Metod Saniga
email: msaniga@astro.sk

Michel Planat
email: michel.planat@femto-st.fr

Petr Pracna
email: pracna@jh-inst.cas.cz

Hans Havlicek
email: havlicek@geometrie.tuwien.ac.at

The Veldkamp Space of Two-Qubits

Keywords: generalized quadrangles, Veldkamp spaces, Pauli operators of two-qubits 2000 Mathematics Subject Classification: 51Exx, 81R99

Given a remarkable representation of the generalized Pauli operators of twoqubits in terms of the points of the generalized quadrangle of order two, W (2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements -the so-called Veldkamp space of W (2). An intriguing novelty is the recognition of (uni-and tri-centric) triads and specific pentads of the Pauli operators in addition to the "classical" subsets answering to geometric hyperplanes of W (2).

Introduction

A deeper understanding of the structure of Hilbert spaces of finite dimensions is of utmost importance for quantum information theory. Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF][START_REF] Saniga | Multiple qubits and symplectic polar spaces of order two[END_REF][START_REF] Planat | On the Pauli graph of N -qudits[END_REF]. The case of two-qubit operator space, N = 2, was scrutinized in very detail [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF][START_REF] Planat | On the Pauli graph of N -qudits[END_REF] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.

Finite generalized quadrangles and Veldkamp spaces

In this section we will briefly highlight the basics of the theory of finite generalized quadrangles [START_REF] Payne | Finite generalized quadrangles[END_REF] and introduce the concept of the Veldkamp space of a point-line incidence geometry [START_REF] Buekenhout | Diagram geometry[END_REF] to be employed in what follows.

A finite generalized quadrangle of order (s, t), usually denoted GQ(s, t), is an incidence structure S = (P, B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I is a symmetric point-line incidence relation satisfying the following axioms [START_REF] Payne | Finite generalized quadrangles[END_REF]: (i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines are incident with at most one point; and (iii) if x is a point and L is a line not incident with x, then there exists a unique pair (y, M ) ∈ P × B for which xIM IyIL; from these axioms it readily follows that |P | = (s + 1)(st + 1) and |B| = (t + 1)(st + 1). It is obvious that there exists a point-line duality with respect to which each of the axioms is self-dual. Interchanging points and lines in S thus yields a generalized quadrangle S D of order (t, s), called the dual of S. If s = t, S is said to have order s. The generalized quadrangle of order (s, 1) is called a grid and that of order (1, t) a dual grid. A generalized quadrangle with both s > 1 and t > 1 is called thick.

Given two points x and y of S one writes x ∼ y and says that x and y are collinear if there exists a line L of S incident with both. For any x ∈ P denote x ⊥ = {y ∈ P |y ∼ x} and note that x ∈ x ⊥ ; obviously, x ⊥ = 1 + s + st. Given an arbitrary subset A of P , the perp(-set) of A, A ⊥ , is defined as A ⊥ = {x ⊥ |x ∈ A} and A ⊥⊥ := (A ⊥ ) ⊥ . A triple of pairwise non-collinear points of S is called a triad; given any triad T , a point of T ⊥ is called its center and we say that T is acentric, centric or unicentric according as |T ⊥ | is, respectively, zero, non-zero or one. An ovoid of a generalized quadrangle S is a set of points of S such that each line of S is incident with exactly one point of the set; hence, each ovoid contains st + 1 points.

The concept of crucial importance is a geometric hyperplane H of a point-line geometry Γ(P, B), which is a proper subset of P such that each line of Γ meets H in one or all points [START_REF] Ronan | Embeddings and hyperplanes of discrete geometries[END_REF]. For Γ = GQ(s, t), it is well known that H is one of the following three kinds: (i) the perp-set of a point x, x ⊥ ; (ii) a (full) subquadrangle of order (s, t ′ ), t ′ < t; and (iii) an ovoid.

Finally, we need to introduce the notion of the Veldkamp space of a point-line incidence geometry Γ(P, B), V(Γ) [START_REF] Buekenhout | Diagram geometry[END_REF]. V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection H 1 H 2 of all geometric hyperplanes H of Γ such that

H 1 H 2 = H 1 H = H 2 H or H = H i (i = 1, 2)
, where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.

