively;

Fig. 9 is a plot of the runtime performance for the OFF problem, comparing the SDP and QC ;

Fig, 10 is a table on the quality and nearest AC feasible solutions of the DC. model for the OPF problems;

Fig. 11 is a table on the runtimes and optimality ga for different relaxations of the OFF problem;

Fig. 12(a) and Fig. 12(b) is a plot of (a) the performance and feasibility of DC- LSGPF and (b) bounds on AC-LSOPF, respectively;

Fig. 13 is a table on the feasibility and optimalit ga of LSOPF with the DC model;

Fig. 14 is a table on the LSOPF optimality gap using the QC relaxation;

Fig. 15 is a table on the sizes of capacitor placement, benchmarks;

Fig. 16(a), Fig. 16(b) and. Fig, 16(c) are plots of the (a) benchmark CI, (b) benchmark C5, and (e) benchmark C6, respectively.

Fig. 17 is a plot of the optimal number of capacitors on all benchmarks with a hound v' = 1.00.

Fig. 18 is a table on the CPP runtime and optitnality gap using the QC relaxation;

Fig. 1 is a schematic diagram of an example structure of a processing device capable of AC power flow analysis.

Nomenclature

The following symbols are used throughout the disclosure.

N Set of buses in. a power network

E Set of lines in a power network

G Set of generators

v. Voltage magnitude on bus i

$. Phase angle on bus ί Susceptatice of Uric 0',./)

Conductance of line

Active power on line (ij)

¾ Reactive power on line < , ) P, Active power on bus i

% Reactive power on bus i

x Approx.1 mats on of x

x" Upper bound of ,¾:

y Lower bound of x

Element of

Detailed Description

Fig. 1 shows an example system 100 that include a processing .device 110 capable of performing AC flow analysis for m electrical -power network. In general, a power network is composed of several types of components such as buses, lines, generators and loads. In the example in Fig. L the .electrical power network comprises four buses 120 (buses 1 to 4) and five transmission lines 130 (lines 1 to 5) connecting the buses .120. Bus .1 and Bus 4 are each connected to a generator 140. Three types of buses are shown, in which:

Bus 1 is known as a slack bus, which is an arbitrary bus in the network that has a generator and with known voltage magnitude and voltage phase.

Buses 2 and 3 are each known as a load bus, which is a bus that is not connected to a generator and with unknown vol tage magnitude and voltage angle.

Bus 4 is a "generator" or voltage-eon trolled bus, which is a bus that has a generator and with known voltage magnitude.

Cen tral to all optimisation problems is the power flow equations that model the steady- state power flow of AC The AC power flow equations form a system of non-convex- nonlinear constraints constituting a challenge when combined with optimisation, problems. Equations defining the AC power flows are generally a system of non-convex, nonlinear equalities which are TSi^'P-hard (i.e. Non-deterministic, Polynomial time hard) to solve and extremely challenging computationally in general. Such non-conve models that are not guaranteed to converge (especially outside of operating conditions).

The power network may be interpreted as a graph {N,E} where the set of buses iV represent tire nodes and the set of lines E represent the edges. Every bus /^' e N has two variables, a voltage magnitude v, and a phase angle θ_; . Lines have two constant electrical properties: the suseeptance b_i} and the conductance _? > All components are governed by two- fundamental, physical laws: (1 ) Ohm's Law yielding the following set of equations:

Pa - g ^v - C0s( -0,)- b v,v_f sin(6» - Θ. ) (0) q_tj ^«

-#,_.) -g#v,. . sin($ - ff_j) (0) where p and q denote the active and reactive power flow on the line (i, j) s E and (2) Kirchoff s Current Law (flo conservation), yielding the following set of equations:

ft =∑ Pi «*> ft = ∑ ¾ (0) where p_{ and <¾ denote the active and reactive power flow at bus i& h

The DC power flow model (Knight 1972) is a common response to the computational challenges of the AC power flow equations. It is a linear approximation derived through a series of assumptions justified by operational considerations. In particular, it .is assumed that the suseeptanc is sufficiently large relative to the conductance I l¾H g,_j I that it is reasonable to consider _¾ - 0 the phase angle difference θ. - Θ. is. small enough to ensure sin.(f¾ ~0_})∞<?. -· <?., cos(#_;—<?,.)∞ 1.0 ; and the voltage magnitudes ν,. are close to I . Under these assumptions, the AC power flow equations reduce to:

= (0) From a computational standpoint, the DC power flow model is much more appealing than the AC model since it forms a system of linear equations that can easily be embedded into MIP solvers. Under normal operating conditions and with some adjustment for line losses, the DC model produces a reasonably accurate approximation of the AC power flow equations for active power. Unfortunately, recent results have demonstrated that the inaccuracies of the DC power flow can he st gnifcant in a number of applications (Stott et al. 2009).

An exampl method for AC power flow analysis in the electrical power network i shown in Fig. 3, The example method 300 is performed by the processor includes the following steps performed by the processing device 110.

At block 310» the processing device 110 determines a relaxation of AC power flows based on information relating to busses and transmission lines in the electrical, power network 100 of this example.

Fig. 2 illustrates a parameter space 200. For simplicity of presentation, parameter space 200 is two-dimensional, where the horizontal dimension may be representative of a power flow in a first line and the vertical dimension, may be representative of a power flow in a second line. In most applications, however, the parameter space has many more dimensions, such, as 100,

Parameter space 200 comprises a physical solution space 202 that contains all solutions that satisf the power flow equations and are therefore physically possible. As can be seen i Fig. 2, the physical solution space 202 is non-convex. Parameter space 200 further comprises a relaxed solution space 206 that is convex and contains all solutions that satisfy the relaxed power flow equations as described below.,

It is difficult for processing device 11.0 to minimise a cost function (not shown in Fig. 2) while ensuring that the solution lies in the non-convex physical solution space 202. Therefore, processing device 11 minimises the cost function while ensuring that the solution likes in the convex relaxed solution space 206, that is, the parameters are constraint by the constraints defining the relaxed solution space 206. Solving this, convex optimisation is computationally more efficient than solving a non-convex optimisation.

