![C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 4
1 Introduction
The U.S. electric utility industry is undergoing major structural changes in an
effort to make it more competitive [21,17,19,11]. One major consequence of the
deregulation will be to decouple the controllers of the network from the power
producers, making it difficult to regulate the levels of power on the network;
consumers as well as producers will eventually be able to negotiate prices to
buy and sell electricity [18]. In practice, deregulation is complicated by the
facts that all power companies will have to share the same power network in the
short term, with the network's capacity being just about sufficient to meet the
current demand. To overcome these problems, most U.S. states have set up an
ISO (independent system operator): a non-profit governing body to arbitrate
the use of the network. The basic questions facing ISOs are how to decide which
contracts to deny (due to capacity constraints), and who is to bear the costs
accrued when contracts are denied. Several criteria/policies have been proposed
and/or are being legislated by the states as possible guidelines for the ISO to
select a maximum-sized subset of contracts that can be cleared simultaneously
[18]. These include: (a) Minimum Flow Denied: The ISO selects the subset of
contracts that denies the least amount of proposed power flow. This proposal
favors clearing bigger contracts first, (b) First-in First-out: The contract that
comes first gets cleared first; this is the least discriminating to the contractors,
(c) Maximum Consumers Served: This clears the smallest contracts first and
favors the small buyers whose interests normally tend to go unheard.
There are three key issues in deciding policies that entail specific mecha-
nisms for selecting a subset of contracts: fairness of a given policy to producers
and consumers; the computational complexity of implementing a policy, and
how sound a given policy is from an economic standpoint. (For instance, does
the policy promote the optimal clearing price/network utilization etc.) Here we
focus on evaluating the efficacy of a given policy with regard to its computa-
tional resource requirement and network resource utilization. It is intuitively
clear that the underlying network, its capacity and topology, and the spatial
locations of the bilateral contracts on the network, will play an important role
in determining the efficacy of these policies. We do not discuss here the fair-
ness and economics aspects of these policies: these are subjects of a companion
paper. The work reported here is done as part of a simulation based analyt-
ical tool for deregulated electrical power industry being developed at the Los
Alamos National Laboratory.
We experimentally analyze several algorithms for simultaneously clearing a
maximal number of bilateral contracts. The qualitative insights obtained in this
paper can be useful to policy makers who carry the ultimate responsibility of
deploying the best clearing mechanism in the real world. The algorithms were
chosen according to provable performance, ability to serve as a proxy for some
of the above-stated policies, and computational requirement. The algorithms
are as follows; see § 3 for their specification. The ILP-RANDOMIZED ROUNDING
(RR) algorithm has a provable performance guarantee under certain conditions.
The computational resource requirement is quite high, but the approach also](https://image.slidesharecdn.com/83447-250405070230-5a110f2d/85/Graph-Algorithms-and-Applications-7-Giuseppe-Liotta-20-320.jpg)
![C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 6
crucially depends on the scenario and the algorithm.
As far as we are aware, this is the first study to investigate the efficacy of
various policies for contract satisfaction in a deregulated power industry. Since
it was done in fairly realistic settings, the qualitative results obtained here have
implications for policy makers. Our results can also be applied in other settings,
such as bandwidth-trading on the Internet. See, e.g., [2]. Also, to our knowledge,
previous researchers have not considered the effect of the underlying network
on the problems; this is an important parameter especially in a free-market
scenario.
The rest of this paper is organized as follows. The problem definitions and
algorithms considered are described in Sections 2 and 3 respectively. Our
experimental setup is discussed in Section 4. Section 5 presents our experimental
results and analyzes them and Section 6 concludes the paper. In the appendix,
we discuss interesting optimization issues that arise from deregulation, and also
show problem instances on which our algorithms do not perform well.
2 Problem Definitions
We briefly define the optimization problems studied here. We are given an
undirected network (the power network) G = (V, E) with capacities ce for each
edge e and a set of source-sink node pairs (si,ti), 1 < i < k. Each pair (si,ti)
has: (i) an integral demand reflecting the amount of power that Sj agrees to
supply to U and (ii) a negotiated cost of sending unit commodity from s, to
tj. As is traditional in the power literature, we will refer to the source-sink
pairs along with the associated demands as a set of contracts. In general, a
source or sink may have multiple associated contracts. We find the following
notation convenient to describe the problems. Denote the set of nodes by N.
The contracts are defined by a relation R C (2V x T
V x 3? x 3?) so that tuple
(v, w, a, /3) e i ? denotes a contract between source v and sink w for a units of
commodity at a cost of /3 per unit of the commodity. For A = (u, w, a, (3) e R
we denote source(A) = v, sink(A) = w, flow(A) = a and cost(A) = (3.
Corresponding to the power network, we construct a digraph H = (VliSUTU
{s, t}, E') with source s, sink node t, capacities u : E' —
> 3? and costs c' : E' —> 3?
as follows. For all A e R, define new vertices VA and WA- Let S = {VA A G R}
and T = {WA A s R}. Each edge {x, y} from G is present in H as the two
arcs (x, y) and (y, x) that have the same capacity as {x, y} has in G, and with
cost 0. In addition, for all A = (v,w,a,[3) € R, we introduce: (i) arcs (VA,V)
and (w, WA) with infinite capacity and zero cost; (ii) arc (S,VA) with capacity
flow{A) = a and cost 0; and (iii) arc (wA,t) with capacity flow(A) = a and
cost equaling cost(A). By this construction, we can assume without loss of
generality that each node can participate in exactly one contract. A flow is
simply an assignment of values to the edges in a graph, where the value of an
edge is the amount of flow traveling on that edge. The value of the flow is
defined as the amount of flow coming out of s (or equivalently the amount of
flow coming in to t). A generic feasible flow f = (fx,y > 0 : (x, y) € E') in](https://image.slidesharecdn.com/83447-250405070230-5a110f2d/85/Graph-Algorithms-and-Applications-7-Giuseppe-Liotta-22-320.jpg)
![C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 7
H is any non-negative flow that: (a) respects the arc capacities, (b) has s as
the only source of flow and t as the only sink. Note that for a given A £ R,
in general it is not necessary that fStVA = fWA,t- For a given contract A £ R,
A is said to be satisfied if the feasible flow f in H has the additional property
that for A = (v,w,a,/3), fs,vA = fwA,t = a
', i-e
-> the contractual obligation of
a units of commodity is shipped out of v and the same amount is received at
w. Given a power network G(V,E), a contract set R is feasible (or satisfied)
if there exists a feasible flow / in the digraph H that satisfies every contract
A £ R. Let B = supply(s) = demand(t) = ^^ejj/^ow;(A).
In practice, it is typically the case that R does not form a feasible set. As
a result we have two possible alternative methods of relaxing the constraints:
(i) relax the notion of feasibility of a contract and (ii) try and find a subset
of contracts that are feasible. Combining these two alternatives we define the
following types of "relaxed feasible" subsets of R. We will concern ourselves
with only one variant here. A contract set R! C R is a 0/1-contract satisfaction
feasible set if, MA = (v,w,a,/3) £ R', fs,VA = fWA,t = a.
Definition 2.1 Given a graph G(V, E) and a contract set R, the (0/1-VERSION
MAX-FEASIBLE FLOW,) problem is to find a feasible flow f in H such that
J^AeR' / ( ^ ) ** maximized where R' forms a 0/1-contract satisfaction feasible
set of contracts. In the related (0/1-VERSION, MAX-#CONTRACTSJ problem,
we aim to find a feasible flow f in H such that R' is maximized, where R!
forms a 0/1-contract satisfaction feasible set of contracts.
Though such electric flow problems have some similarities with those from
other practical situations, there are many basic differences such as reliability,
indistinguishability between the power produced by different generators, short
life-time due to inadequate storage, line effects etc. [22]. The variants of flow
problems related to power transmission studied here are intuitively harder than
traditional multi-commodity flow problems, since we cannot distinguish between
the flow "commodities" (power produced by different generators). As a result,
current solution techniques used to solve single/multi-commodity flow problems
are not directly applicable to the problems considered here.
3 Description and Discussion of Algorithms
We work on the (0/1-VERSION, MAX-#CONTRACTS) problem here. Let n and
m respectively denote the number of vertices and edges in the network G. In
[5], it was shown that (0/1-VERSION, MAX-#CONTRACTS) is JVP-hard; also,
unless NP C ZPP, it cannot be approximated to within a factor of m1
/2 - 6
for
any fixed e > 0, in polynomial time. Thus, we need to consider good heuris-
tics/approximation algorithms. First, there are three simple heuristics. The
SMALLEST-FIRST HEURISTIC considers the contracts in non-decreasing order of
their demands. When a contract is considered, we accept it if it can be feasi-
bly added to the current set of chosen contracts, and reject it otherwise. The
LARGEST-FIRST HEURISTIC is the same, except that the contracts are ordered](https://image.slidesharecdn.com/83447-250405070230-5a110f2d/85/Graph-Algorithms-and-Applications-7-Giuseppe-Liotta-23-320.jpg)
![C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 8
in non-increasing order of demands. In the RANDOM-ORDER HEURISTIC, the
contracts are considered in a random order.