The smallest thick GQ and its Veldkamp space

The smallest thick GQ is obviously the one with s = t = 2, dubbed the "doily." This quadrangle has a number of interesting representations of which we mention the most important two [START_REF] Payne | Finite generalized quadrangles[END_REF]. One, frequently denoted as W 3 (2) or simply W (2), is in terms of the points of P G(3, 2) (i.e., the three-dimensional projective space over the Galois field with two elements) together with the totally isotropic lines with respect to a symplectic polarity. The other, usually denoted as Q(4, 2), is in terms of points and lines of a parabolic quadric in P G [START_REF] Payne | Finite generalized quadrangles[END_REF][START_REF] Saniga | Multiple qubits and symplectic polar spaces of order two[END_REF]. By abuse of notation, any GQ isomorphic to W (2) will also be denoted by this symbol. From the preceding section we readily get that W (2) is endowed with 15 points/lines, each line contains three points and, dually, each point is on three lines; moreover, it is a self-dual object, i.e., isomorphic to its dual. W (2) features all the three kinds of hyperplanes, of the following cardinalities [START_REF] Buekenhout | Diagram geometry[END_REF]: 15 perp-sets, x ⊥ , seven points each; 10 grids (of order (2, 1)), nine points each; and six ovoids, five points each -as depicted in Fig. 1. The quadrangle exhibits two distinct kinds of triads, viz. unicentric and tricentric. A point of W (2) is the center of four distinct unicentric triads (Fig. 2, left); hence, Figure 1. The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by small circles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every intersection of two segments counts for a point of the quadrangle. The upper panel shows the points' perp-sets (yellow bullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouring will become clear later. Each pictureexcept that in the bottom right-hand corner -stands for five different hyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.

Figure 2. Left: -The four distinct unicentric triads (grey bullets) and their common center (black bullet); note that the triads intersect pairwise in a single point and their union covers fully the center's perp-set. Right: -A grid (red bullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); the two triads are also seen to comprise a dual grid (of order (1, 2)).

Figure 3. The five different kinds of the lines of V(W (2)), each being uniquely determined by the properties of its core-set (black bullets). Note that the "yellow" hyperplanes (i.e., perp-sets) occur in each type, and yellow is also the colour of two homogeneous (i.e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also worth mentioning that the cardinality of core-sets is an odd number not exceeding five. The three hyperplanes of any line are always in such relation to each other that their union comprises all the points of W (2).

the number of such triads is 4 × 15 = 60. Tricentric triads always come in "complementary" pairs, one representing the centers of the other, and each such pair is the complement of a grid of W (2) (Fig. 2, right); hence, the number of such triads is 2 × 10 = 20. A unicentric triad is always a subset of an ovoid, which is never the case for a tricentric triad; the latter, in graphcombinatorial terms, representing a complete bipartite graph on six vertices. Now, we have enough background information at hand to reveal the structure of the Veldkamp space of our "doily", V(W (2)). 2From the definition given in Section 2, we easily see that V(W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by grids and six by ovoids. The lines of V(W (2)) feature three points each and are of five distinct types, as illustrated in Fig. 3. These types differ from each other in the cardinality and structure of "core-sets", i.e., the sets of points of W (2) shared by all the three hyperplanes forming a given line. As it is obvious from Fig. 3, the lines of the first three types (the first three rows of the figure) have the core-sets of the same cardinality, three, differing from each other only in the structure of these sets as being unicentric triads, tricentric triads and triples of collinear points, respectively. The lines of the fourth type have as core-sets pentads of points, each being a quadruple of points collinear with a given point of W (2), whereas core-sets of the last type's lines feature just a single point. A much more interesting issue is the composition of the lines. Just a brief look at Fig. 3 reveals that geometric hyperplanes of only one kind, namely perp-sets, are present on each line of V(W (2)); grids and ovoids occur only on two kinds of the lines. We also see that the purely homogeneous types are those whose core-sets feature collinear triples and tricentric triads, the most heterogeneous type -the one exhibiting all the three kinds of hyperplanes -being that characterized by unicentric triads. We also notice that there are no lines comprising solely grids and/or solely ovoids, nor the lines featuring only grids and ovoids, which seems to be connected with the fact that the cardinality of a core-set is an odd number. From the properties of W (2) and its triads as discussed above it readily follows that the number of the lines of type one to five is 60, 20, 15, 45 and 15, respectively, totalling 155. All these observations and facts are gathered in Table 1. We conclude this section with the observation that V(W (2)) has the same number of points (31) and lines (155) as P G(4, 2), the four-dimensional projective space over the Galois field of two elements [START_REF] Hirschfeld | General Galois geometries[END_REF]; this is not a coincidence, as the two spaces are, in fact, isomorphic to each other [START_REF] Buekenhout | Diagram geometry[END_REF].

Pauli operators of two-qubits in light of V(W (2))

As discovered in [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF] (see also [START_REF] Planat | On the Pauli graph of N -qudits[END_REF]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [START_REF] Lawrence | Mutually unbiased binary observable sets on N qubits[END_REF]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF][START_REF] Planat | On the Pauli graph of N -qudits[END_REF] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) (Table 1). As already mentioned, the role of tricentric triads of the operators has Table 1. A succinct summary of the properties of the five different types of the lines of V(W (2)) in terms of the core-sets and the types of geometric hyperplanes featured by a generic line of a given type. The last column gives the total number of lines per the corresponding type. Table 2. Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF][START_REF] Planat | On the Pauli graph of N -qudits[END_REF].

