Systems — availability of a redundant supply

// Availability of a redundant essential supply, as a reliability block
// diagram with the arithmetic attached to the blocks.
//
// The system is the essential electrical supply to an eight-theatre suite:
// grid and standby generator in parallel, a pair of UPS modules in
// parallel behind them, and a single distribution board in series with
// everything. Each item's availability is MTBF / (MTBF + MTTR); parallel
// legs combine as 1 - (1-A1)(1-A2); series items multiply.
//
// The generator carries a second number that a pure availability model
// misses: it is only useful if it starts. Start-on-demand reliability is
// folded into its effective availability, which is why the standby leg is
// worth less than its MTBF suggests. The board, with no redundancy at all,
// then dominates the answer — as a single point of failure always does.

availability_model theatre_supply {
  title: "Essential supply to theatres 1-8"
  scope: "From the 11 kV intake to the theatre isolated-power panels; the IPS transformers themselves are out of scope"
  configuration: "(Grid | standby generator) -> (UPS A | UPS B) -> distribution board"
  operating_hours: 8760 h
  basis: "MTBF from the trust's CMMS 2021-2025; MTTR from contractor attendance records"
  requirement: "HTM 06-01: no single failure to interrupt supply to an occupied theatre"
}

supply_item grid_supply {
  of: theatre_supply
  label: "Grid intake, 11 kV, dual feeder"
  mtbf: 4380 h
  mttr: 3.5 h
  availability: = mtbf / (mtbf + mttr)
  unavailability: = 1 - availability
}

supply_item standby_generator {
  of: theatre_supply
  label: "1.6 MVA standby generator, 15 s start"
  mtbf: 8760 h
  mttr: 12 h
  running_availability: = mtbf / (mtbf + mttr)
  start_reliability: 0.995 source "42 monthly black-start tests, 2021-2025"
  // A generator that is available but does not start is not available.
  availability: = running_availability * start_reliability
  unavailability: = 1 - availability
}

supply_item ups_a {
  of: theatre_supply
  label: "UPS module A, 200 kVA, 20 min autonomy"
  mtbf: 26280 h
  mttr: 6 h
  availability: = mtbf / (mtbf + mttr)
  unavailability: = 1 - availability
}

supply_item ups_b {
  of: theatre_supply
  label: "UPS module B, 200 kVA, 20 min autonomy"
  mtbf: 26280 h
  mttr: 6 h
  availability: = mtbf / (mtbf + mttr)
  unavailability: = 1 - availability
}

supply_item distribution_board {
  of: theatre_supply
  label: "Essential distribution board — no redundancy"
  mtbf: 43800 h
  mttr: 4 h
  availability: = mtbf / (mtbf + mttr)
  unavailability: = 1 - availability
}

availability_result theatre_supply_result {
  of: theatre_supply
  source_leg: = 1 - grid_supply.unavailability * standby_generator.unavailability
  ups_leg: = 1 - ups_a.unavailability * ups_b.unavailability
  system: = source_leg * ups_leg * distribution_board.availability
  unavailability: = 1 - system
  nines: = 0 - log10(unavailability)
  downtime_per_year: = unavailability * theatre_supply.operating_hours
  downtime_minutes_per_year: = unavailability * 525600[min]
  // Where the downtime actually comes from: the unredundant board.
  board_share: = pct(distribution_board.unavailability, unavailability)
  source_share: = pct(1 - source_leg, unavailability)
  ups_share: = pct(1 - ups_leg, unavailability)
}