We next briefly discuss an approximation algorithm of [5]. This has proven
performance only when all source vertices Sj are the same; however, this algo-
rithm extends naturally to the multi-source case which we work on. An integer
linear programming (ILP) formulation for the problem is considered in [5]. We
have indicator variables Xj for the contract between Sj and ti, and variables Zj>e
for each (sj,ij) pair and each edge e. The intended meaning of x» is that the
total flow for (s8, U) is diXf, the meaning of z^e is that the flow due to the con-
tract between (sj,ij) on edge e is Zi>e. We write the obvious flow and capacity
constraints. Crucially, we also add the valid constraint zitB < ceXi for all i and
e. In the integral version of the problem, we will have x* G {0,1}, and the z^e
as non-negative reals. We relax the condition "XJ G {0,1}" to "XJ G [0,1]" and
solve the resultant LP; let y* be the LP's optimal objective function value. We
perform the following rounding steps using a carefully chosen parameter A > 1.
(a) Independently for each i, set a random variable Yi to 1 with probability
Xi/X, and Yi := 0 with probability 1 — Xj/A. (b) If Yi = 1, we will choose to
satisfy (1 — e) of (SJ,£J)'S contract: for all e G E, set Zi<e := ZijB(l — e)/xj. (c)
If Yj, = 0, we choose to have no flow for (si,ti): i.e., we will reset all the z^e to
0. A deterministic version of this result based on pessimistic estimators, is also
provided in [5]; see [5] for further details.
Theorem 3.1 ([5]) Given a network G and a contract set R, we can find
an approximation algorithm for fO/1-VERSION, MAX-#CONTRACTSJ when all
source vertices are the same. Let OPT be the optimum value of the problem,
and m be the number of edges in G. Then, for any given e > 0, we can
in polynomial time find a subset of contracts R' with total weight Q(OPT •
min{(OPT/m)(1_£
)/e
, 1}) such that for all i G R', the flow is at least (1 - e)d*.
4 Experimental Setup and Methodology
To test our algorithms experimentally, we used a network corresponding to a
subset of a real power network along with contracts that we generated using
different scenarios. The network we used is based on the power grid in Colorado
and was derived from data obtained from PSCo's (Public Service Company of
Colorado) Draft Integrated Resources Plan listing of power stations and major
sub stations. The network is shown in Figure 1. We restricted our attention to
major trunks only.
Sources: The location and capacities of the sources was roughly based upon
data obtained from PSCo's Draft Integrated Resources Plan listing of power
stations and major sub stations.
Sinks: The location and capacity of the sinks were roughly based upon the
demographics of the state of Colorado. In order to determine the location and
capacity of the sinks we used the number of households per county obtained from](https://image.slidesharecdn.com/83447-250405070230-5a110f2d/85/Graph-Algorithms-and-Applications-7-Giuseppe-Liotta-24-320.jpg)
![while those on the south are more grotesque. But the capitals of the piers are
the best of all, and the most hurried visitor should spare some time for the
study of these remarkable specimens of sculpture, vigorous and lifelike, yet
always subordinated to their architectural purpose. Those in the transepts[4]
are perhaps the best, but the following in the nave should not be missed:—
North side, Sixth Pier (by north porch): Birds pluming their wings: Beast
licking himself: Ram: Bird with human head, holding knife (?).
“Eighth Pier. Fox stealing goose, peasant following with stick: Birds
pruning their feathers. (Within Bubwith’s Chapel) Human monster with
fish’s tail, holding a fish: Bird holding frog in his beak, which is extremely
long and delicate.
“Ninth Pier. Pedlar carrying his pack on his shoulders, a string of large
beads in one hand. Toothless monster with hands on knees.
“South side, Seventh Pier. Birds with human heads, one wearing a mitre.
“Eighth Pier. Peasant with club, seized by lion: Bird with curious foliated
tail (within St. Edmund’s chapel). Owl: Peasant with mallet (?).”
If we look back towards the west end of the Nave we
note an arcade of five arches, the middle one widest of all
to accommodate the two small arches of the doorway. The
three lancet windows are Perpendicular, remodelled, and
some of their dogtooth moulding, medallions in the
spandrels and little corbel heads of Early English work
remain. There is a gallery below the sill of the window.
The two western towers form two small transepts that
project beyond the aisles. Each is connected with the aisle
by an arch. The Chapel of the Holy Cross under Bubwith’s
Tower (north) is the choir-boys’ vestry. The chapel under
Harewell’s Tower (south) is used by the bell-ringers. An
Early English doorway leads from it into the Cloister.](https://image.slidesharecdn.com/83447-250405070230-5a110f2d/85/Graph-Algorithms-and-Applications-7-Giuseppe-Liotta-37-320.jpg)
Graph Algorithms and Applications 7 Giuseppe Liotta
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- 5. Graph Algorithms and Applications 7 Giuseppe Liotta Digital Instant Download Author(s): Giuseppe Liotta, Roberto Tamassia, Ioannis G. Tollis ISBN(s): 9789812568441, 9812568441 Edition: Kindle File Details: PDF, 20.62 MB Year: 2006 Language: english
- 6. Graph Algorithms and Applications k EDITORS GIUSEPPE LIOTTA ROBERTO TAMASSIA IOANNISGTOLLIS
- 7. Graph Algorithms and Applications k *=* w X~i
- 8. This page is intentionally left blank
- 9. Graph Algorithms and Applications A EDITORS Giuseppe Liotta University of Perugia, Italy Roberto Tamassia Brown University, USA loannis G Tollis University of Crete, ICS-FORTH, Greece and The University of Texas at Dallas, USA JP World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI . CHENNAI
- 10. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. GRAPH ALGORITHMS AND APPLICATIONS 4 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, orparts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-844-1 (pbk) Printed in Singapore.
- 11. Preface This book contains volume 7 of the Journal of Graph Algorithms and Applica- tions (JGAA). Among other papers, the book contains two special issues. Topics of interest for JGAA include: Design and analysis of graph algorithms: exact and approximation graph algo- rithms; centralized and distributed graph algorithms; static and dynamic graph algorithms; internal- and external-memory graph algorithms; se- quential and parallel graph algorithms; deterministic and randomized graph algorithms. Experiences with graph algorithms: animations; experimentations; implementa- tions. Applications of graph algorithms: computational biology; computational geom- etry; computer graphics; computer-aided design; computer and intercon- nection networks; constraint systems; databases; graph drawing; graph embedding and layout; knowledge representation; multimedia; software engineering; telecommunication networks; user interfaces and visualiza- tion; VLSI circuits. JGAA is supported by distinguished advisory and editorial boards, has high scientific standards, and takes advantage of current electronic document tech- nology. The electronic version of JGAA is available on the Web at http://jgaa.info/ We would like to express our gratitude to the members of the advisory board for their encouragement and support of the journal, to the members of the editorial board and guest editors for their invaluable service and to the many anonymous referees for their essential role in the selection process. Finally, we would like to thank all the authors who have submitted papers to JGAA. Giuseppe Liotta Roberto Tamassia Ioannis G. Tollis
- 12. This page is intentionally left blank vi
- 13. Journal of Graph Algorithms and Applications Managing Editor: Giuseppe Liotta, University of Perugia Publication Editor: Emilio Di Giacomo, University of Perugia Editors-in-Chief: Roberto Tamassia, Brown University loannis G. Tollis, University of Crete and ICS-FORTH Advisory Board: J. Chlamtac, CREATE-NET S. Even, Technion G. N. Frederickson, Purdue University T. C. Hu, University of California at San Diego D. E. Knuth, Stanford University C. L. Liu, University of Illinois K. Mehlhom, Max-Planck-Institut fur Informatik T. Nishizeki, Tohoku University F. P. Preparata, Brown University /. H. Sudborough, University of Texas at Dallas R. E. Tarjan, Princeton University M. Yannakakis, Columbia University Editorial Board: S. Albers, Universitat Freiburg L. Arge, University of Aarhus U. Brandes, Universitat Konstanz A. L. Buchsbaum, AT&T Labs - Research G. Di Battista, University of Roma Tre P. Eades, University of Sydney D. Eppstein, University of California at Irvine M. Fiirer, Pennsylvania State University A. Gibbons, King's College M. T. Goodrich, University of California at Irvine
- 14. X. He, State University of New York at Buffalo A. Itai, Technion Y. Kajitani, University of Kitakyushu M. Kaufmann, Universitat Tubingen S. Khuller, University of Maryland S. G. Kobourov, University of Arizona E. W. Mayr, Technischen Universitat Miinchen H. Meijer, Queen's University J. S. B. Mitchell, State University of New York at Stony Brook B. M. E. Moret, University of New Mexico P. Mutzel, Universitat Dortmund B. Raghavachari, University of Texas at Dallas D. Wagner, University of Karlsruhe T. Warnow, University of Texas at Austin S. Whitesides, McGill University vni
- 15. Contents Volume 7:1 (2003) 1 Statistical Analysis of Algorithms: A Case Study of Market-Clearing Mechanisms in the Power Industry. Chris Barrett, Achla Marathe, Madhav Marathe, Doug Cook, Gregory Hicks, Vance Faber, Ar- avind Srinivasan, Yoram Sussmann and Heidi Thornquist. Com- municated by Dorothea Wagner 3 Lower Bounds for the Number of Bends in Three-Dimensional Or- thogonal Graph Drawings. David R. Wood. Communicated by Dorothea Wagner 33 Hamilton Decompositions and (n/2)-Factorizations of Hypercubes. Dou- glas W. Bass and I. Hal Sudborough. Communicated by Balaji Raghavachari 79 Volume 7:2 (2003) 99 Special Issue on Selected Papers from the Seventh International Workshop on Algorithms and Data Structures, WADS 2001. Guest Editor(s): Giuseppe Liotta and loannis G. Tollis. Guest Editors' Foreword. Giuseppe Liotta and loannis G. Tollis. . . . 101 On External-Memory Planar Depth First Search. Lars Arge, Ulrich Meyer, Laura Toma and Norbert Zeh . Communicated by Giuseppe Liotta and loannis G. Tollis 105 Small Maximal Independent Sets and Faster Exact Graph Coloring. David Eppstein. Communicated by Giuseppe Liotta and loannis G. Tollis 131 Deciding Clique-Width for Graphs of Bounded Tree-Width. Wolfgang Espelage, Frank Gurski and Egon Wanke. Communicated by Giuseppe Liotta and loannis G. Tollis 141 Visual Ranking of Link Structures. Ulrik Brandes and Sabine Cor- nelsen. Communicated by Giuseppe Liotta and loannis G. Tollis. 181 An Approach for Mixed Upward Planarization. Markus Eiglsperger, Michael Kaufmann and Frank Eppinger. Communicated by Giuseppe Liotta and loannis G. Tollis 203 Upward Embeddings and Orientations of Undirected Planar Graphs. Walter Didimo and Maurizio Pizzonia . Communicated by Giuseppe Liotta and loannis G. Tollis 221 Volume 7:3 (2003) 243 Crossing Numbers and Cutwidths. Hristo N. Djidjev and Imrich Vrto. Communicated by Giuseppe Liotta 245
