Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

Equations (12.27) are necessary to avoid simultaneous commitment and decommitment of a unit.

The constraints above make clear that multi-period OPF problems that include generator ramp limits, start-up and shut-down constraints, minimum number of up and down periods, etc., are generally modelled as MILP problems. The interested reader can found detailed model in [65, 66, 212]. However, an use of generator ramp limits can be also associated to stability constrained OPF problems. The interested reader can found an example of such use of ramp limits in [356].

12.3Load Constraints

This section discusses constraints related to loads. Subsection 12.3.1 describes demand bid functions. Subsection 12.3.2 describes a model of load daily profile. Finally, Subsection 12.3.3 describes load ramp limits.

12.3.1Demand Bid

In some electricity markets, loads are elastic, i.e., participate to the auction providing bid blocks in the same way suppliers provide o er blocks. The basic parameters for elastic demand bids are shown in Table 12.6 and include maximum and minimum power consumption as well as the coe cient of the bid function. The bid function has the same structure as (12.12), as follows:

cD (pD ) = cD0 + cD1pD + cD2p2D

The reactive power can be a function of the active power demand through a constant power factor cos φD :

As for the constraints, one has:

Similarly to generator bid blocks, one can define a unit commitment discrete variable uC that allows modelling non-dispatched demands with pminD = 0:

Finally, similarly to generator bids, one can define a tie breaking cost kT B . The tie breaking involves a penalty cost kT B prorated by the amount scheduled over the maximum amount that could be scheduled for the load by means of a quadratic function added to the objective function:

p2

cT B = kT B D (12.31)

pmaxD

If the load does not consume power, this cost is zero, whereas if pD is close to the maximum power the tie breaking cost increases quadratically and penalizes the load. Thus, two otherwise tied energy demands will be scheduled to the point where their modified costs are identical, e ectively achieving a prorated result. Generally, the value of kT B is small (e.g., 0.0005).

Table 12.6 Demand bid parameters

12.3.2Demand Daily Profile

Multi-period market clearing procedures require the definition of a daily demand profile. The simplest model is simply a sequence of demand values, one per each period in which the time horizon is subdivided:

For example, for a Δt = 1 h, nT = 24. Each element dt of the demand profile array is a percentage of the load pL0 or, in case of elastic demands, of the demand bid limits pmaxD and pminD . Thus, one has:

Fig. 12.3 Example of daily demand profile

are basically the same as generator ones (see Table 12.5). For example, up and down ramp constraints express the amount of demand power that can be moved up or down during each period Δt and are associated with technical limits in demand facilities, as follows:

These equations are conceptually similar to (12.21) for the generation ramp rate, and uses the same time interval Δt defined in the OPF time horizon. Finally, if pertinent, minimum up and down demand periods can be modelled by means of constraints similar to (12.24)-(12.27).

respect to transient stability and can thus be neglected (see Figure 1.6 of Chapter 1).

Since each device has its own status u, the breaker only has to set to 1 (on-line) or to 0 (o -line) the status of other devices at assigned times. In case of switching transmission lines, each switch also requires rebuilding the admittance matrix as well as checking the network connectivity. Scripts 11.1 and 11.5 of Chapter 11 provide further details on the admittance matrix and the assessment of network connectivity, respectively.

13.3Relay

Relays are devices that, based on certain measures, decide and coordinate the times at which breaker actions occur. The decision logic depends on the relay type. There exist over-current relays, overand under-voltage relays, distance relays, etc. The decision logic is translated into code by if-then-else logic depending on the relay type. For example, for an instantaneous overcurrent relay, one has:

where isp is the relay current set point and top is the operating time at which the relay sends to the breaker the switching signal. If the controlled quantity is measured remotely, a measurement delay tm can be required:

if im(t) ≥ isp then ta = t

where im is the measured current that is delayed with respect to the real current i. Instantaneous relays are a ected by a delay tr which is the time required by the breaker to actually switch after receiving the signal by the relay. Thus the breaker time intervention can be estimated as:

Inverse definite time lag relays have a more interesting model then instantaneous ones. The inverse time function can be defined by a series of (isp, top) pairs. Another possibility is to approximate the operating curve of the relay by means of a function, for example a logarithm [14]:

∞

Fig. 13.1 Relay inverse time characteristic curve

Figure 13.1 illustrates (13.6).

For electro-mechanical relays, the dynamic of the disc travel d can be modelled as:

where spb and stm are the plug bridge and the time multiplier settings, respectively, and Tr is the time required by the disc to reset due to spring action. In (13.7), it is assumed that the initial d(0) after a complete resetting is zero. Then the operating time top is defined as:

In numerical integration, the following formula can be used for approximating (13.8) [14]:

where Δt is the step length of the numerical integration. Finally, the breaker intervention time is defined by (13.5). Table 13.2 defines the parameters required for an electro-mechanical inverse time over-current relay.

13.4Phasor Measurement Unit

A Phasor Measurement Unit (PMU) is a device able to measure the magnitude and the angle of a phasor. Basic theory, definitions and applications about PMUs can be found in [349]. Brief outlines are provided below.

Let us define a sinusoidal quantity:

The phasor is defined for a pure constant sinusoid, but it can also be used for transients, assuming that the phasor is the fundamental frequency component of a waveform over a finite interval (observation window).

PMU devices work on sampled measures (see Figure 13.2). In the case of x(t), we can define the samples signal xk at t = kτs, where τs is the sampling

¯

interval. Using a Discrete Fourier Transform (DFT), the phasor X is given by:

where n is the number of samples in one period of the nominal fundamental frequency fn, and:

k=1

and θ is the sampling angle associated with the sampling interval τs, as follows:

A typical sampling rate used in relaying and measurements functions is 12 times the power system frequency (e.g., 600 Hz for a 50 Hz system or 720 Hz for a 60 Hz system).

Fig. 13.2 Data sampling windows for phasor measurements

Equation (13.12) represents a non-recursive DFT calculation. A recursive

¯ r calculation is an e cient method for time varying phasors. Let X be the

phasor corresponding to the data set x{k = r, r + 1, . . . , n + r − 1}, and let a new data sample be obtained to produce a new data set x{k = r + 1, r +

A recursive calculation through a moving window data sample is faster than a non-recursive one, needs only two sample data at each calculation (xn+r and xr ) and provides a stationary phasor.

If the quantity x(t) undergoes a transient, the moving window detects the amplitude and angle variations with a delay which depends on the time sample rate τs. If the system frequency fn undergoes a variation Δf , at the