// What a second board would buy, which is the only question the model was
// commissioned to answer.
availability_option second_board {
  of: theatre_supply_result
  label: "Second distribution board with automatic changeover"
  board_pair: = 1 - distribution_board.unavailability * distribution_board.unavailability
  system: = theatre_supply_result.source_leg * theatre_supply_result.ups_leg * board_pair
  downtime_minutes_per_year: = (1 - system) * 525600[min]
  minutes_saved: = theatre_supply_result.downtime_minutes_per_year - downtime_minutes_per_year
  capital_cost: 148000
  currency: "GBP"
  cost_per_minute_saved: = capital_cost / (minutes_saved * 15)
  verdict: "A second board removes nineteen twentieths of the expected downtime, because the board is where nearly all of it is. Costed over a fifteen-year life that is about GBP 206 per theatre-minute recovered, which is the cheapest minute of availability on the estate."
}

figure supply_rbd {
  title: "Reliability block diagram — theatre essential supply"
  width: 900
  height: 400
  background: "transparent"
}

label_2d {
  text: "Source (parallel)"
  x: 215
  y: 42
  anchor: middle
  size: 12
  weight: bold
  color: "#334155"
}

label_2d {
  text: "UPS (parallel)"
  x: 475
  y: 42
  anchor: middle
  size: 12
  weight: bold
  color: "#334155"
}

label_2d {
  text: "Distribution (series)"
  x: 735
  y: 42
  anchor: middle
  size: 12
  weight: bold
  color: "#334155"
}

shape rbd_grid {
  kind: rect
  x: 120
  y: 62
  w: 190
  h: 80
  r: 6
  fill: "#dbeafe"
  stroke: "#1d4ed8"
  strokeWidth: 1.4
  label: "Grid intake"
}

shape rbd_generator {
  kind: rect
  x: 120
  y: 202
  w: 190
  h: 80
  r: 6
  fill: "#dbeafe"
  stroke: "#1d4ed8"
  strokeWidth: 1.4
  label: "Standby generator"
}

shape rbd_ups_a {
  kind: rect
  x: 390
  y: 62
  w: 170
  h: 80
  r: 6
  fill: "#dcfce7"
  stroke: "#15803d"
  strokeWidth: 1.4
  label: "UPS A"
}

shape rbd_ups_b {
  kind: rect
  x: 390
  y: 202
  w: 170
  h: 80
  r: 6
  fill: "#dcfce7"
  stroke: "#15803d"
  strokeWidth: 1.4
  label: "UPS B"
}

shape rbd_board {
  kind: rect
  x: 640
  y: 132
  w: 190
  h: 80
  r: 6
  fill: "#fee2e2"
  stroke: "#b91c1c"
  strokeWidth: 1.6
  label: "Distribution board"
}

arrow_2d {
  x1: 40
  y1: 172
  x2: 112
  y2: 172
  color: "#64748b"
  head: none
}

arrow_2d {
  x1: 112
  y1: 102
  x2: 112
  y2: 242
  color: "#64748b"
  head: none
}

arrow_2d {
  x1: 112
  y1: 102
  x2: 118
  y2: 102
  color: "#64748b"
}

arrow_2d {
  x1: 112
  y1: 242
  x2: 118
  y2: 242
  color: "#64748b"
}

arrow_2d {
  x1: 310
  y1: 102
  x2: 388
  y2: 102
  color: "#64748b"
}

arrow_2d {
  x1: 310
  y1: 242
  x2: 388
  y2: 242
  color: "#64748b"
}

arrow_2d {
  x1: 560
  y1: 102
  x2: 638
  y2: 162
  color: "#64748b"
}

arrow_2d {
  x1: 560
  y1: 242
  x2: 638
  y2: 182
  color: "#64748b"
}

arrow_2d {
  x1: 830
  y1: 172
  x2: 886
  y2: 172
  color: "#64748b"
  label: "Theatres"
}

annotation {
  target: rbd_board
  text: "Single point of failure — most of the expected downtime is here"
  offsetX: -40
  offsetY: 96
  style: footnote
}

note availability_language {
  text: "Availability answers how much of the year the supply is there. It says nothing about when: four nines spread as one minute a month reads very differently in an operating theatre from four nines spent in one 53-minute outage. The mitigation for the second is the UPS autonomy, not the availability figure."
  anchor: theatre_supply_result
}

view rbd: figure(supply_rbd)