The physical solution space 202 and the relaxed solution space 206 contain an (unknown) optimum solution 208 that is physically feasible because it lies within the physical solution, space 202, Since processing device 110 uses the relaxed solution space 206 instead of physical solution space 202., processing device 11.0 may find another solution 210 with a lower cost that lies outside the physical solution space 202 and is therefore physically infeasible.

The construction of the relaxed solution space 206 guarantees that the entire physical solution space 202, and therefore optimum solution 208, is contained in the relaxed solution space 206. Therefore, the solution 210 must have a lower cost than physical optimal solution 208. As result, after finding the solution 210 by minimising the cost function over the relaxed solution space 206, it can be guaranteed that no solution exists with a lower cost, or in other words, that, the physical optimal solution, while still unknown, will have an equal or higher cost tha solution 210.

In this example, the convex quadratic relaxation is based on exploiting convex envelopes of quadratic and trigonometric terms appearing in the AC power flow equations and is motivated by the narrow bounds observed, on decision variables such as- hase angle differences involved in power systems. A quadraticaUy relaxation model is first determined and power flow feasibility problems presented. The relaxation of the AC power flow equ tions comprises of the following:

(a) Quadratic relaxation of the cosine function

Let denote an upper bound on the phase angle difference between any pair of buses::

Hf^' < , -0_t) < Θ^Η , V(i, ) e E.

An approximation of the cosine function inside the operational bounds

-π! 2≤-θ*≤&≤ Θ" < % 12 is shown in Fig, 4(a),

This quadratic function over-estimates the original function,

PROPOSITION .1. V0€ Θ* ] : cm(0)≥cos(0).

Proof. Based on the symmetry of both functions, one can only consider interval

. Observe that c s(0) = cos(0) and cos(0^e) = eos((9*)..- One. can also prove that c $(0^u / 2) > co&(ff^l / 2) based on the tri onometric identity

indeed, observe that cm{£> 1 2) ~ 1 s= -

4

We have

(co ff' ) ~lf≥0 => cost ) ² - 2 eos(0" H 1 > 0

=> co$(0")² + 6cosC(9" ) 9 > 8cos((9") + 8

cos(0" Y + 6 cos(< ) + 9 , cosi Θ" + 1

16 2

(cosir)+3)^J

= i — —≥ COR(€" I Tf

16 => ^■ cost, & / 2).

4

Finally, suppose that both functions intersect at a point such that

— < &" . Since cos(#) h convex and strictly decreasing on this interval, the line linking points p^' and p" = { ff eos(@")} would have to be above cosii?) which contradicts the convexity property of this function.

As shown in Fig. 4(a) and as described above, the relaxation of AC power flows comprises the upper constraint .co.s(0) 402. The relaxation further comprises a lower constraint 404 for each of multiple cosine terms associated with active power components and reacti ve power components of the AC power flows.

The lower constraint 404 defines a feasibility space 406 with the upper constraint 402 for the cosine term, such that values of the cosine .term, are between tlie upper constraint 402 and the lower constraint 404. In this example, the lower constraint 404 is simpl a horizontal line cos( ?^w ) . It is noted that in most applications, tlie lower constraint 404 represents an upper bound on the cost, which means that a cost optimisation will move away from the lower constraint 404 towards the upper constraint 402, As a result, the exact configuration of the lower constraint 404 is less relevant for the quality of the result, However, choosing the lower constraint 404 such that the feasibility space 406 is convex has the advantage that convex solvers can be used. For this reason, in the present examples the lower constraint 404 is the least complex constraint that defines a convex feasibility space 406,

As will be described further below, an objective function associated with tlie electrical power network can be optimised to determine a relaxed optimum by constraining function variables of the objecti ve function to the feasibility space 406. Estimation Error

An upper bound on the maximum^' estimation error can be expressed in terms, of 0' . PROPOSITION 2.

cos(<9) - CDS(6>)≤ ( ' )² / 2 +cos(0" ) - 1 , VB e [-Θ^* , θ"] ^' — (fff

Proof. Consider the function cos(#) = .1—— . First, one can show that

2

coke)≥ co&{0) > eos(#), Vff€ j ff 8" ]

Based- on the symmetry of both functions, this only needs to be proven on the interval

[0, r ^~| . Consider the function / : ft→ J f(0) = c (0) - \ + ~~ , observe that

/{0) = 0 and ν/(0) = θ-&ϊη(θ) , since V/( ) > 0_! Vie I^< , / is positive on this interval, thus cos(0) > eos(0), V6>e [-6»" , 0^U ] .

Now, given the bounds [θ, 0" [ , one can compute the maximum, difference ,. this is equivalent to solving the following optimization

program: l ~ co${ff' )

max.

Since the objective function i strictly increasing, the optimal solution, is attained at the upper hound Θ" with the corresponding optimal value (Θ"† 12+ em(0") - 1 .

(b) Quadratic and polyhedral relaxations of non-convex expressions

Quadratic terms v? and v of equations (1) and (2) are relaxed into tiieir convex envelopes as shown in Fig. 4(b): This is denoted as (x^{^})^R. A polyhedral relaxation of the sine function is defined based on the following inequalities and illustrated in Fig. 4 (c):

^' θ ^' j . ( θ"^Λ

θ +sm —

2 2 . 2

This is denoted as {sin( r)) . It is worth emphasising that although the convex envelope of this function is polyhedral; its linear description involves solving a trigonometric equation and thus contains non-rational, coefficients. In one example, a tangent line i added at the upper and lower bounds to reducing the shaded region in Figure 4(c).

