Severn_04: Simulation of a regulated withdrawal (closed-loop control)

David Dorchies

library(airGRiwrm)

Presentation of the case study

Starting from the network and the calibration set in the vignette “V03_Open-loop_influenced_flow,” we add 2 intake points for irrigation.

The following code chunk resumes the procedure of the vignette “V03_Open-loop_influenced_flow”:

data(Severn)
nodes <- Severn$BasinsInfo[, c("gauge_id", "downstream_id", "distance_downstream", "area")]
nodes$distance_downstream <- nodes$distance_downstream
nodes$model <- "RunModel_GR4J"
griwrm <- CreateGRiwrm(nodes, list(id = "gauge_id", down = "downstream_id", length = "distance_downstream"))
griwrmV03 <- griwrm
griwrmV03$model[griwrm$id == "54002"] <- NA
griwrmV03$model[griwrm$id == "54095"] <- NA
griwrmV03
#>      id  down length         model    area
#> 1 54057  <NA>     NA RunModel_GR4J 9885.46
#> 2 54032 54057     15 RunModel_GR4J 6864.88
#> 3 54001 54032     45 RunModel_GR4J 4329.90
#> 4 54095 54001     42          <NA> 3722.68
#> 5 54002 54057     43          <NA> 2207.95
#> 6 54029 54032     32 RunModel_GR4J 1483.65

Network configuration

The intake points are located:

We have to add this 2 nodes in the GRiwrm object that describes the network:

griwrmV04 <- rbind(
  griwrmV03,
  data.frame(
    id = c("Irrigation1", "Irrigation2"),
    down = c("54001", "54032"),
    length = c(35, 10),
    model = NA,
    area = NA
  )
)
plot(griwrmV04)

Blue-grey nodes figure upstream basins (rainfall-runoff modeling only) and green nodes figure intermediate basins, coupling rainfall-runoff and hydraulic routing modeling. Nodes in red color are direct injection points (positive or negative flow) in the model.

It’s important to notice that even if the points “Irrigation1” and “Irrigation2” are physically located on a single branch of the Severn river as well as gauging stations “54095,” “54001” and “54032,” nodes “Irrigation1” and “Irrigation2” are not represented on the same branch in this conceptual model. Consequently, with this network configuration, it is not possible to know the value of the flow in the Severn river at the “Irrigation1” or “Irrigation2” nodes. These values are only available in nodes “54095,” “54001” and “54032” where rainfall-runoff and hydraulic routing are actually modeled.

Irrigation objectives and flow demand at intakes

Irrigation1 covers an area of 15 km² and Irrigation2 covers an area of 30 km².

The objective of these irrigation systems is to cover the rainfall deficit (Burt et al. 1997) with 80% of success. Below is the calculation of the 8th decile of monthly water needed given meteorological data of catchments “54001” and “54032” (unit mm/day) :

# Formatting climatic data for CreateInputsModel (See vignette V01_Structure_SD_model for details)
BasinsObs <- Severn$BasinsObs
DatesR <- BasinsObs[[1]]$DatesR
PrecipTot <- cbind(sapply(BasinsObs, function(x) {x$precipitation}))
PotEvapTot <- cbind(sapply(BasinsObs, function(x) {x$peti}))
Qobs <- cbind(sapply(BasinsObs, function(x) {x$discharge_spec}))
Precip <- ConvertMeteoSD(griwrm, PrecipTot)
PotEvap <- ConvertMeteoSD(griwrm, PotEvapTot)