PO set of five mutually set of six operators nine operators of non-commuting operators commuting with a given one a Mermin's square GQ ovoid perp-set\{reference point} grid been recognized in disguise of complete bipartite graphs on six vertices [START_REF] Planat | On the Pauli graph of N -qudits[END_REF]. A true novelty here is obviously unicentric triads and pentads of the generalized Pauli operators as these are all intimately connected with single-point perp-sets; given a point of W (2) (i.e., a generalized Pauli operator of two-qubits), its perp-set fully encompasses four unicentric triads (Fig. 2, left) and three pentads (Fig. 3, 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [START_REF] Saniga | Mutually unbiased bases and finite projective planes[END_REF], because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u, v} ⊥⊥ , where u, v ∈ x ⊥ with u = v [START_REF] Payne | Finite generalized quadrangles[END_REF].

Identifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle. Here we go one level higher, and identifying these subsets with the points of the associated Veldkamp space leads to recognition of nother remarkable subsets of the Pauli operators, viz. unicentric triads and pentads. It is really intriguing to see that these are the core-sets of the two kinds of lines that both feature grids alias Mermin squares. As it is well known, Mermin squares, which reveal certain important aspects of the entanglement of the system, play a crucial role in the proof of the Kochen-Specker theorem in dimension four and our approach gives a novel geometrical meaning to this [START_REF] Planat | On the Pauli graph of N -qudits[END_REF][START_REF] Saniga | Projective line over the finite quotient ring GF (2)[x]/(x 3 -x) and quantum entanglement: the Mermin "magic" square/pentagram[END_REF]. At the Veldkamp space level it turns out of particular importance to study relations between eigenvectors of the above-mentioned unicentric triads and pentad of operators in order to reveal finer, hitherto unnoticed traits of the structure of Mermin squares. These seem to be intimately connected with the existence of outer automorphisms of the symmetric group on six letters, which is the full group of automorphisms of our quadrangle; as this group is the only symmetric group possessing (non-trivial) outer automorphisms, this implies that two-qubits have a rather special footing among multiple qubit systems. All these aspects deserve special attention and will therefore be dealt with in a separate paper.

Concerning three-qubits, our preliminary study indicates that the corresponding finite geometry differs fundamentally from that of W (2) in the sense that it contains multi-lines, i.e., two or more lines passing through two distinct points [START_REF] Planat | Pauli graph and finite projective lines/geometries, Optics and Optoelectronics[END_REF]. As we do not have a full picture at hand yet, we cannot see if it admits hyperplanes and so lends itself to constructing the corresponding Veldkamp space. If the latter does exist, it is likely to differ substantially from that of twoqubits, which would imply the expected difference between entanglement properties of the two kinds of systems; if not, this will only further strengthen the above-mentioned uniqueness of two-qubits.

Conclusion

By employing the concept of the Veldkamp space of the generalized quadrangle of order two, we were able to recognize other, on top of those examined in [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF][START_REF] Saniga | Multiple qubits and symplectic polar spaces of order two[END_REF][START_REF] Planat | On the Pauli graph of N -qudits[END_REF], distinguished subsets of generalized Pauli operators of two-level quantum systems, namely unicentric triads and pentads of them. It may well be that these two kinds of subsets of the two-qubit Pauli operators hold an important key for getting deeper insights into the nature of finite geometries underlying multiple higher-level quantum systems [START_REF] Planat | Pauli graph and finite projective lines/geometries, Optics and Optoelectronics[END_REF][START_REF] Thas | Pauli operators of N-qubit Hilbert spaces and the Saniga-Planat conjecture[END_REF], in particular when the dimension of Hilbert space is not a power of a prime [START_REF] Thas | The geometry of generalized Pauli operators of N -qudit Hilbert space[END_REF].

It is important to mention here that the definition of Veldkamp space given by Shult in[START_REF] Shult | On Veldkamp lines[END_REF] is more restrictive than that of Buekenhout and Cohen[START_REF] Buekenhout | Diagram geometry[END_REF] adopted in this paper.

As this paper is primarily aimed at physicists rather than mathematicians, in what follows we opt for an elementary and self-contained exposition of the Veldkamp space of W (2); this explanation is based only upon some very simple properties of W (2) readily to be grasped from its depiction as "the doily", and does not presuppose/require any further background from the reader.

Acknowledgements

This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA grant agency projects # 2/6070/26 and # 7012 (all from Slovak Republic), the trans-national ECO-NET project # 12651NJ "Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases" (France), the CNRS-SAV Project # 20246 "Projective and Related Geometries for Quantum Information" (France/Slovakia) and by the Action Austria-Slovakia project # 58s2 "Finite Geometries Behind Hilbert Spaces".