- 16. A Multilevel Algorithm for Force-Directed Graph-Drawing. Chris Walshaw . Communicated by Michael Kaufmann 253 Finding Shortest Paths With Computational Geometry. Po-Shen Loh. Communicated by Joseph S. B. Mitchell 287 Volume 7:4 (2003) 305 Advances in Graph Drawing. Special Issue on Selected Papers from the Ninth International Symposium on Graph Drawing, GD 2001. Guest Editor(s): Petra Mutzel and Michael Jiinger. Guest Editors' Foreword. Petra Mutzel and Michael Jiinger. 307 Polar Coordinate Drawing of Planar Graphs with Good Angular Reso- lution. Christian Duncan and Stephen Kobourov. Communicated by Petra Mutzel and Michael Jiinger 311 Orthogonal Drawings of Plane Graphs Without Bends. Md. Saidur Rahman, Takao Nishizeki and Mahmuda Naznin. Communicated by Petra Mutzel and Michael Jiinger 335 Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions. Stefan Felsner, Giuseppe Liotta and Stephen Wis- math. Communicated by Petra Mutzel and Michael Jiinger. . . . 363 Low-Distortion Embeddings of Trees. Robert Babilon, Jin Matousek, Jana Maxovd and Pavel Valtr . Communicated by Petra Mutzel and Michael Jiinger 399 On Cotree-Critical and DFS Cotree-Critical Graphs. Hubert de Frays- seix and Patrice Ossona de Mendez. Communicated by Petra Mutzel and Michael Jiinger 411
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- 19. Journal of Graph Algorithms and Applications http://jgaa.inf0/ vol. 7, no. 1, pp. 3-31 (2003) Statistical Analysis of Algorithms: A Case Study of Market-Clearing Mechanisms in the Power Industry Chris Barrett Achla Marathe Madhav Marathe Los Alamos National Laboratory Doug Cook Colorado School of Mines Gregory Hicks North Carolina State University Vance Faber Mapping Sciences Inc. Aravind Srinivasan University of Maryland, College Park Yoram Sussmann State University of West Georgia Heidi Thornquist Rice University Abstract We carry out a detailed empirical analysis of simple heuristics and provable algorithms for bilateral contract-satisfaction problems. Such problems arise due to the proposed deregulation of the electric utility industry in the USA. Given a network and a (multi)set of pairs of vertices (contracts) with associated demands, the goal is to find the maximum number of simultaneously satisfiable contracts. Four different algorithms (three heuristics and a provable approximation algorithm) are considered and their performance is studied empirically in fairly realistic settings us- ing rigorous statistical analysis. For this purpose, we use an approximate electrical transmission network in the state of Colorado. Our experiments are based on the statistical technique Analysis of Variance (ANOVA), and show that the three heuristics outperform a theoretically better algorithm. We also test the algorithms on four types of scenarios that are likely to occur in a deregulated marketplace. Our results show that the networks that are adequate in a regulated marketplace might be inadequate for satisfying all the bilateral contracts in a deregulated industry. Communicated by Dorothea Wagner: submitted April 2002; revised December 2002.
- 20. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 4 1 Introduction The U.S. electric utility industry is undergoing major structural changes in an effort to make it more competitive [21,17,19,11]. One major consequence of the deregulation will be to decouple the controllers of the network from the power producers, making it difficult to regulate the levels of power on the network; consumers as well as producers will eventually be able to negotiate prices to buy and sell electricity [18]. In practice, deregulation is complicated by the facts that all power companies will have to share the same power network in the short term, with the network's capacity being just about sufficient to meet the current demand. To overcome these problems, most U.S. states have set up an ISO (independent system operator): a non-profit governing body to arbitrate the use of the network. The basic questions facing ISOs are how to decide which contracts to deny (due to capacity constraints), and who is to bear the costs accrued when contracts are denied. Several criteria/policies have been proposed and/or are being legislated by the states as possible guidelines for the ISO to select a maximum-sized subset of contracts that can be cleared simultaneously [18]. These include: (a) Minimum Flow Denied: The ISO selects the subset of contracts that denies the least amount of proposed power flow. This proposal favors clearing bigger contracts first, (b) First-in First-out: The contract that comes first gets cleared first; this is the least discriminating to the contractors, (c) Maximum Consumers Served: This clears the smallest contracts first and favors the small buyers whose interests normally tend to go unheard. There are three key issues in deciding policies that entail specific mecha- nisms for selecting a subset of contracts: fairness of a given policy to producers and consumers; the computational complexity of implementing a policy, and how sound a given policy is from an economic standpoint. (For instance, does the policy promote the optimal clearing price/network utilization etc.) Here we focus on evaluating the efficacy of a given policy with regard to its computa- tional resource requirement and network resource utilization. It is intuitively clear that the underlying network, its capacity and topology, and the spatial locations of the bilateral contracts on the network, will play an important role in determining the efficacy of these policies. We do not discuss here the fair- ness and economics aspects of these policies: these are subjects of a companion paper. The work reported here is done as part of a simulation based analyt- ical tool for deregulated electrical power industry being developed at the Los Alamos National Laboratory. We experimentally analyze several algorithms for simultaneously clearing a maximal number of bilateral contracts. The qualitative insights obtained in this paper can be useful to policy makers who carry the ultimate responsibility of deploying the best clearing mechanism in the real world. The algorithms were chosen according to provable performance, ability to serve as a proxy for some of the above-stated policies, and computational requirement. The algorithms are as follows; see § 3 for their specification. The ILP-RANDOMIZED ROUNDING (RR) algorithm has a provable performance guarantee under certain conditions. The computational resource requirement is quite high, but the approach also
- 21. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 5 provides us with an upper bound on any optimal solution and proves useful in comparing the performance of the algorithms. The LARGEST-FIRST HEURISTIC (LF) is a proxy for the Minimum Flow Denied policy. The SMALLEST-FIRST HEURISTIC (SF) serves as a proxy for the Maximum Contracts Served policy. The RANDOM-ORDER HEURISTIC (RO) clears the contracts in the random or- der. This algorithm was chosen as a proxy for the First-in First-out policy. Such a policy is probably the most natural clearing mechanism and is currently in place at many exchanges. To compare the algorithms in a quantitative and (semi-)rigorous way, we employ statistical tools and experimental designs. Many of the basic tools are standard in statistics and their use is common in other fields. But to the best of our knowledge, the use of formal statistical methods in experimental algo- rithmics for analyzing/comparing the performance of algorithms has not been investigated. Analysis of Variance (ANOVA) is one such technique that can help identify which algorithms and scenarios are superior in performance. We believe that such statistical methods should be investigated further by the experimen- tal algorithmics community for deriving more (semi)-quantitative conclusions when theoretical proofs are hard or not very insightful. For instance, consider a given approximation algorithm that has a worst-case performance guarantee of p. First, the algorithm may perform much better on realistic instances that are of interest. Quantifying the special structure of such instances is often hard; this often makes it difficult to develop further theoretical improvements on the per- formance of the algorithm. Second, many heuristics that have poor worst-case performance perform very well on such instances. Statistical methods such as ANOVA can facilitate the comparison of such heuristics and provable algorithms in settings that are of interest to the users of such algorithms. We used a coarse representation of the Colorado electrical power network (see § 4) to qualitatively compare the four algorithms discussed above in fairly realistic settings. The realistic networks differ from random networks and struc- tured networks in the following ways: (i) Realistic networks typically have a very low average degree. In fact, in our case the average degree of the network is no more than 3. (ii) Realistic networks are not very uniform. One typically sees one or two large clusters (downtown and neighboring areas) and small clusters spread out throughout, (iii) For most empirical studies with random networks, the edge weights are chosen independently and uniformly at random from a given interval. However, realistic networks typically have very specific kinds of capacities since they are constructed with particular design goal. From our preliminary analysis, it appears that although the simple heuris- tic algorithms do not have worst-case performance guarantees, they outperform the theoretically better randomized rounding algorithm. We tested the algo- rithms on four carefully chosen scenarios. Each scenario was designed to test the algorithms and the resulting solutions in a deregulated setting. The em- pirical results show that networks that are capable of satisfying all demand in a regulated marketplace can often be inadequate for satisfying all (or even a acceptable fraction) of the bilateral contracts in a deregulated market. Our re- sults also confirm intuitive observations: e.g., the number of contracts satisfied