Proof. Observe that the left hand side of the inequality represents the equation of the

decreasing functions on the interval [0, Jtt 2] , thus V is strictly decreasing on that interval, Since V (0) - 0 and V/(¾"/2) > 0 one has V/(0) > .0, V0e [0, ir/ 2]_., thus / is convex on this interval. On the other hand, consider the function g : E→ R, g(0) = ύη(θ) + ύη(θ/ 2} , observe that ^p(0) = () and ¥gC#) = eos(#)+^CQs|^. Observe that Vg(#) >0,¥(9e [0,/T 2], therefore g is also convex on this interval. Note that f{0) - g(ff) ~ 0 and (π/2)> $(π/2) j (π/2) > g are convex on this interval, one can_. deduce that

Finally, note that f(0)-g{&)~ n(0)+ttn(0f2)-3f4 (e/2)&~y_Vl-y_* , thus }',, >>^> _p> . A similar proof can he derived for the pair of points (p,p ) . (c) Multilinear terms

Multilinear terms, such as t¾v_{# t} are relaxed using sequential bilinear MacCormick relaxations introditced in (McCormick 1976, Al- hayyal and Falk 1983). Fig.5(a) and Fig.5(b) shows the general MacCoimick relaxation and MacCoimick relaxation in power flows respectively.

ν!?- '* ^+ν/^ν.·~^ν.·^ν

This is denoted as (vu)^M . hi this framework, the relaxation of the product ν,.ν^ becomes ~ (ν_;ν_;-) and v.._¾. - <«^>^¾ . It will, be appreciated that this example can he extended to trilinear envelopes as introduced by Meyer and Floudas (2003, 2004). Nevertheless, on this particular model, numerical experiments have shown, that the sequential MacCormick relaxation offers comparably tight bounds.

Power flow relaxation

The complete imtial quadratic relaxation of the power flow equations is defined by the followine set of conve constraints.

if ,€ <)- )^fi

vi¾ e (ν_;ν_;)^'

¾ s ^in(^. -^. )

Where refer to the convex envelopes, variable products

including (v_tv,)^M refex to the MacConiiick relaxation a d ^sin^ represents the polyhedral relaxation of the sine function introduced. The set of constraints QC_t defines a relaxation of the feasible region corresponding to the AC power flow equations (l -{2).

Proof, This is a direct result of Propositions 1. 2 and 3. (d) Relaxation strengthening

The idea of introducing redundancy to improve the relaxation of non-convex programs was formalised by Liberti (2004) focusing o the particular case of bilinear expressions. Ruiz and Grossraann: (2011) present the class of general reductions constraints" obtained b intersecting different formulations based on tlie physical interpretation of the problem. A new procedure for creating redundancy based on generating linear combinations of nonlinear constraint in the original formulation is introduced.

For example, one can strengthen the relaxation of the power flow equations (1 ) and (2) by introducing redundancy: - g_y<v? + v -2vyv, < (% - ø,))

= -b,_y(v +v ~2v_iv_i cm(% -0·))

It will be appreciated that other linear combinations are also possible, nevertheless, numerical results have shown that the latter is sufficient for having a tight relaxation. Furthermore, ?„ + /- ,, and q t £/ .. offer a nice factorization and have a physical interpretation representing respectively the active and reactive power loss on line (j, /) . The factorization is obtained by replacing the cosine f unction by its relaxation: Ρ, + Ρ_β = + vj - 2ψ .¾ό> -ø.) = g₉((y. ~v_ff + 2%)

¾ +¾ ·- -hi_jiy + j - Ιν,ν.ΰο& -#.) = -!>.. ({ ?. - v. f + 2vt.^

Introducing redundant power loss constraint will be effective for other formulations of AC power flow that does not include sinusoidal terms,

The con vex, quadratic: relaxation in this model is: (1) quadratic relaxation of the cosine terms; (2) quadratic and polyhedral relaxations of quadratic and sine terras; (3) acCofmick relaxation of bilinear terms; and (4) relaxation strengthening with redundancy. The set of constraints therefore define the relaxation, of the feasible region corresponding to the AG power flow equations.

Referring back to Fig, 3, at block 320, the processing device 110 optimises an objective function relating to the power network constrained by convex quadratic relaxation of AC power flows to determine an optimum. In this example, that is the relaxed optimum is identified as 210 in Fig, 2 in the relaxed solution space 206. Optimisation at block 320 can be performed based on a set of nonlinear constraints Le. QC model and network operational constraints, all of which are associated with active power components and reactive power components of the AC power flows. Processing device 1 10 may optimise the objective function relating to the power network constrained by non-relaxed AC power flows to determine a candidate optimum. In this example, that is the candidate optimum is identified as 208 in Fig. 2 in the physical solution space 202, Optimisation can be performed based on a set of non-relaxed nonlinear non-convex constraints, all of which are associated with active power components and reactive power components of the AC power flows to generate the candidate optimum with no guarantees on optimality. The optimality of the candidate optimum or the bound of the candidate optimum 210 relative to the global optimum of the problem can be determined by the processing device 1 10 by comparison with the relaxed optimum 208 generated based on the QC model. QC is computationally efficient with orders of magnitude of improvement when compared with SDP formulation. Going from the scale of several minutes to few seconds in terms of computational time, the reduction in computational complexity will underpin computer automation such as the autonomous real-time reconfiguration of electrical grids that is an extension to the present case study.

Suitable applications of the optimisation include but not limited to: optimal power flow, node pricing market _.calculations, transmission switching, distribution network configuration, capacitor placement, expansion planning, vulnerabilit analysis, and power system restoration. In this example, based on the method 300. the processor operates to caus a real-time modification to the power network 100, For example, the processor sends an instruction message to the power network to cause changes in network configuration, such as disconnecting a line 130 in. the network 100.