Vasileios Sougkakis

Johann Meulemans

Florent Alzetto

Chris Wood

Mark Gillott

Tom Cox

AN ASSESSMENT OF THE QUB METHOD FOR PREDICTING THE WHOLE BUILDING THERMAL PERFORMANCE UNDER ACTUAL OPERATING CONDITIONS

Keywords: QUB method, Heat Loss Coefficient, in situ measurements

The performance gap between the design and the as-built thermal performance of buildings is an issue of high importance for the construction industry and reducing this gap has been the subject of significant research. Assessing the design thermal performance is straightforward; however estimating the as-built performance of a building presents several technical challenges that often restrict its wider implementation. One of the most common challenges is the time required to perform such testing. The QUB method is a transient method developed to estimate the Heat Loss Coefficient (HLC) in a single night without occupancy. The ability of the method to provide reliable results was demonstrated experimentally in a climate chamber with controlled conditions in a previous work.

The work presented in this paper reports on the findings from a long-term field study conducted in order to assess the applicability of the QUB method under actual operating conditions and to identify key parameters that may affect its performance. A series of tests were performed during the whole heating period in a detached house located in the University Park campus, University of Nottingham considering two distinct conditions: as-built and with increased air tightness. The results suggest that the QUB method was able to estimate the HLC of the house with reasonable repeatability in both cases. The reliance of the results to the experimental conditions that may affect the test is also discussed in order to improve the robustness of the method.

INTRODUCTION

Energy consumption of the domestic sector in the UK accounted for 27% of the nation's final energy consumption in 2015 (DBEIS, 2016). It therefore comes as no surprise that building regulations have been setting increasingly stricter requirements on the insulation levels of buildings as a result of the UK policy on climate change. Under the Climate Change Act, the UK government set the legally binding target to reduce its greenhouse gas emissions by 80% in 2050 compared to the 1990 emissions levels (DECC, 2011). Despite the increasing regulatory requirements, there exists a growing body of evidence suggesting that there is a significant difference between the design and the as-built performance of new UK buildings; this is commonly called 'performance gap' [START_REF] Zero | Closing the Gap Between Design and As-built Performance -Evidence Review Report[END_REF].

Assessing the design thermal performance of the building fabric is quite straightforward; the UK methodology for this is provided by the Government's Standard Assessment Procedure (SAP) (DECC, 2014). The calculation is based on the energy balance of the dwelling and it takes into account factors such as U-values and respective areas of building elements, thermal bridges and the dwelling infiltration characteristics to determine the Heat Loss Coefficient (HLC), expressed in W/K, of the building fabric. A typical energy formula of the different mechanisms that determine the HLC is given in Eq.1.

HLC= ∑ U i •A i + H TB +ρ•c p •Q v , with Q v = √ Q s 2 + Q w 2 = A L 1000 • √ C s •|ΔT|+C w •U w 2 (1)
Where, U i is the U-value (W/m 2 K) of element i with area A i (m 2 ) and H TB is the heat loss from thermal bridges (W/K). Collectively, these terms account for the transmission losses.

Infiltration losses are determined by the product of the air density (kg/m 3 ), the specific heat capacity of air, c p , (J/(kgK)) and the volumetric flow rate, Q v , (m 3 /s). Q v is a combination of the stack driven flow rate, Q s , and the wind driven one, Q w . Here, the model developed by Sherman and Grimsrud (1980), also referred to as the 'Basic model' by ASHRAE is considered (ASHRAE, 2013). These properties are dependent on the stack, C s ((L/s) 2 /(cm 4 K)), and wind coefficient, C w ((L/s) 2 /[cm 4 (m/s) 2 ]) , respectively A L (cm 2 ) is the effective leakage area at 4Pa, U w is the wind speed (m/s) and ∆T (K) is the internal to external temperature difference.

On the other hand, several methods have been developed to assess the thermal performance of a building's fabric experimentally [START_REF] Janssens | Overview of Methods to Analyse Dynamic Data[END_REF]. The most widely used method is the coheating method, a quasi-steady state method that provides an estimate of the HLC of a building. The procedure for the coheating test is described by [START_REF] Johnston | Whole House Heat Loss Test Method (Coheating)[END_REF]. In practice, determining the as-built HLC of a building with a coheating test is quite challenging due to the required duration of the method (a testing period of a minimum of two weeks is usually required) and the fact that it is restricted to the heating season where external temperatures remain consistently low and the effect of solar gains is minimised.