- 22. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 6 crucially depends on the scenario and the algorithm. As far as we are aware, this is the first study to investigate the efficacy of various policies for contract satisfaction in a deregulated power industry. Since it was done in fairly realistic settings, the qualitative results obtained here have implications for policy makers. Our results can also be applied in other settings, such as bandwidth-trading on the Internet. See, e.g., [2]. Also, to our knowledge, previous researchers have not considered the effect of the underlying network on the problems; this is an important parameter especially in a free-market scenario. The rest of this paper is organized as follows. The problem definitions and algorithms considered are described in Sections 2 and 3 respectively. Our experimental setup is discussed in Section 4. Section 5 presents our experimental results and analyzes them and Section 6 concludes the paper. In the appendix, we discuss interesting optimization issues that arise from deregulation, and also show problem instances on which our algorithms do not perform well. 2 Problem Definitions We briefly define the optimization problems studied here. We are given an undirected network (the power network) G = (V, E) with capacities ce for each edge e and a set of source-sink node pairs (si,ti), 1 < i < k. Each pair (si,ti) has: (i) an integral demand reflecting the amount of power that Sj agrees to supply to U and (ii) a negotiated cost of sending unit commodity from s, to tj. As is traditional in the power literature, we will refer to the source-sink pairs along with the associated demands as a set of contracts. In general, a source or sink may have multiple associated contracts. We find the following notation convenient to describe the problems. Denote the set of nodes by N. The contracts are defined by a relation R C (2V x T V x 3? x 3?) so that tuple (v, w, a, /3) e i ? denotes a contract between source v and sink w for a units of commodity at a cost of /3 per unit of the commodity. For A = (u, w, a, (3) e R we denote source(A) = v, sink(A) = w, flow(A) = a and cost(A) = (3. Corresponding to the power network, we construct a digraph H = (VliSUTU {s, t}, E') with source s, sink node t, capacities u : E' — > 3? and costs c' : E' —> 3? as follows. For all A e R, define new vertices VA and WA- Let S = {VA A G R} and T = {WA A s R}. Each edge {x, y} from G is present in H as the two arcs (x, y) and (y, x) that have the same capacity as {x, y} has in G, and with cost 0. In addition, for all A = (v,w,a,[3) € R, we introduce: (i) arcs (VA,V) and (w, WA) with infinite capacity and zero cost; (ii) arc (S,VA) with capacity flow{A) = a and cost 0; and (iii) arc (wA,t) with capacity flow(A) = a and cost equaling cost(A). By this construction, we can assume without loss of generality that each node can participate in exactly one contract. A flow is simply an assignment of values to the edges in a graph, where the value of an edge is the amount of flow traveling on that edge. The value of the flow is defined as the amount of flow coming out of s (or equivalently the amount of flow coming in to t). A generic feasible flow f = (fx,y > 0 : (x, y) € E') in
- 23. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 7 H is any non-negative flow that: (a) respects the arc capacities, (b) has s as the only source of flow and t as the only sink. Note that for a given A £ R, in general it is not necessary that fStVA = fWA,t- For a given contract A £ R, A is said to be satisfied if the feasible flow f in H has the additional property that for A = (v,w,a,/3), fs,vA = fwA,t = a ', i-e -> the contractual obligation of a units of commodity is shipped out of v and the same amount is received at w. Given a power network G(V,E), a contract set R is feasible (or satisfied) if there exists a feasible flow / in the digraph H that satisfies every contract A £ R. Let B = supply(s) = demand(t) = ^^ejj/^ow;(A). In practice, it is typically the case that R does not form a feasible set. As a result we have two possible alternative methods of relaxing the constraints: (i) relax the notion of feasibility of a contract and (ii) try and find a subset of contracts that are feasible. Combining these two alternatives we define the following types of "relaxed feasible" subsets of R. We will concern ourselves with only one variant here. A contract set R! C R is a 0/1-contract satisfaction feasible set if, MA = (v,w,a,/3) £ R', fs,VA = fWA,t = a. Definition 2.1 Given a graph G(V, E) and a contract set R, the (0/1-VERSION MAX-FEASIBLE FLOW,) problem is to find a feasible flow f in H such that J^AeR' / ( ^ ) ** maximized where R' forms a 0/1-contract satisfaction feasible set of contracts. In the related (0/1-VERSION, MAX-#CONTRACTSJ problem, we aim to find a feasible flow f in H such that R' is maximized, where R! forms a 0/1-contract satisfaction feasible set of contracts. Though such electric flow problems have some similarities with those from other practical situations, there are many basic differences such as reliability, indistinguishability between the power produced by different generators, short life-time due to inadequate storage, line effects etc. [22]. The variants of flow problems related to power transmission studied here are intuitively harder than traditional multi-commodity flow problems, since we cannot distinguish between the flow "commodities" (power produced by different generators). As a result, current solution techniques used to solve single/multi-commodity flow problems are not directly applicable to the problems considered here. 3 Description and Discussion of Algorithms We work on the (0/1-VERSION, MAX-#CONTRACTS) problem here. Let n and m respectively denote the number of vertices and edges in the network G. In [5], it was shown that (0/1-VERSION, MAX-#CONTRACTS) is JVP-hard; also, unless NP C ZPP, it cannot be approximated to within a factor of m1 /2 - 6 for any fixed e > 0, in polynomial time. Thus, we need to consider good heuris- tics/approximation algorithms. First, there are three simple heuristics. The SMALLEST-FIRST HEURISTIC considers the contracts in non-decreasing order of their demands. When a contract is considered, we accept it if it can be feasi- bly added to the current set of chosen contracts, and reject it otherwise. The LARGEST-FIRST HEURISTIC is the same, except that the contracts are ordered
- 24. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 8 in non-increasing order of demands. In the RANDOM-ORDER HEURISTIC, the contracts are considered in a random order. We next briefly discuss an approximation algorithm of [5]. This has proven performance only when all source vertices Sj are the same; however, this algo- rithm extends naturally to the multi-source case which we work on. An integer linear programming (ILP) formulation for the problem is considered in [5]. We have indicator variables Xj for the contract between Sj and ti, and variables Zj>e for each (sj,ij) pair and each edge e. The intended meaning of x» is that the total flow for (s8, U) is diXf, the meaning of z^e is that the flow due to the con- tract between (sj,ij) on edge e is Zi>e. We write the obvious flow and capacity constraints. Crucially, we also add the valid constraint zitB < ceXi for all i and e. In the integral version of the problem, we will have x* G {0,1}, and the z^e as non-negative reals. We relax the condition "XJ G {0,1}" to "XJ G [0,1]" and solve the resultant LP; let y* be the LP's optimal objective function value. We perform the following rounding steps using a carefully chosen parameter A > 1. (a) Independently for each i, set a random variable Yi to 1 with probability Xi/X, and Yi := 0 with probability 1 — Xj/A. (b) If Yi = 1, we will choose to satisfy (1 — e) of (SJ,£J)'S contract: for all e G E, set Zi<e := ZijB(l — e)/xj. (c) If Yj, = 0, we choose to have no flow for (si,ti): i.e., we will reset all the z^e to 0. A deterministic version of this result based on pessimistic estimators, is also provided in [5]; see [5] for further details. Theorem 3.1 ([5]) Given a network G and a contract set R, we can find an approximation algorithm for fO/1-VERSION, MAX-#CONTRACTSJ when all source vertices are the same. Let OPT be the optimum value of the problem, and m be the number of edges in G. Then, for any given e > 0, we can in polynomial time find a subset of contracts R' with total weight Q(OPT • min{(OPT/m)(1_£ )/e , 1}) such that for all i G R', the flow is at least (1 - e)d*. 4 Experimental Setup and Methodology To test our algorithms experimentally, we used a network corresponding to a subset of a real power network along with contracts that we generated using different scenarios. The network we used is based on the power grid in Colorado and was derived from data obtained from PSCo's (Public Service Company of Colorado) Draft Integrated Resources Plan listing of power stations and major sub stations. The network is shown in Figure 1. We restricted our attention to major trunks only. Sources: The location and capacities of the sources was roughly based upon data obtained from PSCo's Draft Integrated Resources Plan listing of power stations and major sub stations. Sinks: The location and capacity of the sinks were roughly based upon the demographics of the state of Colorado. In order to determine the location and capacity of the sinks we used the number of households per county obtained from
- 25. C. Barrett et al., Statistical Analysis of Algorithms, JGAA, 7(1) 3-31 (2003) 9 Figure 1: This shows the network with node numbered as they are referenced in all scenarios and edge capacities labeled at values used for Scenarios 1 & 2. The placement of the nodes and edges are what is probably the final form. The least number of edges cross and the nodes in the upper right are spread out a little bit maintaining the general feel of the distribution while allowing easier reading.
