ANÁLISIS ESTADÍSTICO

In this section, the MPC controller is formulated as a controller that creates a sequence of decision-variables to maximise or minimise a cost function over a future time horizon under adjustments with every time step [33]. The MPC controller, also known as the generalised predictive control or receding horizon control, is a time horizon optimisation model that determines a series of optimal control decisions over a specific future time period [33] [129]. Furthermore, the receding horizon controller is widely used nowadays in smart grids and energy applications. Generally, the cost function for energy, peak demand and costs in a real application is usually pre-defined and commonly known [33]. In the first control action, the MPC controller computes the decision for the first-time period based on the demand forecast

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and updates of other variables. In the subsequent time intervals, the predictive control uses updates of the forecast data and other operation variables to adjust the optimal control signal at every future time step and solve the optimisation defined by the cost function. This is then repeated for all time steps [130].

Figure 5-5 presents the exemplified control scheme of the MPC system for a network of electrified RTG cranes with an ESS. The crane’s measurements, updated demand forecast data, the electricity price term and storage measurements are fed to the MPC controller in order to generate a control signal. The control decision uses the cost function in Equation (5-9) to minimise the electricity energy cost and peak demand. The demand prediction of the crane network was developed in Chapter 4. The energy prediction model, (ANN, Model B.2) has been used to predict the future crane network demand, as the Model B.2 only uses one known external variable (X₁(n), container gross weight) and was shown to be an accurate forecast model with MAPEs 12%, as discussed in Chapter 4 (see Section 4.3.3). The optimal energy controllers based on load forecasting in this research will be evaluated and compared with perfect and worst case forecast scenarios as will be shown in Chapter 7.

Figure 5-5: The scheme of MPC for the electrified RTG crane network equipped with an ESS.

In this section, the cost function in Equation (5-9) is optimised by generating a control signal to the ESS that aims to minimise the peak demand. The Matlab optimisation solver has been used to minimise the cost function. This cost formulation is subject to the constraints presented in Section 5.1.1 for the crane and ESS models. Recall from the previous chapter, in this thesis,

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the forecast and historical or actual data are presented in the equations with and without (^) notation, respectively. arg min ∆E ∑ max (L̂k(n) + ∆E(n)) 2 , N n=1 (5-9)

The cost function in Equation (5-9) is optimised by generating a control signal to the ESS that aims to minimise the peak demand over the prediction horizon period subject to ESS and RTG model Equations (5-3) to (5-6), where ∆E(n) is the change in the ESS energy, L̂_k(n) is an estimated of the RTG crane demand future (forecast) at the current time step, n, for day k, and N is the number of half the hour time steps in one day (N = 48). In this thesis, the reduction in the energy cost of the network of cranes is achieved by finding the optimal operation of the ESS that minimise the peak demand under the following constraints:

SoC(n set ) = SoCmax (5-10)

SoC(N) = SoCmin _(5-11)

The constraints, Equations (5-10) and (5-11), aims to fully charge the ESS during the low tariff period and fully discharge during the high tariff period in order to achieve the maximum energy cost saving based on the electricity price term, C(n), at Port of Felixstowe, as described in Equation (5-12). The MPC controller as a real time controller is computationally expensive and the above control procedure help to achieve the maximum cost saving and reduce the computational cost by simplifying the cost function. However, this procedure could affect the peak reduction term in order to satisfy the cost constraints.

C(n) ≔ {C _C D ∀ n ≥ n set

N ∀n < n set ,

(5-12)

where n ∊ {1,2, . . . ,48} is the half hour period, the high electricity tariff (C _D) is between n _set = 14 (7:00 am) and midnight, and the lower price is during the rest of the day (C _N). The controller model obtains the updated demand prediction data for the period between the current time interval (n), and n + i, where i is the prediction horizon and n + i ≤ N and N is the one-day ahead forecast period (N = 48). The forecast model is designed to predict the load for one day ahead and then updates the MPC plan. After each time step the forecast model will use the actual data for this step and the forecast error to recalculate and update the forecast model. The MPC controller then calculates the optimal control decision by calculating the ESS

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energy in Equation (5-9) to minimise the cost function and sends the control signal to the network system. These steps are repeated at every time step n + 1 by updating the forecast data and other system variables and using the updated forecast data n + 1 + i to compute the control signal, as illustrated in Figure 5-6. This control process is mainly referred to as the receding horizon controller [33] [131] and it is described in more detail in Appendix B. The results section discusses the size of the horizon for an MPC as a significant parameter for the selection of this algorithm and evaluates the ESS performance using an MPC controller compared to a single time step control algorithm without future knowledge (the set-point control). The current literature on LV network applications and microgrids [33] [52] is beginning to investigate the benefits of treating the volatile demand as a stochastic element and develop a stochastic control in order to increase the efficiency performance of the ESS on the distribution network. The following chapter will extend the method by upgrading the control to a stochastic model predictive controller.

Algorithm 5-1: Basic concept of MPC for network electrified RTG crane system model with ESS.

1. Select the time horizon step and prediction horizon. 2. Determine the control objective and constraints. 3. Initialise: the crane, forecast data and ESS data. 4. For n= 1 to N (daily demand operation), do

a. Solve optimal Equation (5-9), subject to:

• RTG crane model Equation (5-2). • ESS model Equations (5-3) to (5-6).

• Energy cost saving Equations (5-10) to (5-12).

• For (n = 1), the model computes the optimal solution based on the RTG crane demand prediction and initial data.

b. Find the optimal signal for (n + 1), Equation (5-9), and apply the control to the system. c. Update the forecast model for time step (n + 1) to N by regenerating the forecast load

profile with the new observation and update the other system variables. d. Set n = n + 1,

5. An optimal solution is achieved for the electrified RTG crane system model with ESS for the specific day.

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Figure 5-6: A simple illustration of the receding horizon used in MPC.