# Bi-level optimization based two-stage market clearing model considering guaranteed accommodation of renewable energy generation – Protection and Control of Modern Power Systems

Aug 10, 2022

### Mathematical model of participation in dual-market clearing

The dual-market clearing mechanism is for the traditional energy to participate in the unified market bidding and the subsidy market bidding, and needs to declare based on two price-electricity curves. The optimization of the market clearing model needs to consider an appropriate objective function and constraints. The objective function should consider the strategic nature of market bids for maximizing the profit of the market players. However, the clearing process also includes the objective of maximizing social welfare. The optimization process of market clearing will influence the next decision-making behavior of the market players. This master–slave structure with distinct, interlinked, and constrained upper and lower objectives can be described by the bi-level market equilibrium problem. The optimal decision process for market players in the upper level of the model relies on the locational marginal prices and winning electrical energy in the lower level of the model, while the market clearing process conducted in the lower level relies on the optimal price-electricity curve obtained in the upper level [25]. A schematic illustrating the functioning of the bi-level optimization model is presented in Fig. 5.

#### Upper level optimization process

The objective function F applied in the optimization model based on the profit maximization of each power generation enterprise obtains its optimal bidding strategy according to the locational marginal price (lambda_{t,n}) of bus (n) at time interval (t), and the winning electric energy (P_{i,t,k}^{G}) obtained from the lower level clearing model for the i-th generation unit in set G during the k-th bidding segment. Neglecting the start-up cost of wind power units, the specific optimization process in the upper level is given as:

$$begin{gathered} min F = sumlimits_{i = 1}^{M} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( { – lambda_{t,n} P_{i,t,k}^{G} } right) + hfill \ sumlimits_{i = M + 1}^{M + N} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( { – overline{beta } – lambda_{t,n} + lambda_{i,t}^{G} } right)P_{i,t,k}^{G} hfill \ end{gathered}$$

(1)

where N and M are the total numbers of conventional and renewable generation units, respectively. (T) is the total number of time intervals, (K) is the total number of bidding segments, (overline{beta }) is the final subsidy price established in the subsidy market, and (lambda_{i,t}^{G}) is the unit marginal cost of generation.

Reasonable locational marginal prices can facilitate optimal electricity system operation [26]. Therefore, the locational marginal price is defined in this paper as the average of the marginal electricity consumption benefit (F_{{}}^{{text{D}}}) on the electricity consumption side and the marginal generation cost (F_{{}}^{{text{G}}}) on the generation side per unit load increment, as:

$$F_{{}}^{{text{G}}} { = }sumlimits_{i}^{{}} {sumlimits_{t}^{{}} {sumlimits_{k}^{{}} {alpha_{i,k}^{{}} P_{i,t,k}^{{text{G}}} } } }$$

(2)

$$F_{{}}^{{text{D}}} { = }sumlimits_{t = 1}^{T} {} lambda_{t}^{D} P_{t}^{D}$$

(3)

where (alpha_{i,k}) is the unit bidding price, (lambda_{t}^{D}) is the marginal benefit on the customer side, and (P_{t}^{D}) is the demand for electricity on the customer side. Accordingly, (lambda_{t,n}) is defined as:

$$lambda_{t,n} = (partial F_{{}}^{{text{G}}} /partial P_{t,n}^{{text{D}}} + partial F_{{}}^{{text{D}}} /partial P_{t,n}^{{text{D}}} )/2$$

(4)

The constraints applied in the upper level of the optimization model are defined as follows:

(1) Bidding electric power constraints

$$sumlimits_{k}^{{}} {P_{i,k}^{{{text{Gmax}}}} } = P_{i}^{{{text{Gmax}}}}$$

(5)

$$varsigma P_{i}^{{{text{Gmax}}}} le P_{i,k}^{{{text{Gmax}}}}$$

(6)

Equation (5) prevents power generation enterprises from holding reserve electrical energy in an effort to increase the price. It does this by requiring that the sum of the declared maximum electrical energy of each generation unit (i.e., (P_{i,k}^{{{text{Gmax}}}})) over all bidding segments is equal to its upper bidding electrical energy limit (P_{i}^{{{text{Gmax}}}}). Equation (6) ensures that the declared maximum electrical energy of each generation unit must be greater than or equal to a specified proportion (varsigma) of its upper bidding electrical energy limit.

(2) Bidding price constraint

$$alpha_{{}}^{max } ge alpha_{i,k}^{{}} ge alpha_{i,k – 1}^{{}} { > 0,}forall {text{k}} ge 2$$

(7)

Equation (7) ensures that the unit price-electricity curve must be monotonically increasing, with (alpha_{{}}^{max }) representing the upper unit bidding price limit.