- 26. Discovering Diverse Content Through Random Scribd Documents
- 27. In 1318 the canons voluntarily offered a fifth of their salaries to raise the central tower, which was carried up three more stages and finished in 1321; and in 1325 they began new stalls, each canon having agreed to pay for his own stall. In 1337 and 1338 the whole church was thrown into dismay on account of fractures in the tower; for the tower appears to have sunk deeply into the earth, owing to pressure on the arches. All the masonry was disturbed; and in order to remedy this trouble, the curious double arches were inserted, to help support the strain. The original arches were also patched up and filled in with great blocks of stone and strengthened in various ways. Much was due to Bishop Ralph of Shrewsbury (1329- 1363), who was buried before the High Altar in the Choir he had founded. He also finished the Palace begun by Jocelin. Bishop Harewell, who died in 1386, gave two- thirds of the cost of the south-west tower called by his name; and the executors of Bishop Bubwith finished the northwest tower that bears his name. Bishop Beckington built the lovely gateways, and Dean Gunthorpe (died 1498), the Deanery. The eastern walk of the Cloister and the Library above date from between 1407 and 1424; and the western and southern Cloister walks, between 1443 and 1464. “Late researches have shown that Bishop Reginald began the present church and that the Early English work should be divided into four periods: (1) The three western arches of the choir, with the four western bays of its aisles, the transepts and the four eastern bays of the nave, which are
- 28. Reginald’s work (1174-1191), and so early as to be still in a state of transition from the Norman. It is a unique example of transitional building, and Willis calls it ‘an improved Norman, worked with considerable lightness and richness, but distinguished from the Early English by greater massiveness and severity.’ The characteristics of this late Twelfth Century work are bold round mouldings, square abaci, capitals, some with traces of the classical volute, others interwoven with fanciful imagery that reminds us of the Norman work of Glastonbury; while in the north porch, which must be the earliest of all, we even find the zigzag Norman moulding. (2) The rest of the nave, which was finished in Jocelin’s time—that is to say, in the first half of the Thirteenth Century—preserves the main characteristics of the earlier work, though the flowing sculptured foliage becomes more naturalistic, and lacks the quaint intermingling of figure subjects. (3) The west front, which is Jocelin’s work, and alone can claim to be of pure Early English style. (4) The chapter-house crypt, which is so late as to be almost Transitional, though, curiously enough, it contains the characteristic Early English dog- tooth moulding which is found nowhere else except in the west window. From this, we reach the Early Decorated of the staircase, the full Decorated of the chapter-house itself, the later Decorated of the Lady-Chapel, the transitional Decorated of the presbytery, and the full Perpendicular of the western towers. Much of the masonry in the transepts, choir, choir aisles, and even in the eastern transepts, bears the peculiar diagonal lines which are the marks of Norman tooling. This does not, of course, prove that any part of Bishop Robert’s church is standing, for mediæval builders were notoriously economical in using up old masonry, but it does show that there are more remains of his work in the building than was generally supposed.”—(P. D.) The Cathedral was much damaged during the Reformation and also during Monmouth’s rebellion in 1685, when the Duke’s followers stabled their horses in it and enjoyed a barrel of beer on the high altar. There is a nave of nine bays, a space under the tower, a choir opening eastward of it and two transepts (each of four bays) with aisles opening north and south. The choir from
- 29. the screen to the high altar occupies six bays; a retro-choir of two bays lies behind the altar; and beyond it again is an apsidal Lady-Chapel. The west front has been much admired, but some critics consider it too heavy for the short towers that abut on it. The windows of the nave and transepts are Decorated. The windows of the choir are more ornate, although in the same style, and those of the Lady-Chapel are still more so. The central tower (Perpendicular) is entirely covered with panelling. There is no spire. On the south side large cloisters open from the south-western tower and from the western aisle of the south transept; but there are only three walks, there being none on the north side. The Chapter-House is approached from the north side of the choir by a short passage and a flight of steps: a crypt lies under it. A beautiful porch, with parvise, opens into the sixth bay of the north aisle. From the eastern aisle of the north transept the Chain-Gate passes to the Vicars’ College, a double row of picturesque houses, dating from 1360. “The Chain-Gate, in its association with the Chapter-House and the Vicars’ Close, is unique. The incline of the steps, easily to be distinguished from without, gives the corner a character quite its own. And the entrance to the Green by this gate, with the Cathedral on one side, balanced by the varied gables and roofs of the houses opposite, is particularly striking. The exterior of the Chapter-House comes into full view; the great central tower stands boldly up against the sky; the eastern gable presents its curious apex, and the Lady-Chapel below stands like a thing separate from the rest. Beyond, and under the Chain-Gateway, an arch admits to the Vicars’ Close —a charming street, lined on either side with diminutive dwelling-houses, once the separate residences of the vicars choral. At the top of the close is a
- 30. small Perpendicular chapel with a library above. The interior is profusely— almost grotesquely—decorated in a manner to remind one to some extent of those strange little oratories so frequently met with in other parts of Europe. But to many it will possess a certain charm, despite its florid adornments, not often realised in this country. The Vicars’ Hall, a considerable portion of which is of the Fourteenth Century, with additions of a tower and other features, probably by Bishop Beckington, stands at the bottom of the street and communicates through the gallery of the Chain-Gate with the Chapter- House staircase, and thus with the cathedral. By this gallery the choristers passed into the church.”—(A. A.) The celebrated West Front “consists of a centre, in which are the three lancets of the western window and above them a gable receding in stages, with small pinnacles at the angles; and of two wings or western towers, projecting beyond the nave, as at Salisbury. The upper part of these towers is of Perpendicular character. That to the north-west was completed by Bishop Bubwith (1407-1424), whose statue remains in one of the niches: that to the south-west was the work of Bishop Harewell (1366-1386). Both these towers, fine as are their details, have a somewhat truncated appearance; and it is probable that the original Early English design terminated at the uppermost band of sculpture. The three western doors are of unusually small dimensions, perhaps in order to leave ample room for the tiers of figures which rise above them. Six narrow buttresses at the angles of which are slender shafts of Purbeck marble, supporting canopies, divide the entire front into five portions. The whole of the statues which fill the niches are of Doulting stone.”—(R. J. K.) Many visitors are at the first sight disappointed at the mutilated and archaic expression of the figures; but they have commanded the greatest admiration ever since old Fuller wrote: “The west front of Wells is a masterpiece of art indeed, made of imagery in just proportion, so that we may call them vera et spirantis signa. England affordeth not the like.”
- 31. The West Front should be considered as a great screen intended for the display of statuary rather than as the west termination of the nave. The stone population, numbering about three hundred life-size or colossal figures, is only equalled by that of Rheims and that of Chartres. All critics agree that these statues, so notable for their graceful draperies and spiritual expressions, rank with the contemporary masterpieces of Italy and France. They are thought to have been made by Italian sculptors at the time when Niccola Pisano was reviving sculpture in Italy under the inspiration of classical models. The kings, queens, princes, knights and nobles wear the costume of the Thirteenth Century. The other figures are prophets, angels, martyrs and “the holy church throughout the world.” Unlike the monumental west fronts of France, with their splendid porches and doors, the doors of Wells have been compared to “rabbit-holes on a mountain-side.” The western towers projecting beyond the aisles of the nave give additional breadth to the west front. The arrangement resembles that of Rouen. The two towers are very similar. Both have two belfry windows on each side and a stair turret on the outer western angle. The spires were never added. The Central Tower is Early English to the level of the roof, and the two upper stages are Decorated. From its summit a beautiful view is to be enjoyed. The North Porch (Norman) is the oldest part
- 34. Wells: North Porch of the church. Some architects consider it the finest piece of architecture at Wells. “The entrance is doubly recessed and has the zigzag ornament among its mouldings, an indication, if not of its early construction, at least of lingering Norman traditions among its builders. These mouldings deserve the most careful attention. The outer or dripstone, is formed of a very beautiful combination of Early English foliage. Square panels on either side of the arch contain figures of mystic animals, one of which is a cockatrice. The gable above has a blind arcade, in the centre of which a small triplet gives light to a parvise chamber. From the buttress at the angles rise slender spire- capped pinnacles. The buttresses themselves are flat and narrow. “The interior of the porch is divided into two bays, and its walls are lined with a double arcade, the upper row of arches being more deeply recessed than the tower. The vault springs from a central group of triple shafts. The sculptures of the capitals on the east side possibly represent the death of King Edmund the Martyr (A.D. 870),—bound to a tree as a mark for the Danish arrows and afterwards beheaded. The figures are well designed, and full of life and character. The double doorway leading into the nave displays, like the exterior arch, the Norman zigzag.”—(R. J. K.) On entering the Nave the visitor is at once struck by the noble proportions, the impression of great length, the broad horizontal band of the triforium, and the wealth of spirited and varied carving of the capitals and corbels; but the most striking feature of all is the great inverted, or double, arch that struts across the central piers forming a St. Andrew’s Cross, by which name it is generally known, and giving a grotesque (we are almost tempted to say Chinese) appearance.
- 35. “Undoubtedly the first thing that the stranger notices in Wells Cathedral, and the last that he is likely to forget, is the curious contrivance by which the central tower is supported. Of the three pairs of arches (the upper arch resting inverted upon the lower) which stretch across the nave and each of the transepts, that in the nave is seen at once, and lends a unique character to the whole church. At first these arches give one something of a shock, so unnecessarily frank are they, so excessively sturdy, so very English, we may think. They carry their burden as a great-limbed labourer will carry a child in a crowd, to the great advantage of the burden and the natural dissatisfaction of the crowd. In fact, they seem to block up the view, and to deform what they do not hide. “That is the first impression, but it does not last for long. Familiarity breeds respect for this simple, strong device, which arrested the fall of the tower in the Fourteenth Century, and has kept its walls ever since in perfect security, so that the great structure has stood like a rock upon the watery soil of Wells for nearly seven centuries, with its rents and breaks just as they were when the damage was first repaired. The ingenuity, too, of these strange flying-buttresses becomes more and more evident; the ‘ungainly props’ are seen to be so worked into the tower they support, that they almost seem like part of the original design of the first builders. One discovers that it is the organ, and not the arches, that really blocks the view, and one marvels that so huge a mass of masonry can look so light as to present, with the great circles in the spandrels where the arches meet, a kind of pattern of gigantic geometrical tracery. Indeed I think no one who has been in Wells a week could wish to see the inverted arches removed. “To appreciate the work fully, it should be looked at from some spot, such as the north-east corner of the north transept, whence the three great pairs of arches can be seen together. The effect from here is very fine, especially when the nave is lighted up and strong shadows are cast. The extreme boldness of the mouldings, the absence of shafts and capitals and of all ornament, give them a primitive vigour, and their great intermingling curves, which contrast so magnificently with the little shafts of the piers beyond, seem more like a part of some great mountain cavern than a mere device of architectural utility.”—(P. D.)