# Calculation of the water need at the sub-basin scale
dailyWaterNeed <- PotEvap - Precip
dailyWaterNeed <- cbind(as.data.frame(DatesR), dailyWaterNeed[,c("54001", "54032")])
monthlyWaterNeed <- SeriesAggreg(dailyWaterNeed, "%Y%m", rep("mean",2))
monthlyWaterNeed <- SeriesAggreg(dailyWaterNeed, "%m", rep("q80",2))
monthlyWaterNeed[monthlyWaterNeed < 0] <- 0
monthlyWaterNeed$DatesR <- as.numeric(format(monthlyWaterNeed$DatesR,"%m"))
names(monthlyWaterNeed)[1] <- "month"
monthlyWaterNeed <- monthlyWaterNeed[order(monthlyWaterNeed$month),]
monthlyWaterNeed
#>       month     54001     54032
#> 25        1 0.2400000 0.2365627
#> 1721      2 0.5918416 0.6188486
#> 2804      3 1.2376475 1.2384480
#> 3745      4 2.1809664 2.2036046
#> 4721      5 3.0230456 3.0555408
#> 5716      6 3.4775922 3.5096402
#> 6722      7 3.5305228 3.5769480
#> 7644      8 2.6925228 2.7331421
#> 8637      9 1.8323644 1.8489218
#> 9588     10 0.8626301 0.8867260
#> 10554    11 0.2707842 0.2497664
#> 11491    12 0.1488753 0.1665834

We restrict the irrigation season between March and September As a consequence, the crop requirement can be expressed in m3/s as follows:

irrigationObjective <- monthlyWaterNeed
# Conversion in m3/day
irrigationObjective$"54001" <- monthlyWaterNeed$"54001" * 15 * 1E3
irrigationObjective$"54032" <- monthlyWaterNeed$"54032" * 30 * 1E3
# Irrigation period between March and September
irrigationObjective[-seq(3,9),-1] <- 0
# Conversion in m3/s
irrigationObjective[,c(2,3)] <- round(irrigationObjective[,c(2,3)] / 86400, 1)
irrigationObjective$total <- rowSums(irrigationObjective[,c(2,3)])
irrigationObjective
#>       month 54001 54032 total
#> 25        1   0.0   0.0   0.0
#> 1721      2   0.0   0.0   0.0
#> 2804      3   0.2   0.4   0.6
#> 3745      4   0.4   0.8   1.2
#> 4721      5   0.5   1.1   1.6
#> 5716      6   0.6   1.2   1.8
#> 6722      7   0.6   1.2   1.8
#> 7644      8   0.5   0.9   1.4
#> 8637      9   0.3   0.6   0.9
#> 9588     10   0.0   0.0   0.0
#> 10554    11   0.0   0.0   0.0
#> 11491    12   0.0   0.0   0.0

We assume that the efficiency of the irrigation systems is equal to 50% as proposed by Seckler, Molden, and Sakthivadivel (2003). as a consequence, the flow demand at intake for each irrigation system is as follows (unit: m3/s):

# Application of the 50% irrigation system efficiency on the water demand
irrigationObjective[,seq(2,4)] <- irrigationObjective[,seq(2,4)] / 0.5
# Display result in m3/s
irrigationObjective
#>       month 54001 54032 total
#> 25        1   0.0   0.0   0.0
#> 1721      2   0.0   0.0   0.0
#> 2804      3   0.4   0.8   1.2
#> 3745      4   0.8   1.6   2.4
#> 4721      5   1.0   2.2   3.2
#> 5716      6   1.2   2.4   3.6
#> 6722      7   1.2   2.4   3.6
#> 7644      8   1.0   1.8   2.8
#> 8637      9   0.6   1.2   1.8
#> 9588     10   0.0   0.0   0.0
#> 10554    11   0.0   0.0   0.0
#> 11491    12   0.0   0.0   0.0

Restriction of irrigation in case of water scarcity

Minimal environmental flow at the intakes

In the UK, abstraction restrictions are driven by Environmental Flow Indicator (EFI) supporting Good Ecological Status (GES) (Klaar et al. 2014). Abstraction restriction consists in limiting the proportion of available flow for abstraction in function of the current flow regime (Reference taken for a river that is highly sensitive to abstraction classified “ASB3”).

restriction_rule <- data.frame(quantile_natural_flow = c(.05, .3, 0.5, 0.7),
                               abstraction_rate = c(0.1, 0.15, 0.20, 0.24))

The control of the abstraction will be done at the gauging station downstream all the abstraction locations (node “54032”). So we need the flow corresponding to the quantiles of natural flow and flow available for abstraction in each case.