QUB method

The QUB method is a dynamic method developed by Saint-Gobain to determine the as-built HLC of dwellings within a much shorter period. The test commences after sunset and finishes before sunrise of the following day (Figure 1). The principle of the QUB method is based on a single resistance and capacity model (left)) [START_REF] Mangematin | Quick measurements of energy efficiency of buildings[END_REF][START_REF] Pandraud | QUB: Validation of a Rapid Energy Diagnosis Method for Buildings[END_REF]Pandraud et al., 2013) where the building is represented by a global thermal resistance R (the reciprocal of HLC of the building) and a global capacitance, C (the internal heat capacity) (Figure 1 (left)). Internal and external temperature nodes (T IN and T OUT respectively) are considered homogeneous and heat exchange between the two nodes occurs through the thermal resistance, R. Therefore, the energy input, P(t), is heat lost through the envelope and stored/released by the thermal mass of the fabric (Eq. 2). Temperature evolution is described as a single decaying exponential. (2)

The two unknowns, R and C in Eq.2, can then be determined by using two different constant powers in two different phases (respectively noted 1 and 2 in Figure 1 (right)).The HLC can then be determined by the following formula:

HLC= T 2 ' P 1 -T 1 ' P 2 T 2 ' ΔT 1 -T 1 ' ΔT 2 (3)
Where P i is the power input during phase i, T' i is the slope of the temperature profile at the end of phase i (defined as the t i -min(t i /2,τ), where t i is the duration of the i th phase and τ = 2 hours) and ΔT i is the internal to external temperature difference (T IN -T OUT ) at the end of phase i [START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF].The Taylor series method for uncertainty propagation is used to compute the relative uncertainty associated to the HLC. For increased accuracy it is aimed that the test is carried in an empty unoccupied dwelling without additional heat sources and in most cases there is no power input in phase 2 (P 2 = 0 W), i.e. this is the free cooling phase.

The HLC of a building can be estimated in one night using the QUB method [START_REF] Pandraud | Experimental optimization of the QUB method[END_REF]. This makes it suitable for large scale use by industry to test the actual performance of buildings. Furthermore, it is a research tool since it allows a large number of consecutive tests to be performed in a short period of time. However, there are some issues that need to be considered for increasing confidence on a measurement. As described earlier, the model assumes homogeneous internal temperature. In practice, this means that care should be taken when arranging the power input and location of heaters inside the building. Furthermore, as the method is able to estimate the HLC within one night, the result may be affected by the weather conditions within that night; evaluating the effect of certain weather parameters was the scope of this work.

The accuracy of the method is dependent on a dimensionless parameter inherent to the QUB method, the α-criterion, α [START_REF] Pandraud | Experimental optimization of the QUB method[END_REF]. This is determined by: α=1-

HLC ref ΔT 0 P 1 (4)
Where HLC ref is the reference Heat Loss Coefficient (theoretical or determined experimentally) (W/K), ΔT 0 is the initial temperature difference between the interior and the external environment (K) and P 1 is the power input during the heating phase (W). It has been demonstrated that the α-criterion has no effect on the resulting HLC for values 0.4 < α < 0.7, while for α > 0.7 there is usually an overestimation of the HLC mostly at very short tests [START_REF] Pandraud | Experimental optimization of the QUB method[END_REF][START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF]. Hence, the α-criterion is considered as a measure of confidence level for test accuracy with recommended values 0.4-0.7.

Aim of work

The ability of the QUB method to provide reliable results, has been demonstrated numerically [START_REF] Alzetto | Technical Standard L1. Measuring Air Permeability in the Envelopes of Dwelling[END_REF] and experimentally in a climate chamber under steady conditions [START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF]. The aim of the work presented here was to evaluate the performance of the method under actual operating conditions, i.e. assess reproducibility of results and identify reliance of the method on specific weather and testing conditions.

MATERIALS AND METHOD

The

Monitoring equipment

In order to perform a QUB test the following parameters need to be monitored: zone air temperatures, external air temperature and the power input. Air temperatures were monitored with platinum RTD sensors (PT100) and the energy consumed by the heaters was monitored by pulse meters. Data were stored in two Datataker DT85 loggers and a DT80 logger at 1 minute intervals. Weather conditions were recorded with a WSD1 sensor by EML and a Skye rht + sensor installed on the roof of a neighbouring house.

Testing protocol

The HLC of the house was determined under two different airtightness configurations: a) baseline (as-built including minor interventions, i.e. repairing of broken window glass by applying boarding) and b) with increased airtightness by applying polyethylene films across the entire window reveals, in order to prevent air leakage through the old window frames (Figure 3). The air permeability at 50 Pa, q 50 (in m 3 /m 2 h), was measured by performing a blower door test (ATTMA, 2016). It was found to be 9.3 and 6.5 m 3 /m 2 h for the baseline configuration and for the increased airtightness configuration, respectively.