- 36. The general effect of the Nave is that of length rather than height, largely due to the continuous arcade of the triforium which leads the eye irresistibly eastwards, and the comparatively restricted height of the Cathedral has been increased by bold vaulting, and by the way the lantern arches fit into the vault. A little study will show the visitor the separation between the late Twelfth Century work of Reginald de Bohun, or Fitz-Jocelyn, and the Thirteenth Century work of Jocelin. These differences lie in the masonry and the carved heads and the capitals. The heads of a king and bishop, projecting from the south side between the fourth and fifth piers, mark the point of change eastward: the masonry of piers, walls and aisle walls is in small courses of stone; westward, the blocks are larger, eastward, small human heads project at the angles of the pier-arches and westward there are none; eastward, the tympana of the triforium arcade are filled with carvings of grotesque animals and small heads at the corners, and westward, the tympana are filled with foliage and ornamented with larger heads. There are also other differences. “Certainly it is an unusual instance of an architect deliberately setting himself to complete the works of an earlier period in faithful accordance with the original plan; and we may well be grateful to him for his modesty. “All the carving is most interesting and beautiful: the caps and corbels of the vaulting shafts; the little heads at the angles of the arches, which are vivid sketches of every type of contemporary character; and the carvings in the tympana, which are best in the seventh, eighth and ninth bays (counting from the west end), those on the north excelling in design and execution,
- 37. while those on the south are more grotesque. But the capitals of the piers are the best of all, and the most hurried visitor should spare some time for the study of these remarkable specimens of sculpture, vigorous and lifelike, yet always subordinated to their architectural purpose. Those in the transepts[4] are perhaps the best, but the following in the nave should not be missed:— North side, Sixth Pier (by north porch): Birds pluming their wings: Beast licking himself: Ram: Bird with human head, holding knife (?). “Eighth Pier. Fox stealing goose, peasant following with stick: Birds pruning their feathers. (Within Bubwith’s Chapel) Human monster with fish’s tail, holding a fish: Bird holding frog in his beak, which is extremely long and delicate. “Ninth Pier. Pedlar carrying his pack on his shoulders, a string of large beads in one hand. Toothless monster with hands on knees. “South side, Seventh Pier. Birds with human heads, one wearing a mitre. “Eighth Pier. Peasant with club, seized by lion: Bird with curious foliated tail (within St. Edmund’s chapel). Owl: Peasant with mallet (?).” If we look back towards the west end of the Nave we note an arcade of five arches, the middle one widest of all to accommodate the two small arches of the doorway. The three lancet windows are Perpendicular, remodelled, and some of their dogtooth moulding, medallions in the spandrels and little corbel heads of Early English work remain. There is a gallery below the sill of the window. The two western towers form two small transepts that project beyond the aisles. Each is connected with the aisle by an arch. The Chapel of the Holy Cross under Bubwith’s Tower (north) is the choir-boys’ vestry. The chapel under Harewell’s Tower (south) is used by the bell-ringers. An Early English doorway leads from it into the Cloister.
- 38. “The nave, as far as the piers of the central tower, consists of ten bays, divided by octangular piers, with clustered shafts in groups of three. The capitals are enriched with Early English foliage, much of which is of unusually classical character,—one of the many indications of a lingering local school, with its Norman traditions. Birds, animals and monsters of various forms—among which is the bird with a man’s face, said to feed on human flesh—twine and perch among the foliage. Above the pier arches runs the triforium, very deeply set, and extending backward over the whole of the side aisles. The roof retains its original position. (The whole arrangement should be compared with the Norman triforia of Norwich and Ely, both of which extend over the side-aisles; but their exterior walls have been raised and Perpendicular windows inserted). The narrow lancet openings toward the nave are arranged in groups of three, with thick wall- plates between them. The head with each lancet is filled with a solid tympanum, displaying foliage and grotesques, of which those toward the upper end of the south side are especially curious. At the angles of the lancets are bosses of foliage and human heads, full of character. In the upper spaces between each arch are medallions with leafage. Triple shafts, with enriched capitals, form the vaulting-shafts, the corbels supporting which deserve examination. A clerestory window (the tracery is Perpendicular, and was inserted by Bishop Beckington (1443-1464)) opens between each bay of the vaulting, which is groined, with moulded ribs and bosses of foliage at the intersections.”—(R. J. K.) In the clerestory of the sixth bay on the south side there is a Music Gallery, early Perpendicular, the front of which consists of three panels with large quartrefoils containing shields. It is very fine, but not equal to the Minstrels’ Gallery in Exeter. It is finished with an embattled cornice. The aisles of the Nave are of the same architectural character as the Nave itself. Among the striking capitals are: Fifth shaft. Peasants carrying sheep, with a dog.
- 39. Ninth shaft. Man in a rough coat carrying foliage on his back. Tenth shaft. Mason carrying a hod of mortar and a mallet; opposite side of arch: Peasant in hood with staff and opposite this two heads, evidently with toothache. The greater part of the glass of the West Window was collected by Bishop Creyghton in 1660-1670, excellent Sixteenth Century representations of the history of John the Baptist. Possibly Creyghton added the figures of King Ina and Bishop Ralph in the other lights, for the southern one also bears his arms. The top and bottom of the middle light are said to have come from Rouen in 1813. Now we will examine the transepts. “The transepts seem to have been built before the nave, but some of the carved work of the capitals and corbels is of later date than the nave. The capitals on the west side of both transepts are among the finest in England. Many refer to the toothache. “North Transept: first Pier.—(Inside the Priest Vicars’ vestry) A prophet(?) with scroll on which there is no name: Man carrying goose. (Outside) Head with tongue on teeth. “Second Pier.—Aaron writing his name on a scroll: Moses with the tables of stone. “Third Pier.—Woman with a bandage across her face. Above this cap the corbel consists of a seated figure, naked, with distorted mouth and an agonised expression. “South Transept, second pier (from the south end). Two men are stealing grapes, one holds the basket full, the other plucks grapes, holding a knife in his other hand: The farmers in pursuit, one carries a spade and
- 40. the other a pitchfork: The man with the fork, a vigorous figure, catches one thief: The man with the spade hits the other (whose face is most woe-begone) on the head. “Third pier.—Woman pulling thorn out of her foot: Man with one eye, finger in his mouth: Baboon head: Cobbler; this figure shows very plainly the method of shoemaking at this time; the cobbler in his apron, sits with the shoe on one knee, his strap passes over the knee and round the other foot, his foot is turned over so as to present the side and not the sole to the strap: Woman’s head with long hair. “Fourth pier.—Head perfectly hairless: Elias P. (the prophet) with hand on cheek as if he, too, has the toothache: Head in hood, with tongue on the one remaining tooth. “It may be well here to say a word about the general classification of these earlier capitals, since their date is a matter of great architectural interest. I would venture to divide them into five groups— “(1) Those of the three western bays of the choir: simple carved foliage of distinctly Norman character, as in the north porch: these belong to the time of Reginald (1174- 1191). “(2) The four eastern bays of the nave and its aisles. Some of these may belong to the first period, though later than the choir: they are more advanced in the foliage, and teem with grotesque birds and beasts. Some, however, of the caps in these bays are of quite different character; they
- 41. contain genre subjects of perfectly naturalistic treatment, very different to the St. Edmund of the north porch capital; but exactly similar to the figure caps of the transepts. They must therefore have been carved later than the death of Saint William Bytton. “(3) The western bays of the nave. These, which are of much less interest, belong to the period of Jocelin’s reconstruction (1220-1242). They are characteristic examples of rich stiff-leaf foliage, freer than that of the earlier work, but much less varied and without either human figures or grotesques. “(4) On the eastern range of transept piers. These would seem also to come within Jocelin’s period, with the exception of the third pier of the south transept. “(5) On the western range of transept piers, with which must be classed those later caps already referred to in the nave under group 2. Their date is settled by the fact that they abound in unmistakable representations of the toothache. Now Saint William Bytton died in 1274, and his tomb became immediately famous for cures of this malady. In 1286, the chapter decided to repair the old work, no doubt because the offerings at his tomb had brought money to the church.”—(P. D.) In studying these fascinating grotesques, however, we have neglected to examine the two chantries in the nave— Bishop Bubwith’s and Dean Sugar’s. They are opposite one another and are alike in general characteristics. The screen work and cornices of Bubwith’s composed of light and elaborate tracery are very much admired. Light doorways permit entrance. The altar here was dedicated to
- 42. St. Saviour. Bishop Bubwith (who built the north-west tower) died in 1424. His arms, containing holly-leaves, are beautifully carved. Sugar’s Chantry, about sixty years later in date, is even more elaborate. Like Bubwith’s, it is hexagonal and the canopy over the altar is vaulted with delicate fan-tracery. Critics now consider it the finer of the two. Adjoining Sugar’s Chantry the stone Pulpit, built in the reign of Henry VIII., calls for attention. In front are the arms of Bishop Knight, who built it and who is buried near it (he died in 1547). Beside it, is a brass lectern presented in 1660; upon this rests a Bible of the same date. In the South transept, we find the Font, interesting because it is the one relic of Bishop Robert’s Norman church. It may have stood in the earlier Saxon cathedral. The cover is Jacobean. In the south end of the south transept is the Tomb of Bishop de Marchia (died 1302). The effigy of the bishop, lying in a recess under a canopy bristling with crockets and finials and brilliant with scarlet and crimson, green and gold, is very striking. Some of the angels surrounding the figure are charming. It is interesting to compare this with the Tomb of Lady Lisle, also adorned with crockets and brightly coloured. Perpendicular stone screens divide the transepts from their small chapels. The chapels of the south transept are St. Martin’s (now the canon’s vestry) and that of St.
