Stochastic Unit Commitment Formulation

The objective of the stochastic scheduling is to minimise the expected operation cost:

∑ 𝜋(𝑛) (∑ 𝐶_𝑔(𝑛) + ∆𝜏(𝑛)(𝑐^𝐿𝑆𝑃^𝐿𝑆(𝑛) + 𝑐^𝐹𝑆𝑃^𝐹𝑆(𝑛))

𝑔𝜖𝐺

)

𝑛∈𝑁

(2.18)

Subject constraints as following:

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1. System Constraints

The load balance constraint is formulated as below and applied to bus ib in node n:

∑ 𝑃_𝑔(𝑛)

𝑔∈𝐺_𝑖

+ ∑ 𝑃_𝑠(𝑛)

𝑠∈𝑆_𝑖

+ 𝑃_𝑖^𝑊𝑁(𝑛) − 𝑃_𝑖^𝑊𝐶(𝑛) + 𝑃_𝑖^𝐿𝑆(𝑛) = 𝑃_𝑖^𝐷(𝑛) (2.19) 2. Thermal Generator Constraints

The local constraints pertaining to thermal units are set out in this section. The shutdown and start-up decision variables, 𝑁_𝑔^𝑠𝑑 and 𝑁_𝑔^𝑠𝑡 , are nominally integer variables, while all other decision variables are continuous.

Some of the constraints at node 𝑛 refer to subsets of the ancestors of 𝑛. The subsets are defined as follows. If a generator in group 𝑔 starts generating at node 𝑛, then it must have been started up at a node in the set

𝐴_𝑔^𝑠𝑡(𝑛) = 𝐴(𝑛) ∩ {𝑛^′ ∈ 𝑁 ∪ 𝑃: 𝜏(𝑎(𝑛)) − 𝑇_𝑔^𝑠𝑡 < 𝜏(𝑛^′) ≤ 𝜏(𝑛) − 𝑇_𝑔^𝑠𝑡) (2.20) If a generator in group g is shut down at node n, it cannot have started generating at any node in the set

𝐴_𝑔^𝑚𝑢(𝑛) = 𝐴(𝑛) ∩ {𝑛^′ ∈ 𝑁 ∪ 𝑃: 𝜏(𝑛) − 𝑇_𝑔^𝑚𝑢 < 𝜏(𝑛^′) ≤ 𝜏(𝑛)} (2.21) If a generator in group g is started up at node n, it cannot have been shut down at any node in the set

𝐴_𝑔^𝑚𝑜(𝑛) = 𝐴(𝑛) ∩ {𝑛^′ ∈ 𝑁 ∪ 𝑃: 𝜏(𝑛) − 𝑇_𝑔^𝑚𝑜 < 𝜏(𝑛^′) ≤ 𝜏(𝑛)} (2.22) Total power output and operating costs in each group can be written as

𝑃_𝑔(𝑛) = 𝑃_𝑔^𝑚𝑠𝑔(𝑁_𝑔^𝑢𝑝(𝑛) − 𝑁_𝑔^{𝑖𝑑𝑙𝑒}(𝑛)) + 𝑃_𝑔^𝑥(𝑛) (2.23) 𝐶_𝑔(𝑛) = 𝐶_𝑔^𝑠𝑡𝑁_𝑔^𝑠𝑔(𝑛) + ∆𝜏(𝑛) (𝐶_𝑔^𝑛𝑙(𝑁_𝑔^𝑢𝑝(𝑛) − 𝑁_𝑔^{𝑖𝑑𝑙𝑒}(𝑛)) + 𝐶_𝑔^{𝑖𝑑𝑙𝑒}𝑁_𝑔^{𝑖𝑑𝑙𝑒}(𝑛) + 𝐶_𝑔^𝑚𝑃_𝑔(𝑛))

(2.24) Total output above MSG is limited by the number of generating units and the range of power output of each unit:

𝑃_𝑔^𝑥(𝑛) ≤ (𝑁_𝑔^𝑢𝑝(𝑛) − 𝑁_𝑔^{𝑖𝑑𝑙𝑒}(𝑛)) (𝑃_𝑔^𝑚𝑎𝑥− 𝑃_𝑔^𝑚𝑠𝑔) (2.25) The number of generators that start generating at node n is equal to the number of generators that was started up 𝑇_𝑔^𝑠𝑡 previousely:

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𝑁_𝑔^𝑠𝑔(𝑛) = ∑ 𝑁_𝑔^𝑠𝑡(𝑎)