The house was heated by means of STANLEY 2 kW fan heaters controlled by timers. The timers were programmed to switch on the heaters after the sunset of each day and switch them off several hours later at the end of the heating phase (Phase 1 in Figure 1). Each test was then completed at the end of Phase 2 (of same duration as Phase 1) before the sunrise of the following day. A constant power input, P 1 , of approximately 11 kW was used throughout the whole testing period. This was the maximum allowable power input imposed by the electrical installation of the house. In order to maintain a desired temperature inside the house prior to each test, a second set of STANLEY heaters controlled by thermostats and timers was also installed. It was attempted that each test would be conducted with an α-value within the recommended range (0.4 < α < 0.7) in order to ensure measurement accuracy and avoid systematic overestimation of the HLC. As the power input was kept constant due to restrictions of the electrical installation, this effectively meant that the starting internal temperature was adjusted with respect to the external one.

The house was tested on a daily basis from the end of September 2016 until the end of April 2017 with minor interruptions during this period. Due to the fact that the duration of the night changed over the different months, the duration of the QUB test was also amended. However, the minimum duration of the heating phase at any test was five hours (i.e. the minimum test duration was ten hours as the test consists of two phases of equal duration); it has been demonstrated that for heating duration higher than 4 hours the results were not affected by the α-criterion for 0.4 < α < 0.7 [START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF]. The different configurations as well as the different test durations are shown in Table 1. In total 161 tests were conducted, out of which 146 (i.e. 91%) lied within the recommended values.

RESULTS AND DISCUSSION

The analysis focused on assessing the ability of the method to provide robust estimates of the HLC. The effect of α-criterion, wind conditions and external temperature were investigated.

α-criterion

Results from the first set of tests (baseline) and the associated uncertainties of measurements were calculated with the use of Eq. 3 and are presented in Figure 4. In order to determine the α-criterion, the value of the mean HLC was used as the reference HLC (HLC ref in Eq.4). In total 52 QUB tests were performed for the baseline configuration that successfully met the α-criterion requirement. The mean HLC of these tests with 0.4<α<0.7 was 522.3 ± 5.2 W/K. This is shown by the red solid horizontal line in Figure 4, while the black solid lines represent the ±10% band. Some degree of fluctuation between the results was observed and the majority of tests were within ±10% from the mean value. However, there have been few occasions when the resulting HLC was slightly above this limit, while in one instance it reached 20%. For values of the α-criterion higher than 0.8 the HLC was significantly overestimated. This result is consistent with previous studies [START_REF] Pandraud | Experimental optimization of the QUB method[END_REF][START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF]. It can be seen that there is no effect of the α-criterion on the HLC when the former lied within the range 0.4 to 0.6. A rising trend is observed with the calculated HLC when the value of the α-criterion is greater than 0.6. At α > 0.7, the HLC was consistently within the +10% region. In addition, relative uncertainty lied within ± 10 -15% for the vast majority of the measurements. It was also seen that both durations yielded very similar HLC values.

The respective results for the increased airtightness configuration are presented in the right hand side of Figure 4. With regards to the increased airtightness configuration the mean HLC of 94 tests with 0.4 < α < 0.7 was found to be 508.2 ± 4.6 W/K. Reduction in the mean HLC as a result of increasing the air tightness of the building was approximately 14.1 ± 12.8 W/K. This could be attributed to the fact that the building is fairly sheltered from three sides, with just the West façade exposed. It should be noted that a simplified theoretical calculation showed an estimated reduction of the infiltration heat losses by 19.5 W/K of the increased airtightness compared to the baseline configuration (infiltration losses from 64.5 to 45 W/K). This was consistent with the reduction observed through the testing procedure. Results again appear to fluctuate around the mean HLC as with the baseline configuration and again the majority of the results falls into the ± 10% band. However, in this case they appear to be more dispersed; there are some results extending to ± 15%, with a couple of points approaching ± 20%. Again, it can be seen that there was no correlation between the α-criterion and the HLC for α values even up to 0.8. It should be noted that measurements at the increased airtightness configuration presented larger relative uncertainties than at the baseline configuration; approximately 15% on average compared to 12% for the baseline configuration. With regards to the duration of the tests it can be seen that tests with heating duration of 6 and 7 hours presented very similar results. Tests with 5-hour heating duration appeared to result similar HLC values, albeit somewhat at the lower band. However, this is the result of just 13 tests. In any case it can be derived that longer tests result in consistent HLC that have minimum reliance on the α-criterion. This is of particular importance since the α-criterion relies in turn on the HLC ref (Eq. 4). Poor estimation of the HLC ref may lead to high values of the α-criterion and lead to the overestimation of HLC. Therefore longer test duration provides confidence on the accuracy of the test results. In practical terms this means that a 12 hour QUB test (i.e. heating duration of 6 hours) may be performed from end of October until end of March with great confidence that the result will not been overestimated by the α-criterion. 10-hour tests also provide confidence on the results as they were affected only at very high values of the α-criterion (close to 0.85). These can be performed from end of September until end of April in the UK.