- 43. Calixtus, enclosed on the side of the choir-aisle by some beautiful ironwork from Beckington’s tomb. On the south side of St. Calixtus’s chapel we must pause to examine Dean Husse’s tomb, of alabaster, and noted for its carved panels even in this cathedral of splendid carvings. St. David’s Chapel in the north transept compels us to pause again to look at the capital of the second transept pier —a handsome head with curls and a smile on his face— and a fine corbel carved into the form of a lizard eating leaves of a plant with berries. In this chapel lies an interesting effigy of Bishop Still (1543-1607) in a red robe lined with white fur. Next comes the Chapel of the Holy Cross in which is the tomb of Bishop Cornish (died 1513), thought also to have been used as the Easter Sepulchre, where the Host was laid during Holy Week. The north transept contains a relic of the past that delights every one who happens to be there at the striking of the hour. The famous clock that once belonged to Glastonbury Abbey is still in working order. A little figure known locally as “Jack Blandiver” kicks the quarters with his heels on two little bells and at the hour four figures on horseback above the clock rush around and charge each other. The curious clock was made by Peter Lightfoot, a monk of the abbey. It was said to have been in constant use at Glastonbury for 250 years before it was removed to Wells at the Dissolution of the monasteries.
- 44. From the east aisle of the north transept a door opens to the Staircase that leads to the Chapter-House and also to the celebrated Chain-Gate, or carved bridge that connects the Vicars’ College with the Cathedral. Through this gallery the Vicars could pass from their own Close into the Cathedral. The common hall of their college (1340) opens from it. “There are few things in English architecture that can be compared with it for strange impressive beauty; the staircase goes upward for eighteen steps and then part of it sweeps off to the Chapter-house on the right, while the other part goes on and up till it reaches the chain-bridge; thus the steps lie, worn here and there by the tread of many feet, like fallen leaves, the last of them lost in the brighter light of the bridge. Here one is still almost within the cathedral, and yet the carts are passing underneath, and their rattle mixes with the sound of the organ within. “The main gallery of the Chain-Gate is shut off by a door, which, if it were kept open, would make the prospect even more beautiful than it is. Two corbels which support the vaulting-shafts of the lower staircase should be noticed; they both represent figures thrusting their staves into the mouth of a dragon, but that on the east (wearing a hood and a leathern girdle round his surcoat) is as vigorous in action as the figure on the west side is feeble. A small barred opening in the top of the east wall lights a curious little chamber, which is reached from the staircase that leads to the roof.”—(P. D.) The Chapter-House is famous among these beautiful adjuncts to English cathedrals. It has been called “a glorious development of window and vault.” It was built in the latter half of the Geometrical period (1280-1315). Note the profusion of ball-flower ornament round the windows and the ogee dripstones outside.
- 45. “Of octagonal plan, its vaulting ribs branch out from sixteen Purbeck shafts which cluster round the central pillar, typifying the diocesan church with all its members gathered round its common father, the bishop. Each of the eight sides of the room is occupied by a window of four lights, with graceful tracery of an advanced geometrical type. These windows, which are among the finest examples of the period, have no shafts, but their arch mouldings are enriched with a continuous series of the ball-flower ornament. Most of the old glass in which ruby and white are the predominant colours, remains in the upper lights. Under the windows runs an arcade which forms fifty-one stalls, separated into groups of seven by the blue lias vaulting- shafts at the angles, but in the side which is occupied by the doorway there are only two stalls, one on either side of the entrance. Two rows of stone benches are under the stalls, and there is a bench of Purbeck round the base of the central pier.”—(P. D.) Another authority says: “At the springs of the arches are sculptured heads full of expression, kings, bishops, monks, ladies, jesters; and at the angles, grotesques of various kinds. A line of the ball-flower ornament is carried round above the canopies. “The double arches at the entrance show traces of a door on the exterior. Remark the curious boss in the vaulting, composed of four bearded faces. The diameter of the chapter-house is fifty feet, its height forty-one feet. Its unusual, and indeed unique, features are—its separation from the cloisters from which the chapter-house generally opens; and its crypt, or lower story, which rendered necessary the staircase by which it is approached. “A most striking view of the chapter-house is obtained from the fourth angle of the staircase, close to the doorway of the Vicars’ College. The effect of the double-door arches with their tracery, of the central pier, the branched ribs of the vaulting, and the fine windows is magnificent; and when the latter were filled with stained glass, must have been quite unrivalled. The chapter- house is by no means the least important of the many architectural masterpieces which combine to place Wells so high in the ranks of English cathedrals.”—(R. J. K.)
- 46. The Crypt, finished by 1286, represents the last development of the Early English style. It was used as the treasury where valuables were kept. It is reached by a dark passage from the north-choir-aisle. The odd corbels should be noted. The walls are very thick, the windows narrow with wide splays and the vaulting-ribs spring from round and massive pillars with much effect. This Crypt is unusually high, because the many springs at Wells would not permit of a subterranean chamber. But again we have been led astray from the main body of the Cathedral. Returning the same way, we again enter the north transept and stand beneath the splendid fan-tracery vault of the tower, a vault, beautiful as it is, that hides the lantern with its arcades. These, however, can be seen during the ascent of the tower. The Screen dates from the Fourteenth Century. “The first impression on entering the choir will not readily be forgotten. Owing to the peculiar and most beautiful arrangement of the Lady-chapel and the retro-choir, to the manner in which the varied groups of arches and pilasters are seen beyond the low altar screen, to the rich splendours of the stained glass, to the beautiful architectural details of the choir itself, and to the grace and finish of the late restorations, it may safely be said that the choir of no English cathedral affords a view more impressive or more picturesque. It is difficult to determine whether the effect is more striking at early morning, when the blaze of many-coloured light from all the eastern windows is reflected upon the slender shafts of Purbeck and upon the vaulted roof, or at the late winter services, when the darkened figures of saints and prophets in the clerestory combine with the few lights burning at the choristers’ stalls to add something of mystery and solemn gloom to the maze of half-seen aisles and chapels.
- 47. “The first three piers and arches of the choir are Early English, of the same character as those of the nave and transepts, and are probably the work of Bishop Jocelin. The remaining portion, including the whole of the vaulting as well as the clerestory above the first three bays, is very rich early Decorated (geometrical) and deserves the most careful study. “The tabernacle work and the window tracery of the first three bays, although of the same date, are less rich than those of the eastern half of the choir. In this latter portion remark the triple banded shafts of Purbeck, carried quite to the roof as vaulting-shafts, and the tabernacle-work occupying the place of the triforium, deeper and wider than in the lower bays. Under each arch is a short triple shaft, supporting a bracket richly carved in foliage. The sculpture of the capitals and of these brackets is very good and should be noticed. The foliage has become unconventional, and has evidently been studied from nature. Its diminutive character, as compared with the Early English work in the nave, is very striking. “The east end of the choir is formed of three arches divided by slender piers above which is some very rich tabernacle-work, surmounted by an east window of unusual design. At the back of the altar, and between the piers, is a low diapered screen, beyond which are seen the arches and stained windows of the retro-choir and Lady-chapel.”—(R. J. K.) The stone vault is unusual, a sort of “coved roof,” Freeman calls it, “with cells cut in it for the clerestory windows.” The three western bays are Bishop Reginald’s of the Twelfth Century. Here we are in the very oldest part of the Cathedral. Triple vaulting-shafts of Purbeck marble are carried down to the floor. “The clerestory windows contain flowing tracery of an advanced and not very good type. In some the plain mullions are carried on through the head of the window and intersect each other. Above the tabernacle-work of the east end is the EAST WINDOW of seven lights, the last bit of the Fourteenth Century reconstruction, the last flicker of Decorated freedom. Its curious
- 48. tracery is still beautiful, doubly so for the glass it enshrines, but the rule and square of Perpendicular domination have already set their mark upon it; the two principal mullions run straight up to the window head, and part of the tracery between them is rectangular.”—(P. D.) The Cathedral possesses sixty-four Misericords, from the old choir-stalls, regarded as among the best examples of mediæval wood-carving in England. The skilful hand of the carver has wonderfully represented griffins fighting, mermaids, apes, goats, dragons, wyverns, popinjays, cats, foxes, peacocks, monsters, angels, eagles, hawks, rabbits, kings, peasants—and many other birds, animals and grotesques. The soft yet brilliant light sifts in from the Jesse Window above the high altar. We lift our eyes and with some pains discern the twining branches of the vine with the recumbent figure of Jesse at the base, resting his head on his hand. From him rises the leading shoot of the tree, with the figures of the Virgin and the Child each with radiant nimbus and beneath a golden canopy. The tendrils of the vine enwreath prophets, priests and kings,—the ancestors of the Babe of Bethlehem. Above is a representation of the Crucifixion; and at the very
- 50. Wells: South-west top of the window, the outstretched wings of the Holy Spirit. The choir-aisles are of the same character as the choir itself and are entered from the transepts through ogee arches, ornamented with crockets and finials. The south-choir-aisle contains the Tomb of Saint William Bytton, at which (the oldest incised slab in England) offerings were made by those suffering from toothache, as we have already seen. Further away is the Tomb of Beckington, surrounded by a beautiful iron- screen of the same date as the tomb (1452). The carving is very fine, especially the wings of the angels. A little colour
- 51. is left here and there. His effigy rests upon it, with old and wrinkled face. This bishop said mass for his own soul here in January, 1452, thirteen years before he died. In the south-east transept, we find the Chapel of St. John Baptist, where a Decorated piscina with canopy deserves attention. At the extreme end of the north-choir-aisle is Saint Stephen’s Chapel and at the extreme end of the south- choir-aisle is the corresponding Saint Catherine’s Chapel. Both contain effigies of bishops, tombs and monuments. Between and back of these is the Lady-Chapel. We now return to the Retro-choir. Four slender piers of Purbeck marble bear up the vault. The arrangement of the columns should be particularly noticed here. It is hard to realise that this Retro-choir was merely a device for connecting the Lady-Chapel with the Choir, it seems so entirely a part of the scheme. “The beauty of the retro-choir, or ‘procession aisles,’ the arrangement of its piers and clustered columns, and the admirable manner in which it unites the Lady-chapel with the choir should be here remarked. It is throughout Early Decorated. The foliage of the capitals and the bosses of the vaulting will repay careful examination. Many of the vaulting ribs appear to spring from two grotesque heads—one on either side of the low choir-screen— which hold them between their teeth. The four supporting pillars and shafts are placed within the line of the choir-piers, thus producing the unusual intricacy and variety of the eastward view from the choir. At Salisbury, and in all other English cathedrals, the piers of the procession-aisles are placed in a line with those of the choir.”—(R. J. K.)