According to the example method in Fig, 3, convex quadratic relaxation of AC power flows is determined and used in the subsequent optimisation. As such, optimisation based on the convex quadratic relaxation may be performed more accurately and the determined optimum offers a provable bound on the optimaiity of candidate optimum, Further, since a quadratic model is used, discrete optimisatio technology may be used to solve decision support problems in power systems more efficiently. This leads to more accurate modeling of AC power flows, as well as more efficient and cost- effective electrical power networks. The model (^!QC model^* ) includes convex quadratic relaxation of AC power flows and AC power loss constraints generated at block 310, where the AC power loss constraints may include convex quadration relaxations of AC power loss, that in turn is based on linear combinations of the conve quadratic relaxations of AC power flows, According to at least one example in the present disclosure, the QC model may have one or more of the following properties:

1. reasonable accuracy both inside and outside normal operating conditions;

2. reasonable computation l efficiency and scalability;

3. the ability to reason about voltage magnitudes and reactive, power which become critical outside normal operating conditions;

4. the ability to offer tight lower bounds tight lower bounds to optimal power flows

5< the ability to be integrated in general purpose discrete optimisation solvers, in particular Mixed-Integer Programming (MIP) or Mixed-Integer Nonlinear

Programming (M1NLP) solvers (e.g. CPLEX, Gurobi, and Bonmin) and their hybridizations with Constraint Programming (CP) and Large Neighbourhood Search (LNS).

This section, describes three case studies assessing the potential of the QC model in decision-support applications in optimal power flow, line switching power flow and placement of reactive power compensation devices, such as capacitors. The goal is to provide evidence that the QC model strikes an appealing trade-off between efficiency and accuracy. The QC model is compared to the standard field practices including the DC power flow approximation and the state-of-the-art SDP relaxation (Lavaei and Low 2012). Note that, a linear version of the QC model is also included in the analysis denoted byjgC,._.. The outer- approximation is generated using a set of uniformly distributed linearization points. The number of tangent lines was set to 10 in all numerical experiments. It is important to emphasise that all of the power flow models under investigation (i.e. DC, SDP relaxation and QC) are only models of the AC equations. To understand the real-world practicality of these models, their outputs should be assessed on the accuracy to the global optimum. This is achieved by first solving an optimisation problem using the approximate model, followed by a ste to determine the candidate optimum heuristieaUy constrained by the non-convex nonlinear AC equations, see 330 of Fig. 3. The details of this procedure vaiy depending on .the problem and are described in each case study. It should be noted that in the assessement of the accuracy to the global minimum, the relative optimality ga with respect to a lower bound is computed a the scalar difference between optimum of the models of the AC equation and optimum determined based on the true non-convex nonlinear AC equations. The outcomes of which are normalised by the optimum of the models of the AC equations. The bound is the value of the objective function at the optimum of the models of AC equations as shown in 340 of Fig. 3 . The nonlinear Branch and Bound implemented in Bonmin (Boiiami 200S) was used as a heuristic for solving non-convex MINLPs. CPLEX 12.5 (IBM 2012) was used to solve convex Mixed-Integer Quadrati eali Constrained Programs (MiQCP) along with continuous Quadratic Programs (QP) and Ipopt (Wachter and Biegler 2006) was used to solve general continuous ^"Nonlinear Programs (NLP). All experimental results were per.fon.ned on a Intel Xeon 2.0 GHz CPU and the time limit was set to one hour tor all runs. Thirteen standard -power system benchmarks provided in MatPower (Zimmerman et ai. 2011) are used. The size of the instances' is presented in Table 1 in Fig. 6. It should be mentioned in some of the results, computations simply exceeded the one hour time limit or out of memory and this is denoted as ⁴T, L.' or Ό, M' respectively.

It is important to introduce a number of operational constraints commo to all power optimizatio problems. Every node i.e Nin the network has a voltage magnitude that should be maintained around a nominal value of 1.0 per unit. The acceptable deviations from this value (usually ±20% ) are captured by the voltage bounds ( vf_»v" ^').

The network contains generators i e G c: N , which represent the sources of power. These components can produce active and reactive power, but their size and design enforce upper and lower bounds ( p[ _t p" ) and ( q' q" ) on the quantities t ey can. manage. A. line (i, j)≡E has two operational properties, a bound $" on the phase angle difference \.θ. - Θ. 1 and a thermal limit s on the apparent power P + q^ -

Together these operational constraints, denoted (PF), along with the power flow equations (l)-(4) form a common basis for all power flow decision problems. The phase angle bound Θ" is set to^r / 1 . p₍'≤ p,≤ _i , Vi e V, v₍'≤v.≤v M€ N,

r ≤v_;≤v , V/e N, (PF) -if 0_t- < &' , V(i, j) e £,

pf_j + q_y ^' ≤sf_} , V(i; j) e E. 99

(a) Case Study 1 - Optimal Power Flow

^'The Optimal Power Flow problem (OPF) is extensively studied in (Momoh et al. 1999, Lavaei and Low 2012), The goal in this decision-support problem is to find the cheapest power generation solution satisfying customers' demand, see 320 and 330 of Fig. 3. Given the set of generators G , the OPF problem consists of minimizing the generation, cost c. {pf† + e, ( p ) subject to network operations and power flow (l )-(4) is denoted as OPF constraints;

In this case study, four formulations are presented: (1 ) the original non-convex model (OFF) (2) the DC model (3) the SDP relaxation (4) and the QC formulation proposed. The complete numerical results are presented in Tables 2-3 and demonstrate three key points:

1. The classic DC approximation underestimates the (OPF) optimal objective value.

2. The QC model benefits greatly from the relaxation strengthening technique presented.

3. The strengthened QC model is order of magnitudes faster than the state-of-the- art SDP relaxation with minimal accuracy losses.

First, in Figure 9, one can measure a maximum quality ga of 7% when, using the DC approximation;, underlining inaccuracies in this formulation. Nevertheless, Table 2 in Fig. 10 shows that a DC solution can easily be converted into an AC feasible point, with relatively small cost losses.

Second, Figure 12(a) along with results reported in Table 3 in Fig. 1.1 highlight the efficiency of the strengthening procedure, where more than 96% of the gap is closed by adding the new redundant constraints. Last, Figures 12(b). 12(c) along with, the performance curve illustrated in Figure .16 sustain the third claim, demonstrating a computational gam up to two orders of magnitude when comparing QC to the SDP formulation, with an average optimality gap loss of 2%> .