Wind

Wind conditions are likely to affect the measured heat loss coefficient of a building as expected from Eq.1. Several research studies investigated the impact of wind conditions on infiltration losses during coheating tests [START_REF] Everett | Linford Low Energy Houses[END_REF], Miles-Shenton et al., 2011, Siddal et al., 2011). For this reason, the effect of wind was examined under the baseline and the increased airtightness configurations in order to determine whether there is correlation between the measured HLC and the average wind speed measured.

Figure 5 shows the relationship between the HLC and the respective average wind speed for both configurations for tests where 0.4 < α < 0.7. With regards to the baseline configuration (left) it can be seen that there is no correlation between wind speed and HLC. The relationship between HLC and the average wind speed for the increased airtightness configuration is presented in the right hand side of Figure 5. In this case, more data points were available and there were some tests within a higher region of average wind speed recorded, i.e. at average wind speeds higher than 3 m/s. Similarly to the baseline configuration, there was no correlation between the average wind speed and the HLC. In both cases it was found that the HLC was not related to the average wind speed and therefore, there was no scope for correcting the HLC based on this parameter. Infiltration heat losses are fairly complex and they are not solely linearly related to wind speed. Other factors such as temperature, the airtightness of the building, air leakage paths and sheltering are also affecting heat as well as mechanisms such as wind washing and thermal bypass (Sidall, 2011, Johnston et al., 2013[START_REF] Stamp | Assessing Uncertainty in co-heating tests: Calibrating a whole building steady state heat loss measurement method[END_REF]. The weak relationship observed here can be attributed to the fact that the building is fairly sheltered from three sides. The wind coefficient, C w , that determines the wind flow rate in Eq. 1 depends on the shelter factor of the building and the building height (ASHRAE, 2013). Furthermore, it can be seen in Eq.1 that wind driven flow is not linear to the wind speed and becomes dominant at high wind speeds. Recorded average wind speeds were relatively low and therefore infiltration losses were mainly driven by the stack effect. [START_REF] Stamp | Assessing Uncertainty in co-heating tests: Calibrating a whole building steady state heat loss measurement method[END_REF] reported the findings of several field tests attempting to find correlation between wind speed and the whole building HLC. It was found that in some cases there was no correlation and in some cases a relationship between these two variables could be established. The findings of this study therefore are in agreement with the ones reported by [START_REF] Stamp | Assessing Uncertainty in co-heating tests: Calibrating a whole building steady state heat loss measurement method[END_REF].

Temperature and ground floor losses

The effect of external temperature on the obtained results was also investigated. The temperature difference between the internal and external environment is included in the formula of the α-criterion (Eq.4) and therefore the external temperature was already considered an important environmental parameter on the QUB measurement. In addition, the HLC of a building is expressed in W/K with respect to the temperature difference between the internal environment and the external air temperature. However, heat loss in a building occurs not only to the external air temperature but also to the ground temperature (which does not follow the seasonal variation of the air temperature) through the floor. This issue was highlighted by [START_REF] Everett | Linford Low Energy Houses[END_REF] and floor losses (as well as infiltration losses) were isolated from the HLC in the 'Thermal calibration' method. To identify any potential relationship between these parameters, HLC for the baseline and the increased airtightness configurations were plotted together with the initial external temperature (Figure 6 and Figure 7). The HLC and the external temperature followed very similar fluctuation throughout the testing period. This was the case for both airtightness configurations. A clear trend is shown in both graphs where the HLC appears to increase when the external temperature increases.

Correlation between these two variables is consistent in both cases (i.e. the adjusted R-squared R² was 0.21 for the baseline and 0.16 for the increased airtightness configuration) and it is suggesting that some of the variance in the HLC is attributed to the variance in the external temperature. This can be explained by the way the test was carried out. As seen in Eq. 4 the two testing parameters that can be controlled to adjust the value of α are the temperature difference and the power input. As described in the 'Testing protocol' section the power input was kept the same throughout the testing period while the internal starting temperature was changed according to the external temperature on any given test day such that the α-criterion lied within the recommended range (i.e. between 0.4 and 0.7).

Effectively, this meant reducing the internal temperature on colder nights and increasing it on warmer nights by adjusting the thermostats on the second set of heaters.

However a complication exists in relation to the heat loss via the ground floor. The ground floor temperature is influenced by a large thermal mass and therefore does not follow the air temperature variation. The consequence of this is that the floor temperature is very slow to change (over weeks rather than days) and therefore the floor heat loss, and the subsequent HLC, is shown to be higher on warmer nights. It should be noted that the HLC plotted in Figure 6 and Figure 7 is the raw HLC of the whole dwelling (i.e. including the ground floor losses). Heat flux on the floor was monitored with heat flux plates. Data from these sensors will be analysed further in order to correct for the floor losses and investigate whether this correlation is then eliminated and the consistency of the results is improved.