𝑎∈𝐴_𝑔^𝑠𝑡(𝑛)

(2.26)

The number of generators that are generating at node 𝑛 is equal to the number of generators that were generating at node 𝑛^′𝑠 parent, plus the number that started generating at node 𝑛, less the number that are shut down at node n:

𝑁_𝑔^𝑢𝑝(𝑛) = 𝑁_𝑔^𝑢𝑝(𝑎(𝑛)) + 𝑁_𝑔^𝑠𝑔(𝑛) − 𝑁_𝑔^𝑠𝑑(𝑛) (2.27) The number of generators that are off at node 𝑛 is equal to the number of generators that were off at node 𝑛^′𝑠 parent, plus the number that are shut down at node 𝑛, less the number that are started up at node n:

𝑁_𝑔^𝑜𝑓𝑓(𝑛) = 𝑁_𝑔^𝑜𝑓𝑓(𝑎(𝑛)) + 𝑁_𝑔^𝑠𝑑(𝑛) − 𝑁_𝑔^𝑠𝑡(𝑛) (2.28) Total number of units which is allow to be shut down at node n is limited to the total number of units which were generating at node 𝑛^′𝑠 parent, less the number of units that have been generating for less than 𝑇_𝑔^𝑚𝑢 hours:

𝑁_𝑔^𝑠𝑑(𝑛) ≤ 𝑁_𝑔^𝑢𝑝(𝑎(𝑛)) − ∑ 𝑁_𝑔^𝑠𝑔(𝑎)

𝑎∈𝐴^𝑚𝑢_𝑔 (𝑛)

(2.29)

Total number of units which allow to be started up at node n is limited to the total number of units which were off at node 𝑛^′𝑠 parent, less the number of units that have been off for less than 𝑇_𝑔^𝑚𝑜hours:

𝑁_𝑔^𝑠𝑡(𝑛) ≤ 𝑁_𝑔^𝑜𝑓𝑓(𝑎(𝑛)) − ∑ 𝑁_𝑔^𝑠𝑑(𝑎)

𝑎∈𝐴_𝑔^𝑚𝑜(𝑛)

(2.30)

The number of units which is allowed to be in idle state is limited to the total number of units which are online at node n:

𝑁_𝑔^{𝑖𝑑𝑙𝑒}(𝑛) ≤ 𝑁_𝑔^𝑢𝑝(𝑛) (2.31) Ramp rate limits can be modelled as:

𝑃_𝑔^𝑥(𝑛) − 𝑃_𝑔^𝑥(𝑎(𝑛)) ≤ ∆𝜏(𝑎(𝑛))∆𝑃_𝑔^𝑟𝑢𝑁_𝑔^𝑢𝑝(𝑛) (2.32) 𝑃_𝑔^𝑥(𝑛) − 𝑃_𝑔^𝑥(𝑎(𝑛)) ≥ −∆𝜏(𝑎(𝑛))∆𝑃_𝑔^𝑟𝑑𝑁_𝑔^𝑢𝑝(𝑎(𝑛)) (2.33) As shown in Figure 2-2 , the amount of frequency response that each generator can deliver is limited by its maximum response capability and the slope 𝑓_𝑔^𝐹 that links the frequency response provision with the spinning headroom [30]:

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0 ≤ 𝑅_𝑔(𝑛) ≤ 𝑅_𝑔^𝑚𝑎𝑥 (2.34) 𝑅_𝑔(𝑛) ≤ 𝑓_𝑔^𝐹(𝑁_𝑔^𝑢𝑝(𝑛)𝑃_𝑔^𝑚𝑎𝑥 − 𝑃_𝑔(𝑛)) (2.35)

Figure 2-2 Example of response characteristic of conventional thermal plants.