(b) Case Study 2 - Line Switching Optimal Power Flow

The Line-Switching Optimal Power Flow problem (LSOPF) was originially tntrodueed in (Fischer et al 2008). it is a simple extension of the OPF problem where lines can be disconnected from, the network (i.e., "switched off'). Controlling the topology of the -network^' by removing lines changes the flow of power and can reduce the generation costs. A complete formulation is presented, in (LSOPF). The binar variable z_(j indicates whether a line is included in the network or discarded. The introduction of discrete variables naturally increases the complexity of the problem in the literature, the LSOPF problem i typically studied under the DC model relying on strong industrial, mixed-integer linear solvers. The exact AC model can be formulated as a MINL subject to network operations (PF) and power flow (3)-(4) constraints, see 330 of Fig- 3:

V(iJ e E,

¾ .= _½H¼v +b^v _i cos«?₍ - θ_})- g_t^v_J ήη ₍ -<¾)), V(/, _, )€ E, (LSOPF)

T the best of our knowledge, this is the first attempt for exactly solving the LSOPF problem in the AC power flow equations. Three formulations are considered: (1) the original non-con vex model (LSOPF) (2) the MiP model based on the DC model (Fisher et al. 2008) (3) and the mixed-integer extension of the quadratic relaxation introduced previously. In this case study, the feasibility of the DC model is tested by fixing the z_;j variables in accordance with the integer optimal solution and. solving the induced OPF problem (OPF), The new quadratic relaxation is used to prove infe&sibility of some optimal DC topologies.

The numerical experiments demonstrate the f llo j.ng key points:

1. The DC approximation underestimates the (LSOPF_.) optimal objective value.

2, In several cases, the DC approximation produces infeasibl network topologies with respect to the original (OPF) constraints.

3. Feasible network topologies proposed by the DC approximation are sub- optimal compared to solu tions produced by MINLP heuristics.

4, The bounds provided by the quadratic relaxation assess the high-quality of these heuristic solutions.

First, in Table 4 in Fig. 13, one can measure a maximum cost gap of 6.32% when using the DC approximation, underlying inaccuracies in this formulation. Second, the table indicates that the DC approximation cannot always be converted into an AC- feasible solution. Six. instances out of eight are proved to he infeasible. Third, the table reveals that feasible DC solutions can incur a cost increase up to 4.45% relative to a direct MINLP solution. The performance and feasibility of DC approximation is summarized in Figure .17(a). Last, Figure 17(b) and Table 5 in Fig. 14 report art average optimality gap of 2.6% when combining MINLP heuristics with QC 's lower bound.

(c) Case Study 3 - Capacitor Placement

The Capacitor Placement Problem (CPP) is another well-studied application with different variants (Aguiar and Cuervo 2005, Delfanti et at 2000, Huang et al 1996). The CPP is a particularly challenging problem since reactive power and voltage variables play an essential role, therefore linear formulations including only active power flows, such as the DC model, are naturally discarded. Informally speaking, the CPP consists of placing capacitors throughout a power network to improve the voltage profile. The version studied here aims at minimizing the number of installed capacitors, while meeting a voltage lower bound ' , satisfying a capacitor injection limit g" while subject to network operations (PF) and power flow (l)-(3) constraints

min∑¾

qj < ^" , Vi e N , (CAP) ,≥v' ₍ Vi e N,

z. e Z, Vie N.

The integer variable z,- represents the number of installed capacitors on node ;^' and q. the amount of injected reactive power.

In this study, the quadratic relaxation QC is used to assess the quality of feasible solutions obtained using MINLP heuristics (Bonmin et al, 2008), see 340 of Fig. 3, The numerical results demonstrate three key points:

1. MI LP heuristics implemented in Bonmi return near-optimal solutions.

2. The QC model offers tight lower bounds supporting the previous claim. 3. The QC model efficientl handles mixed-integer variables scaling u to medium-size instances (see Table 6 in Fig. 15),

To investigate the quality of feasible solutions produced by the MINLP solver,, two types of experiments are conducted: (1.) a parameter sensitivity study, varying the voltage lower bound v¹ for a given instance and (2) a scalability study, fixing the voltage lower bound v* and varying the size of the instance. Together, these experiments will highlight tlie robustnes of the MINLP solutions and the efficiency of the QC relaxation on six capacitor placement benchmarks (Tabic 6).

Figure 9 summarizes the sensitivity study and investigates how the voltage bound affects the quality of the MINLP solutions over .the parameter range 0.90 < v*≤ 1.05 . On. the smallest benchmark in Figure 9(a), one can see that the relaxation is off by at most one capacitor, proving optrrnality in two configurations. For the highest settings of the voltage bound (1.025≤ v'≤ 1.05 ), Bonmin finds no solution in the time limit, and the relaxation is able to prove the problem is truly infeasible. On the larger benchmarks,. Figures 9(b)-9(e) , a similar trend can be observed, although QC is unable to prove infeasibliliry on the high voltage configurations.

Finally, in the scalability study, with a voltage lower bound of 1.0, Figure 10 and Table 7 in Fig. 18 highlight the relative growth of the optimally gap proportionally to the instance size, validating the main key point of this experiment.

It is noted here that other reactive power compensation devices than capacitors may be included into the power network to control the voltage at particular points in the network.

Processing Unit 10

The example method in Fig. 3 can be implemented by hardware, software or firmware or a combination thereof. Referring to Fig. 19, an example structure of a processing device 1900 capable of acting as a processing unit 2.10 is shown in Fig. 2,

The example device 19100 includes a processor 1910, a memor 1920 and a network interface device 1.940 that communicate with each other via a communication bus 1930. Information may be transmitted and received via the network, interface device 1940,· which may include one or more logical or physical ports that connect the device 1900 to another network device. Example information received at the port are details of the power network 200. Example information sent from the port is a message that cause a modification to the power network 200.

For example, the various methods, processes and functional units described herein, may be implemented by the processor 1910. The term 'processor' is to be interpreted broadly to include CPU, processing unit, ASIC, logic unit, or programmable gate array etc. The processes, Methods and functional units ma all be performed by a single processor 1930 or split between several processors (not shown in Fig. 19 for simplicity), A reference in this disclosure or the claims to a 'processor' should thus be interpreted to mean 'one or more processors'.