quant_m3s32 <- quantile(
  Qobs[,"54032"] * griwrmV04[griwrmV04$id == "54032", "area"] / 86.4,
  restriction_rule$quantile_natural_flow,
  na.rm = TRUE
)
restriction_rule_m3s <- data.frame(
  threshold_natural_flow = quant_m3s32,
  abstraction_rate = restriction_rule$abstraction_rate
)

matplot(restriction_rule$quantile_natural_flow,
        cbind(restriction_rule_m3s$threshold_natural_flow,
              restriction_rule$abstraction_rate * restriction_rule_m3s$threshold_natural_flow,
              max(irrigationObjective$total)),
        log = "x", type = "l",
        main = "Quantiles of flow on the Severn at Saxons Lode (54032)",
        xlab = "quantiles", ylab = "Flow (m3/s)",
        lty = 1, col = rainbow(3, rev = TRUE)
        )
legend("topleft", legend = c("Natural flow", "Abstraction limit", "Irrigation max. objective"),
       col = rainbow(3, rev = TRUE), lty = 1)

The water availability or abstraction restriction depending on the natural flow is calculated with the function below:

# A function to enclose the parameters in the function (See: http://adv-r.had.co.nz/Functional-programming.html#closures)
getAvailableAbstractionEnclosed <- function(restriction_rule_m3s) {
  function(Qnat) approx(restriction_rule_m3s$threshold_natural_flow,
                        restriction_rule_m3s$abstraction_rate,
                        Qnat,
                        rule = 2)
}
# The function with the parameters inside it :)
getAvailableAbstraction <- getAvailableAbstractionEnclosed(restriction_rule_m3s)
# You can check the storage of the parameters in the function with
as.list(environment(getAvailableAbstraction))
#> $restriction_rule_m3s
#>     threshold_natural_flow abstraction_rate
#> 5%                15.09638             0.10
#> 30%               30.19276             0.15
#> 50%               50.85096             0.20
#> 70%               90.57828             0.24

Restriction rules

The figure above shows that restrictions will be imposed to the irrigation perimeter if the natural flow at Saxons Lode (54032) is under around 20 m3/s.

Applying restriction on the intake on a real field is always challenging since it is difficult to regulate day by day the flow at the intake. Policy makers often decide to close the irrigation abstraction points in turn several days a week based on the mean flow of the previous week.

The number of authorized days per week for irrigation can be calculated as follows. All calculations are based on the mean flow measured the week previous the current time step. First, the naturalized flow \(N\) is equal to

\[ N = M + I_l \] with:

  • \(M\), the measured flow at the downstream gauging station
  • \(I_l\), the total abstracted flow for irrigation for the last week

Available flow for abstraction \(A\) is:

\[A = f_{a}(N)\]

with \(f_a\) the availability function calculated from quantiles of natural flow and related restriction rates.

The flow planned for irrigation \(Ip\) is then:

\[ I_p = \min (O, A)\]

with \(O\) the irrigation objective flow.

The number of days for irrigation \(n\) per week is then equal to:

\[ n = \lfloor \frac{I_p}{O} \times 7 \rfloor\] with \(\lfloor x \rfloor\) the function that returns the largest integers not greater than \(x\)

The rotation of restriction days between the 2 irrigation perimeters is operated as follows:

restriction_rotation <- matrix(c(5,7,6,4,2,1,3,3,1,2,4,6,7,5), ncol = 2)
m <- do.call(
  rbind,
  lapply(seq(0,7), function(x) {
    b <- restriction_rotation <= x
    rowSums(b)
  })
)
# Display the planning of restriction
image(1:ncol(m), 1:nrow(m), t(m), col = heat.colors(3, rev = TRUE),
      axes = FALSE, xlab = "week day", ylab = "number of restriction days",
      main = "Number of closed irrigation perimeters")
axis(1, 1:ncol(m), unlist(strsplit("SMTWTFS", "")))
axis(2, 1:nrow(m), seq(0,7))
for (x in 1:ncol(m))
  for (y in 1:nrow(m))
    text(x, y, m[y,x])

Implementation of the model

As for the previous model, we need to set up an GRiwrmInputsModel object containing all the model inputs:

# A flow is needed for all direct injection nodes in the network
# even if they may be overwritten after by a controller
# The Qobs matrix is completed with 2 new columns for the new nodes
QobsIrrig <- cbind(Qobs[, c("54002", "54095")], Irrigation1 = 0, Irrigation2 = 0)

# Creation of the GRiwrmInputsModel object
IM_Irrig <- CreateInputsModel(griwrmV04, DatesR, Precip, PotEvap, QobsIrrig)
#> CreateInputsModel.GRiwrm: Treating sub-basin 54001...
#> CreateInputsModel.GRiwrm: Treating sub-basin 54029...
#> CreateInputsModel.GRiwrm: Treating sub-basin 54032...
#> CreateInputsModel.GRiwrm: Treating sub-basin 54057...

Implementation of the regulation controller

The supervisor

The simulation is piloted through a Supervisor that can contain one or more Controller. This supervisor will work with a cycle of 7 days: the measurement are taken on the last 7 days and decisions are taken for each time step for the next seven days.

sv <- CreateSupervisor(IM_Irrig, TimeStep = 7L)

The control logic function

We need a controller that measures the flow at Saxons Lode (“54032”) and adapts on a weekly basis the abstracted flow at the two irrigation points. The supervisor will stop the simulation every 7 days and will provide to the controller the last 7 simulated flow values at Saxons Lode (“54032”) (measured variables) and the controller should provide “command variables” for the next 7 days for the 2 irrigation points.

A control logic function should be provided to the controller. This control logic function processes the logic of the regulation taking measured flows as input and returning the “command variables.” Both measured variables and command variables are of type matrix with the variables in columns and the time steps in rows.

In this example, the logic function must do the following tasks:

  1. Calculate the objective of irrigation according to the month of the current days of simulation
  2. Calculate the naturalized flow from the measured flow and the abstracted flow of the previous week
  3. Calculate the number of days of restriction for each irrigation point
  4. Return the abstracted flow for the next week taking into account restriction days
fIrrigationFactory <- function(supervisor,
                               irrigationObjective,
                               restriction_rule_m3s,
                               restriction_rotation) {
  function(Y) {
    # Y is in m3/day and the basin's area is in km2
    # Calculate the objective of irrigation according to the month of the current days of simulation
    month <- as.numeric(format(supervisor$ts.date, "%m"))
    U <- irrigationObjective[month, c(2,3)] # m3/s
    meanU <- mean(rowSums(U))
    if(meanU > 0) {
      # calculate the naturalized flow from the measured flow and the abstracted flow of the previous week
      lastU <- supervisor$controllers[[supervisor$controller.id]]$U # m3/day
      Qnat <- (Y - rowSums(lastU)) / 86400 # m3/s
      # Maximum abstracted flow available
      Qrestricted <- mean(
        approx(restriction_rule_m3s$threshold_natural_flow,
               restriction_rule_m3s$abstraction_rate,
               Qnat,
               rule = 2)$y * Qnat
      )
      # Total for irrigation
      QIrrig <- min(meanU, Qrestricted)
      # Number of days of irrigation
      n <- floor(7 * (1 - QIrrig / meanU))
      # Apply days off
      U[restriction_rotation[seq(nrow(U)),] <= n] <- 0
    }
    return(-U * 86400) # withdrawal is a negative flow in m3/day on an upstream node
  }
}

You can notice that the data required for processing the control logic are enclosed in the function fIrrigationFactory, which takes the required data as arguments and returns the control logic function.