3. Storage Unit Constraints:

The constraints for each storage unit at each node are formulated as below:

Energy constraints

𝐸_𝑠^𝑚𝑖𝑛 ≤ 𝐸_𝑠(𝑛) ≤ 𝐸_𝑠^𝑚𝑎𝑥 (2.36) Operation state constraint (pumping or generating)

𝑁_𝑠^𝐺𝑒𝑛(𝑛) ∈ {0,1} (2.37) Power output constraints

𝑃_𝑠(𝑛) = 𝑃_𝑠^𝑑(𝑛) − 𝑃_𝑠^𝑐(𝑛) (2.38) (1 − 𝑁_𝑠^𝐺𝑒𝑛(𝑛))𝑃_𝑠^{𝑐𝑚𝑖𝑛} ≤ 𝑃_𝑠^𝑐(𝑛) ≤ (1 − 𝑁_𝑠^𝐺𝑒𝑛(𝑛))𝑃_𝑠^{𝑐𝑚𝑎𝑥} (2.39) 𝑁_𝑠^𝐺𝑒𝑛(𝑛)𝑃_𝑠^{𝑑𝑚𝑖𝑛} ≤ 𝑃_𝑠^𝑑(𝑛) ≤ 𝑁_𝑠^𝐺𝑒𝑛(𝑛)𝑃_𝑠^{𝑑𝑚𝑎𝑥} (2.40) Energy balance constraint

𝐸_𝑠(𝑛) = 𝐸_𝑠(𝑎(𝑛)) + ∆𝜏(𝑛) (𝜂_𝑠^𝑐𝑃_𝑠^𝑐(𝑛) −𝑃_𝑠^𝑑(𝑛)

𝜂_𝑠^𝑑 ) (2.41) Frequency response provision constraints:

0 ≤ 𝑅_𝑠 ≤ 𝑅_𝑠^𝑚𝑎𝑥 (2.42) 𝑅_𝑠(𝑛) ≤ (𝑁_𝑠^𝐺𝑒𝑛(𝑛)𝑃_𝑠^𝑚𝑎𝑥 − 𝑃_𝑠(𝑛)) (2.43) 4. Modelling of Demand Side Response

Demand side response (DSR) model is developed by incorporating constraints regarding maximum energy shifted in or out in each time step and total amount of

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shifted energy in each day. Maximum energy shifted in or out in one time step can be defined as a proportion of the demand in that step or a proportion of the total demand in the day which that step belongs to. For DSR scheme, the total amount of shifted energy in each day should be zero. The proposed DSR model allows the user to choose a time during each day, when the total amount of shifted energy return to be zero.

A generic model for storage, DSR and combined heat and power (CHP) is developed as shown Figure 2-3. If the red circle and internal demand are ignored, this model can be used to describe the traditional storage. If the discharge route is ignored, this model can be used as CHP storage. If the red circle and discharge route are ignored, this model can be used to simulate flexible EV charging. (Note: Internal demand in the figure represents the original demand before shifting)

Figure 2-3 A generic model for storage, DSR and CHP 5. Risk Constraints:

Modern power systems are operated in a risk-averse fashion and system operators have different risk attitudes. Robust optimisation approach [46] [47] [48] utilises a user-defined uncertainty set to describe the uncertain elements and optimises the system operation against worst case situation. This approach provides robust solution which is feasible to all the realisations of uncertain elements. However, robust optimisation ignores the different possibilities for each realisation and tends to be conservative, since the worst case happens rarely. A combined stochastic and robust UC is proposed in [49], which allows users-specified weights on stochastic optimisation part and robust optimisation part. Chance constrained SUC is proposed in [50] [51] to enforce a low probability of load shedding. Conditional value-at-risk

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(CVaR) [52] has been widely implemented in finance sector to measure risk. It can be formulated as a linear constraint [24], making it more computationally attractive. In this thesis, a simple risk constraint is adopted and incorporated into the model. The risk constraint limits the probability of the load shedding when it is larger than 𝑃_𝑗^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡) below 𝑃𝑟𝑜𝑏_𝑗^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡) at hour 𝑡:

𝑃𝑟𝑜𝑏(𝑃^𝐿𝑆(𝑡) > 𝑃^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡)) ≤ 𝑃𝑟𝑜𝑏^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡) (2.44) The above risk constraint is implemented using the following MILP formulation:

𝑃𝑟𝑜𝑏(𝑃^𝐿𝑆(𝑡) > 𝑃^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡)) = ∑ 𝜋(𝑛) ∗

𝑛∈𝑁(𝑡)

𝑅𝑖(𝑛) ≤ 𝑃𝑟𝑜𝑏^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑡) (2.45) 𝑃^𝐿𝑆(𝑛) ≤ 𝑃^{𝐿𝑆𝑎𝑙𝑙𝑜𝑤𝑒𝑑}(𝑛) + 𝑅𝑖(𝑛) ∗ 𝑀 (2.46) where M is a constant number [53] and 𝑅𝑖(𝑛) is a binary variable.

2.3 Modelling of Inertia-dependent Frequency Regulation Requirements