Although one network interface device 1940 is shown in Fig. 19, processes performed by the network interface device 1940 may be split between several network interface devices. As such, reference in this disclosure to a 'network interface device' should be interpreted to mean 'one or more network interface devices".

The processes, methods and functional units may be implemented as machine-readable instructions executable by one or more processors, hardware logic circuitry of the one or more processors or a combination thereof. In the example in Fig. 19, the machine- readable instructions 1.924 for analysing AC power flows are stored in the memory 1920. Other information 1 22 such as input information, constraints and/or variables computed, such as relaxed and candidate optimum, by the processing unit 210 may be stored in the memory 1920, or remote data stores (not shown i Fig, 2), , Further, the processes, methods and functional units described in this disclosure may be implemented in the form of a computer program product. The computer program product i stored in a computer- readable storage medium and comprises a plurality of computer-readable instruction for making a device 210 implement the methods, recited in the examples of the present disclosure. Note that there is another example for equations (l)-(2), which is based on a variable substitution::

Pa = %j xf+y} -XiXj - -* > ^la)

% = (^■ +y!~^xi^xf-yi y? ^} ~ % c¾ ^'< ~ ^x> ; ) (^2a>

where x, - v,cm($_i ^") and y_; = v_i am(0_t). This may be referred to as the rectangular form of the power flow equations, in contrast with the previously defined polar form.

Network Operation Constraints In one exmaple, in addition to physical properties, power system optimization problems share a set of common operational constraints. The voltage magnitude v. at ever node i^'e N should be matotained around a nominal value of 1.0. The acceptable deviations from this value (usually ±20% ) are captured by the voltage bounds ( v. , v" ). The network contains generators ie G czN _f which represent the sources of power. These components can produce active arid reactive, power, respectively denoted p aud q, but their size and design enforce upper and lower bounds (p'.p*) and (q'.q*) on the quantities they can manage. A line (i, j) e E has two operational properties: A. bound q" on the phase angle difference 10,— θ, I and a thermal limit £.. on the line flow /¾+¾J. Together, these operational constraints (6)-f 1.0), along with the power flow equations (l)-(4) form a common basis for most power network optimization problems.

p;< ₍.<p",Vi6N, (7) q!≤¾≤q .,V eN, (8)

--Θ"≤ Q_t - θ,≤θ", V{i, /» e E, (9)

▼J Sv, iV. (10) In one example, the relaxations described herein are valid for a bound q"_.≤#72 However, in. practice, the design of the power network can make the acceptable phase angle difference smaller, such as ?r/36.

In another example, based on: Proposition 1 the relaxation of the cosine function is defined as

CS i ™ if

(Θ"

Oi>COS(#")

The complete initial quadratic relaxation of the power flow equations is defined by the following set of convex constraints QC-Init =

Pa = -b_lsws,. (11)

(12)

¾6-<C0S(0t -0_})}* (13)

¾e<sin(6>. -#.)>* (14)

(16)

(17)

(18 where

/ 2) + mi(ff^J 12)

1 2) -sin( / 2)

In a further example in relation to the strengthening of ^'the relaxation described above, one can strengthen the relaxation of die power flow equations (l)-(2) by including ^■constraints defining the power loss on a line (ji, j) :

p^l. + q

Note that — is a convex function as shown in Proposition 4. The function

/(E^; xl' - E) : /(i, j, ;) - ^' is convex .

Introducing auxiliary variables I.. s , bounding the square of the current magnitude p + q^*

^{■' IJ} „ ^' " , the strengthened (QC) relaxation, is defined as:

A second-order cone relaxation of (21) can be derived by^' reformulation,

^≥ti +4 (21 )

This gives rise to the following model: (11) - (20)

(QC-SOCP) -- (21α)

Extension to Mixed-Integer Nonlinear Programs An extension for various preblems in power systems is to allow topological Change by means of line -switching. The introduction of binary variables to model corresponding discrete variables leads to an additional level of complexity . Another major challenge is the modeling of on/off constraints appearing in these problems. This disclosure leverages recent developments in disjunctive programming to develop strong formulations of power systems problems featuring line switching.

On/Off Constraints

Given convex functions. / : W^,+'"→ , h : *^m→ R* and g_k : R"→ R, VJfc e K , we are interested in optimisation problems of the form

mm f(x, ^'i)

s.t. A(¾ ) < 0, (Pr)

¾ ( )≤0if ¾ = l, Vfee K

is S\ z s S^w.

Each g_k.(x) <⁽) represents an ^""on/off constraint, with ¾ as its corresponding indicator variable, h x,z)≤ gathers the remaining constraints. Bounds on variables are assumed t be finite. (Pr) can be reformulated as a disjunctive program

min /(x, ij

&.t.h(x_>z)≤ ,

f = { ( ,¾) : ¾ = 1. g_k (5)≤ 0, P < x < u Given this approach, one can define the best convex relaxation of each disjunctive constraint g_k to be the conve hull Γο«ν(Γ¾ Fj' ) . When the set r reduces to a single point (I ¹¹ - w° ~ 0 ), convfc U l^' ) can. be formulated in the space of original variables. The main result is the following (superscript k is dropped for clarity purposes) c nv(T₀ u F., } = closure il^''^, where

Let us emphasise that for quadratic functions

Γ^* is closed, and the convex hull is

This result can be extended to the general case ( j °≠ιΐ^α ) when functions g are monotonia [Hrjasd, H„ Bonami, P., Cor iejois, G., Quorou, A.: Mixed-integer nonlinear programs featuring "on/off" constraints. Computational Optimization and Applications 52(2), 537-558 (20.12)]- Specifically, in the linear case where g(x) = A ' ^rx ^' -b , the convex hull is defined in [fJijazi, H.L., Bonarni, P., Ouorou, A.: A note on linear on/off constraints. Australian National University technical report (2014)] as:

where m is the number of vartabks with nonzero coefficients in the linear constraint a^r.i ~l> , and S any subset of [l,...,m) . Observe that for .5 = 0 one gets the bigM- like constraint.