Creating fIrrigation by calling fIrrigationFactory with the arguments currently in memory saves these variables in the environment of the function:

fIrrigation <- fIrrigationFactory(supervisor = sv,
                                  irrigationObjective = irrigationObjective,
                                  restriction_rule_m3s = restriction_rule_m3s,
                                  restriction_rotation = restriction_rotation)

You can see what data are available in the environment of the function with:

str(as.list(environment(fIrrigation)))
#> List of 4
#>  $ supervisor          :Classes 'Supervisor', 'environment' <environment: 0x000000001d334160> 
#>  $ irrigationObjective :'data.frame':    12 obs. of  4 variables:
#>   ..$ month: num [1:12] 1 2 3 4 5 6 7 8 9 10 ...
#>   ..$ 54001: num [1:12] 0 0 0.4 0.8 1 1.2 1.2 1 0.6 0 ...
#>   ..$ 54032: num [1:12] 0 0 0.8 1.6 2.2 2.4 2.4 1.8 1.2 0 ...
#>   ..$ total: num [1:12] 0 0 1.2 2.4 3.2 3.6 3.6 2.8 1.8 0 ...
#>  $ restriction_rule_m3s:'data.frame':    4 obs. of  2 variables:
#>   ..$ threshold_natural_flow: num [1:4] 15.1 30.2 50.9 90.6
#>   ..$ abstraction_rate      : num [1:4] 0.1 0.15 0.2 0.24
#>  $ restriction_rotation: num [1:7, 1:2] 5 7 6 4 2 1 3 3 1 2 ...

The supervisor variable is itself an environment which means that the variables contained inside it will be updated during the simulation. Some of them are useful for computing the control logic such as:

The controller

The controller contains:

CreateController(sv,
                 ctrl.id = "Irrigation",
                 Y = "54032",
                 U = c("Irrigation1", "Irrigation2"),
                 FUN = fIrrigation)
#> The controller has been added to the supervisor

Running the simulation

First we need to create a GRiwrmRunOptions object and load the parameters calibrated in the vignette “V03_Open-loop_influenced_flow”:

IndPeriod_Run <- seq(
  which(DatesR == (DatesR[1] + 365*24*60*60)), # Set aside warm-up period
  length(DatesR) # Until the end of the time series
)
IndPeriod_WarmUp = seq(1,IndPeriod_Run[1]-1)
RunOptions <- CreateRunOptions(IM_Irrig,
                               IndPeriod_WarmUp = IndPeriod_WarmUp,
                               IndPeriod_Run = IndPeriod_Run)
ParamV03 <- readRDS(system.file("vignettes", "ParamV03.RDS", package = "airGRiwrm"))

For running a model with a supervision, you only need to substitute InputsModel by a Supervisor in the RunModel function call.

OM_Irrig <- RunModel(sv, RunOptions = RunOptions, Param = ParamV03)

Simulated flows can be extracted and plot as follows:

Qm3s <- attr(OM_Irrig, "Qm3s")
Qm3s <- Qm3s[Qm3s$DatesR > "2003-05-25" & Qm3s$DatesR < "2003-10-05",]
oldpar <- par(mfrow=c(2,1), mar = c(2.5,4,1,1))
plot(Qm3s[, c("DatesR", "54095", "54001", "54032")], main = "", xlab = "", ylim = c(0,40))
plot(Qm3s[, c("DatesR", "Irrigation1", "Irrigation2")], main = "", xlab = "", legend.x = "bottomright")

par(oldpar)

We can observe that the irrigation points are alternatively closed some days a week when the flow at node “54032” becomes low.

References

Burt, C. M., A. J. Clemmens, T. S. Strelkoff, K. H. Solomon, R. D. Bliesner, L. A. Hardy, T. A. Howell, and D. E. Eisenhauer. 1997. “Irrigation Performance Measures: Efficiency and Uniformity.” Journal of Irrigation and Drainage Engineering 123 (6): 423–42. https://doi.org/10.1061/(ASCE)0733-9437(1997)123:6(423).
Klaar, Megan J., Michael J. Dunbar, Mark Warren, and Rob Soley. 2014. “Developing Hydroecological Models to Inform Environmental Flow Standards: A Case Study from England.” WIREs Water 1 (2): 207–17. https://doi.org/10.1002/wat2.1012.
Seckler, D., D. Molden, and R. Sakthivadivel. 2003. “The Concept of Efficiency in Water-Resources Management and Policy.” In Water Productivity in Agriculture: Limits and Opportunities for Improvement, edited by J. W. Kijne, R. Barker, and D. Molden, 37–51. Wallingford: CABI. https://doi.org/10.1079/9780851996691.0037.