- 52. Mr. Bond thinks the Wells architect got his idea for the octagonal Lady-Chapel by tacking on the elongated octagonal of the Lichfield Chapter-House to the rectangular retro-choir of Salisbury. “The Lady-chapel is an early work of the Curvilinear period; for it seems to have been complete in 1324. The windows have beautiful reticulated tracery of early type. There is lovely carving in the capitals, bosses, reredos, sedilia and piscina. The Curvilinear foliated capitals here and in the choir should be compared with the somewhat earlier capitals of the chapter-house, with the early Geometrical capitals of the staircase, the Lancet capitals of the west front and the late Transitional ones of porch, nave and transepts. The ancient glass here and in the Jesse window of the choir is superb in colour. “As every one knows, it is the most beautiful east end we have in England. It may be worth while to see how this design was arrived at—a design as exceptional as it is effective. The simplest form of an east end in English Gothic is seen at York and Lincoln: it consists merely of a low wall with a big window above it. The next improvement is to build an aisle or processional path behind the east end; at the same time piercing the east wall with one, two or three arches. This was done at Hereford about 1180; and on a magnificent scale in the Chapels of Nine Altars at Durham and at Fountains early in the Thirteenth Century. But the French apsidal cathedrals —of which we have an example in Westminster—have not only an encircling processional aisle, but also a chevet of chapels radiating out from it; thus providing ever-changing vistas of entrancing beauty. The next step in England also was to provide our rectangular choirs with a chevet as well as with a processional aisle. An early example of this plan is to be seen at Abbey Dore, in Herefordshire, about 1190. It occurs early in the Thirteenth Century on a still grander scale at Salisbury, where one finds not one but two processional aisles, as well as chapels to the east of them; and, in addition, a Lady-chapel projecting still farther to the east, thus producing a design of great complexity and beauty. Nevertheless, at Salisbury, since the chief supporting piers of the retro-choir and the chevet are in a line with those of the choir, there is by no means the same changeful intricacy of vista that affords one ever fresh delight in an apsidal church. At Wells, however, the
- 53. architect attained all the success of the Continental builder simply because he built his Lady-chapel not rectangular but octagonal. For to get this octagon, of which only five sides were supported by walls, he had to plant in the retro-choir two piers to support the remaining three sides; and these piers are necessarily out of line with the piers of the choir. He had got the Continental vista. He saw it; but he saw also that it could be improved upon. And he did improve it, by putting up an outer ring of four more piers round the western part of the octagon of the Lady-chapel. It was an intuition of genius: it makes the vistas into the retro-choir and the Lady-chapel a veritable glimpse into fairyland; and provides here alone in England a rival to the glorious eastern terminations of Amiens and Le Mans. And that is not all. We saw in the chapter-house the grand effect of the central stalk branching upward and outward in all directions, like some palm tree transmuted into stone. This beautiful effect he transfers to the retro-choir, but multiplied—four palm trees in place of one; for each of the four external piers of the octagon emulates the chapter-house’s central stalk.”—(F. B.) The large windows are filled with fine specimens of Fourteenth Century glass unfortunately now jumbled together. The East Window is composed of odd pieces put together by Willement. David and other patriarchs occupy the upper tier, and the Virgin, Eve and the Serpent and Moses and the Brazen Serpent, the lower tier. The upper lights display angels with the instruments of the Passion, emblems of the Evangelists and busts of bishops and patriarchs. “From the south-west transept we pass into the CLOISTERS, which occupy an unusual amount of space, but have only three walks instead of the usual four. “The difference between a true monastic cloister and this of Wells should be remarked. The canons of Wells were not monks and did not require a cloister in the ordinary sense. This is merely an ornamental walk around the cemetery. It did not lead to either dormitory, refectory or chapter-house. It
- 54. served as a passage to the Bishop’s Palace; and the wall of the east walk is Early English of the same date as the palace itself. The lavatory in the east walk should be remarked, as well as the grotesque bosses of the roof in the portion built by Bishop Beckington. Over the western cloister is the Chapter Grammar School. The central space is known as the ‘Palm Churchyard,’ from the yew-tree in its centre, the branches of which were formerly carried in procession as palms. From the south-east angle of the cloisters we descend into the open ground within the gateway adjoining the marketplace, and opposite the episcopal palace. This is surrounded by a moat, as well as by strong external walls and bastions, and would have been capable of sustaining a long siege according to the mediæval system of warfare. The moat is fed by springs from St. Andrew’s, or the ‘bottomless well’—the original ‘great well’ of King Ina,—which rise close to the palace and fall into the moat in a cascade at the north-east corner. Both walls and moat were the work of Bishop Ralph of Shrewsbury (1329-1365).”—(R. J. K.) Wells is famous for its ancient houses. The old Palace and the Deanery are still occupied by the bishop and the dean; the canons and vicars also live in the individual houses built for these ecclesiastics. Wells was never a monastery with a common refectory and dormitory: there were always secular priests here and each man lived in his own house. Of all the domestic buildings the Bishop’s Palace is the most beautiful. It is considered the most perfect specimen of an Early English house that exists.
- 55. BATH ABBEY Dedication: St. Peter and St. Paul. A Church served by Secular Canons. Special feature: West Front. Standing before the West Front, we notice, first of all, that upon the angles of the nave on either side of the great window are two turrets, on the face of each of which is carved a ladder with angels ascending or descending. The space above the window is also carved with angels; and, under a canopy above the group, stands a figure of God the Father. Of this strange decoration the following story is told: Oliver King, Bishop of Exeter, was translated to the See of Bath and Wells in 1495. He went at once to Bath, and found the church in a dilapidated condition. While there, he had a repetition of Jacob’s famous dream of a ladder reaching from heaven to earth with angels ascending and descending. Above them stood the Lord, who said: “Let an Olive establish the crown and a King restore the church.” Taking the hint, Bishop Oliver King immediately set to work to rebuild the church and had his dream recorded upon the west front. He also had an olive-tree and crown carved on each of the corner buttresses. Bishop King’s new church was smaller than the old one. It only occupied the site of the former nave. He died before
- 56. it was finished. Prior William Birde continued the work, not forgetting a chantry for himself, which is regarded as the best thing in the church. Birde died in 1525; and the work was still unfinished when it was seized by the king’s commissioners. The roofless and neglected church soon fell into decay; but in 1572 it was patched up a little in order that services might be held in it. The east window was glazed and the choir was roofed. The nave, however, was not roofed until Bishop Montague’s rule (1608-1616). At the beginning of the Nineteenth Century, many mean houses that had clustered around Bath Abbey were removed, and buttresses and pinnacles were added to strengthen the walls. Repeated restorations have made it exceedingly trim in appearance. About 775, Offa, the Mercian king, founded here a college of secular canons, who were expelled by Dunstan in the Tenth Century and superseded by monks. One great event in the abbey church was the coronation of King Edgar on the Feast of Pentecost, 973; and for centuries afterwards it was the custom to select on Whitsunday a “King of Bath” from among its citizens, in honour of this circumstance. John de Villula, a Frenchman from Tours, who was Bishop of Somerset in the reign of William Rufus, greatly preferred Bath to Wells. He was able to merge Bath Abbey into the bishopric; and then he began to rebuild the church dedicated to St. Peter. When it was finished, he transferred
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