Note that (25) is not sufficient for defining the convex hull as shown in [Hijazi 2014], therefore, one can strengthen the relaxation by introducing the remaining non- dominated constraints, On/Off Constraints in Power Systems

In Power Systems, a number of variables and constraints may be affected by line switching. First, consider the phase angle variables θ, , if a line. (i ). is switched off, the phase angle difference ί 0. - # I bound increases to \ Ε \ Θ" . Let e fO, 1 } represent the line switching variable on line (i, ;) »_: then the power flow disjunctions are defined as follows.

Sine Functio Disjunction The sine function relaxatio introduced above can be written as a linear: disjunction (indices are dropped for clarity purposes):

I* » {(,?, θ. ζ)& R³ : - 1 < s < 1. ~ I E S θ^κ≤ Θ <1 E \ Θ" , z = O}

/ 2) -cos(#" 2)iT / 2. /2) ~CQs(0' 12)0· 12, 0^*≤0≤0*, z = l

Based on results in Hijazt 2014], the convex envelope of each disjunction characterised in. the space of original variables using (24).

Cosine Function Disjunction

The quadratic relaxation of the cosine function defined in (13) does not fit the hypothesis of (23). its disjunctive version is given as following:

Note that the existence of a compact representation for convex hulls of on/off constraints featuring ncm-monotonic functions is still an open question. Hence, we propo

Proposition 5. com' (F u Γ'^) c F^* _s

Proof: Since rV. is a convex set it is sufficient to show that Γ", ε Γ^ and Γ',, ς Γζ.. Intersecting with [z = 0} and f ,? = i ] completes the proof.

Current Magnitude Disjunction

We will focus on the second-order cone formulation of the current magnitude constraints defined in (21a), a similar approach applies for the nonlinear version (21). The disjunctive extension of (21a) is given as following: Γ_; = {(p_*.q, I, v, z) K⁵ : ( ¹†≤ v≤ (v^fl f_y p = 0, q = 0J = 0, _z = ø}

Note that Γ" does not fit the hypothesis of (22). in order to follow these assumptions, we introduce a new redundant constraint:

This gives rise to a new disjunction defined as:

I = {(p_<q, I, z) 6 R⁴ : /> « 0, <? « 0, I « 0, z - ø}

Based on (22), the convex-hull formulation is given by

Thermal Limit Disjunction

Let t_¾; denote the thermal limit on line (i, j) , a similar formulation can be applied to the thermal limit constraints p^* _t + ≤ t_it , with the following disjunction:

I = q, z)e K³ : p = 0, = 0. * = ø}

Based on. (22) the convex-buil. formulation is given by:

Observe the squaring of the binary variable, which conflicts with the natural intuition of multiplying the right-hand side with z■ The squared formulation is naturally tighter in the continuous relaxation where z. can take real values.

Power Flow Disjunction

Finally, in the power flow constraints (11)-(12), since v' > 0, v_{ can account for the off-configuration of the line where p., = q =· 0 , This is accomplished by introducing auxiliary variables i>? and enforcing the following set of constraints;

^ = -¾^{i? +£}¾ ^lk (28) v, - { 1 - ¾ X v" )² < £ < ΐ - ( 1 - ¾ X v' )²

(29)

The following five examples may be considered:

1. (AC-MINLP), the original mixed-integer non-convex model.

2. (DC), the mixed-integer linear model based on the DC approximation.

3. ( QC -Strong) the strengthened mixed-integer second-order cone formulation min Tc_; ( pf† 4- c. (p- )

(30) (31,)

(15V(18)

™{1~ ,)(ν ^<^ ₁~·(1~ζ_!ί)(_ν ^ί)\

(32)

¾. -c iF I 2)(θ_ι-θ_ι)< z_tl (sin(^ / 2)-cos(^ / 2)0^* 12)

+{l-¾)(cos(0" / 2) I £ I 0^s +1),

+GOS(^ / 2J(6»: ~0. ) < z..(sin( / 2)-GOS(0" 2)0^B / 2)

+0 - 2_1/)(cos(^" / 2) I £ I ø" + i ),

1 ¾ l< *¾ (sin(0" / 2) + co$(0" / 2)0" / 2) + (1 - ¾^■), cos<0" / 2) I I< ₂($in(0^* / 2) - cos(0* / 2)0* / 2 + */»(0" ))

+(l-z_i)(cos(0^''/2)l£l0^K),

(40) z_if smi-ff¹ ) - (1 - z₉ ) < ¾ < ¾ ¾ø(ø")+ (1 -¾),

(4D

¾ cos{~0" )~<1- ¾)≤ < 1,

(42)

-¾ * -(1 - ¾ ) I £ I 0"≤(0,-ø,)< ¾0^H + (l - z. ) I £ I

(43)

4. (QC-Weak). a weak version of (QC-Strong), where (35), (36) and (39) are discardcdy an (40) is replaced with p~_t + q~≤ t.. z_if■

5. (QC-M1NLP), an extension of (QC-NLP) including tight representations of on/off constraints from above:

rain f-ipfy +c_i(pf) s, {.15)-(18)

30) -(37)

</

z_i(^ytHl-z e ₍₄₅₎

.Another example of a capacitor placement aims at minimizing the number of installed capacitors, while meetin a voltage lower bound v' and satisfying a capacitor injection limit q'l: min y¾

fe'.AT

s.t. (l)-(4), (6H9)

v. = LVfs , (49)

≤v, < v^ ieNXG, (50)

The integer variable z, represents th number of installed capacitors on node i and q_t ^e the amount of reactive power injected by those capacitors. The formulation of the QC- SOCP for CPP is:

min T¾ s.t. (3)(4),(6)-(9_;

(n)-(20),<2-la). (47X51),

Further, the formulation of the Q -MNLP for CPP

rni .^T¾ s.t. (3)(41(:6 (9)₅

(1 1X2.1),

(47)-(5l).