CONCLUSIONS

The findings from a series of in situ tests in a circa 1950s detached dwelling carried to assess its thermal performance were presented in this work. The objectives of the study were to evaluate the performance of the QUB method on site, investigate repeatability of the results and assess reliance of the results to the prevalent test conditions. The QUB method is able to estimate the HLC of a building within just one night, significantly less than the 2-4 week period required for a coheating test or other quasi steady-state methods [START_REF] Janssens | Overview of Methods to Analyse Dynamic Data[END_REF].

The tests were conducted on a daily basis from the end of September 2016 until the end of April 2017 under two distinct airtightness configurations. This was only made possible due to the short duration of the method. Therefore, QUB does not only have significant potential for use as a diagnostics tool in industry; it is also a research tool. The following conclusions were drawn from the analysis:

1. The QUB method was able to provide reasonably consistent results (i.e. with a coefficient of variation of ± 10%)) provided the α-criterion lied within the recommended range (i.e. between 0.4 and 0.7). 2. In the UK a test may be conducted from late October until late March (or even late September until late April) with confidence that results would not be affected by the α-criterion or any assumption made in calculating the HLC ref .

3. No significant correlation was found between wind speed and the building's HLC. This could be attributed to the fact that that the building was sheltered from three sides and the wind speeds experienced were not excessive. 4. External temperature appeared to affect the resulting HLC and this was attributed mainly to the ground floor heat losses.

Results from the first long term in situ monitoring study in the UK with the QUB method were presented in this paper. The aim was to establish specific trends on the performance of the QUB method in situ. Further analysis is required on the particular conditions of each test performed and the heat flux on the building elements to investigate the outliers that appear to affect the level of consistency of the test results. Data from heat flux sensors installed on the floor will help to investigate further the effect of external temperature and assess whether consistency of results is improved when isolating the floor heat losses.

Reliance of the results on additional weather parameters (solar radiation, precipitation, night long-wave radiation) and the thermal performance of building elements (i.e. U-values with the QUB/e method; [START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF] will also be investigated in future studies.

It should be stressed out that these findings apply to measurements performed in an uninsulated dwelling. Therefore additional testing is required to investigate whether the same reproducibility of results would be achieved for well insulated buildings representative of new built or retrofitted properties. Climate chamber testing suggested that similar reliability can be expected [START_REF] Meulemans | QUB/e -A novel transient experimental method for in situ measurements of the thermal performance of building fabrics[END_REF], however testing in field conditions is required to verify this assumption. Testing a well-insulated property will allow greater flexibility in the power input to adjust the α-criterion. In addition, further validation and verification of results will be achieved by comparing results obtained with the QUB method to the ones obtained with coheating or other similar methods [START_REF] Janssens | Overview of Methods to Analyse Dynamic Data[END_REF].

Figure 1 :

 1 Figure 1: RC model used in the QUB method (left) and evolution of temperature and power during a QUB measurement (right) P= T IN -T OUT R +C dT IN dt

  tests was carried in a circa 1950s detached house located at 5, Wortley Close at the University Park Campus, University of Nottingham. The property, designed by Thomas Cecil Howitt, is a two storey L-shaped detached house. The wall construction is a 50 mm unfilled masonry cavity and the floor is solid uninsulated concrete slab. It has a cold pitched roof with insulation at ceiling level and the windows are single glazed. The house has North -East Orientation and ground floor area of approximately 93 m 2 . A view of the house and the plans for the ground and first floor are shown in Figure 2 below.

Figure 2 :

 2 Figure 2: External view (left) and Ground and First Floor plans(right)

Figure 3 :

 3 Figure 3: Polyethylene film applied on windows to increase the air tightness of the fabricTable 1: Test configuration and heating duration

Figure 4 :

 4 Figure 4: Summary of results against the α-criterion obtained for the baseline (left) and the increased airtightness configuration (right)

Figure 5 :

 5 Figure 5: Effect of average wind speed on the measured HLC for the baseline (left) and the increased airtightness (right) configuration

Figure 6 :Figure 7 :

 67 Figure 6: Heat Loss Coefficient against initial external temperature for the baseline configuration in a bar chart (left) and scatterplot (right) for the different test durations

Table 1 : Test configuration and heating duration Configuration Airtightness q 50 (m 3 /m 2 h)

 1

			Number of tests	Period	Heating duration
	Baseline	9.3	58	22.09 -10.11.2016 11.11 -05.12.2016	5 hours 7 hours
	Increased airtightness	6.5	103	07.12.2016 -30.01.2017 31.01 -21.03.2017 04.04 -25.04.2017	7 hours 6 hours 5 hours