The figures are only Illustrations of an example, wherein the units or procedure shown in the figures are not necessarily essential for implementing the present disclosure, Those skilled in the art will understand that the units in the device in the example can be arranged in the device in the examples as described, or can be alternatively located in one or more devices different from that in the examples. The units in the examples described can be combined into one module or further divided into a plurality of sub- units.

Although the flowcharts described show a specific order of execution, the order of execution may differ from that which is depicted. For example, the order of execution of two or more blocks may be changed relative to the order shown. Also, two or more blocks shown in succession may be executed concurrently or with partial concurrence. All such variations axe within the scope of the present disclosure.

It should also be understood that, unless specifically stated otherwise as apparent from the following discussion, it: is appreciated that throughout the description, discussions utilizing terms such as "optimising", "classifying", "constructing", "receiving", "processing", "tettieving", "selecting", "calculating", "determining", "optimising'', "displaying" or the like, refer to the action and processes of a computer system, or similar electronic computing device, that processes and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other mch information storage, transmission or display devices. Unless the context clearly -requires otherwise, words using singular or plural number also include the plural or singular number respecti vely. It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the above-described embodiments, without departing from the broad general scope of the present disclosure. The present embodiment ate, therefore, to be considered in all respects as illustrative and not restrictive.

Aguiar, R„ P. Cuervo, 2005. Capacitor placement in radial distribution networks through a linear detemiinistic optimization model. Proceedings of the 15th Power .Systems Computation Conference (PSCC'05), Lige, Belgium. Al-Khayyal, F.A., J.E. Falk, 1983. Jointly constrained biconvex programming. Mathematics of Operations Research 8(2) 273-286.

Bonaroi, P. .2008. Bonmin,

Delfanti, M.„ G.P. Graneili, P. Marannino, M. Montagna. 2000. Optimal capacitor placement using deterministic and genetic algorithms. IEEE Transactions on Power Systems 15(3) 1041 -1046.

Fisher, E.B.. R.P. O'Neill, M.C. Ferns. 2008. Optimal transmission switching. IEEE Transactions on Power Systems 23(3) 1346- .1355.

Huang, y.-C, H.-T. Yang, C.-L. Huang. 1996. .Solving the capacitor placement problem in a radial distribution system using tabu search approach. IEEE Transactions on Power Systems 1 1(4) 1868-1873,

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Ruiz, J. P., I. E. Grossmann. 201 1 , Using redundancy to strengthen the relaxation for the global optimization of MINLP problems. Computers & Chemical Engineering 35(12) 2729-2740, Stott, B₅ J Jardim, 0 Alsac, 2009. Dc power flow revisited. IEEE Transactions on Power System 24(3) 1290-1300.

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Hijazi, H.L., Bonami, P., Ouorou, A,: A note on linear on/off constraints, Australian National University technical report (2014).

Claims

CLAIMS:

1. A computer-implemented method for alternating current (AC) power flow analysis in an electrical power network, the method comprising:

(a) based on information relating to buses and transmission lines connecting the buses in the electrical power network, determining a relaxation of AC power flows in the electrical power network,

wherein the relaxation of AC power flows comprises a first constraint and a second constraint for each of multiple cosine terms associated with active power components and reactive power components of the AC power flows, the first constraint for that cosine term being mdicative of a quadratic function and the second constraint for that cosine term defining a ieasibility space with the first constraint for that cosine term, such that values of that cosine term are between the first constraint for that cosine term and the second constraint for that cosine term; and

(b) optimising an objective function associated with the electrical power network to determine a relaxed .optimum by constrainin function variables of the objective function to the feasibility space.

2. The method of claim 1, wherei the feasibility space is a convex, feasibility space,

3. The method of claim 1 or 2, wherein the relaxation of A power flows comprises a first constraint and a second constraint for each of multiple sine terms associated with active power components and reactive power components of the AC power flows, the first constraint for that, sine term being mdicative of a firs linear function and the second constraint for that sine term being indicative of a second linear function and defining a feasibility space with the first constraint for that sine term, such that values of that sine term over a range of interest are between the first constraint for that sine term and the second constraint for that, sine term.

4. The method of any one of the preceding claims, wherein the feasibility space further defined by a range of interest in relation to a voltage angle.

5. The method of any one of the preceding claims, further comprising determinin a maximum error value of the relaxation of the AC power flows. , The method of any one of the preceding claims, wherein determining the relaxation of the AC power flows comprises determining one or more redundancies and adding the one or more redundancies into the relaxation.

7. The method of claim 6, wherein the one or more redundancies comprise linear combinations of power flow equations.

8. The method of claim 6 or 7, wherein each o the one or more redundancies is representative of a power loss on a transmission line. 9. The method of any one of the preceding claims, wherein the method further comprises causing a real-time modification to the powe network based on the relaxed optimum.

10. The method of any one of the preceding claims, wherein the method further comprises determining a power network, configuration.

11. The method of any one of the preceding claims, where the objective function is associated with a cost of one or more of:

generating electricity in the electrical power network,

selecting one or more transmission lines to be included in the electrical power network, and

voltage control in the electrical power network,

12. The method of any one of the preceding claims, wherein optimising the objective function comprises determining a minimum cost of generating electricity in the electrical power network.

13. The method of claim 12, wherein optimising the objective function comprises controlling a topology of the electrical power network.

14. The method of claim .13, wherein controlling the topology of the electrical power network comprises selecting one or more transmission lines to be included in the electrical power network.

15. The method of claim 1 wherein the relaxation of AC power flows comprise a binary variable for each transmissio line indicative of whether that transmission line is included in the electrical power network.

16. The method of any one of the preceding claims, wherein optimising the objective function comprises determining an optimum number of reactive power compensation devices in the electrical, power network.

17. Computer program comprising computer-executable instructions t cause a computer to perform the method for alternating current (AC) -power flow analysis m an electrical power network according to any one of claims 1. to 16. 18, A computer system for alternating current (AC) power flow analysis in an. electrical power network, the system comprising a processing device to perform the method according to an one of claims 1 to 16.

19. An electrical power network in which alternating current (AC) power flow analysis is performed using the method according to any one of claims 1 to 1.6.