(3) Subsidy bidding price constraint

$$0 le beta_{i} le beta_{max }$$

(8)

Here, (beta_{i}) is the bidding price of unit (i) in the subsidy market and (beta_{max }) is the upper limit of the subsidy bidding price. This is based on the difference between the long-term average marginal costs of conventional and renewable energy generation units.

#### Lower level optimization process

First, the generation unit start and stop plan is obtained using the security constraint unit commitment procedure based on the power generation enterprise bidding strategy obtained from the upper level of the model. Then, the winning electrical energy and locational marginal price are calculated in the lower level process. As mentioned above, the market clearing process is conducted in the lower level, with the objective of maximizing social welfare. The specific optimization model is given as:

$$begin{gathered} min F = sumlimits_{i = 1}^{M} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( {alpha_{i,k} P_{i,t,k}^{G} } right) + hfill \ sumlimits_{i = M + 1}^{M + N} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( {alpha_{i,k} + beta_{i} } right)P_{i,t,k}^{G} – sumlimits_{t = 1}^{T} {} lambda_{t}^{D} P_{t}^{D} hfill \ end{gathered}$$

(9)

The constraints applied in the lower level process are given as follows:

1. 1)

System power flow balance constraints

$${mathbf{A}} times P^{G} – {mathbf{B}} times P^{D} – {mathbf{S}} times PL = 0$$

(10)

$$PL = {mathbf{X}}^{ – 1} {mathbf{S}}^{rm T} theta$$

(11)

Here, A is the bus-generator association matrix, B is the bus-load association matrix, S is the bus-branch association matrix, (PL) is the transmission line power flow, X is the branch reactance matrix, and θ is the bus voltage phase angle.

2. 2)

Line power flow constraint

$$- P_{n,m}^{max } le PL le P_{n,m}^{max } ,forall nforall m$$

(12)

Here, (P_{n,m}^{max }) is the upper limit of the line transmission power from bus (n) to bus (m).

3. 3)

Bus voltage phase angle constraints

$$- pi le theta_{n,t} le pi$$

(13)

$$theta_{n} = 0,n = 1,forall t$$

(14)

Here, expression (14) assigns the bus 1 to be the reference bus.

4. 4)

Unit output constraints

$$0 le P_{i,k}^{{text{G}}} le P_{i,k}^{{{text{Gmax}}}} { ,}forall i ,$$

(15)

$$u_{i,t} P_{i}^{{{text{Gmin}}}} le sumlimits_{k}^{{}} {P_{i,k}^{{text{G}}} } le u_{i,t} P_{i}^{{{text{Gmax}}}} { ,}forall i , forall t ,$$

(16)

Equation (15) ensures that the winning electrical energy of each unit during bidding is less than the declared electrical energy, while (16) applies a similar constraint to the sum of the winning electrical energies for each unit over the entire bidding process, and ensures that this sum resides somewhere between (P_{i}^{{{text{Gmax}}}}) and the minimum electrical energy (P_{i}^{{{text{Gmin}}}}), where (u_{i,t}) is the binary start-up (ui,t = 1) and shutdown (ui,t = 0) variable of each unit.

5. 5)

Ramping constraints

begin{aligned} &sumlimits_{k}^{{}} {P_{i,t,k}^{{text{G}}} } – sumlimits_{k}^{{}} {P_{i,t – 1,j}^{{text{G}}} } le u_{i,t – 1} R_{i}^{{text{U}}} + left( {u_{i,t} – u_{i,t – 1} } right)P_{i}^{{{text{Gmin}}}} hfill \ &quad+ left( {1 – u_{i,t} } right)P_{i}^{{{text{Gmax}}}} { ,}forall tforall i hfill \ end{aligned}

(17)

begin{aligned} sumlimits_{k}^{{}} {P_{i,t,k}^{{text{G}}} } – sumlimits_{k}^{{}} {P_{i,t – 1,j}^{{text{G}}} } le u_{i,t} R_{i}^{{text{D}}} – left( {u_{i,t} – u_{i,t – 1} } right)P_{i}^{{{text{Gmin}}}} hfill \ &quad+ left( {1 – u_{i,t – 1} } right)P_{i}^{{{text{Gmax}}}} { ,}forall tforall i hfill \ end{aligned}

(18)

Here, (R_{i}^{{text{U}}}) and (R_{i}^{{text{D}}}) are the respective upward and downward ramping rates of each unit.

6. 6)

Spinning reserve constraints

$$min left{ begin{gathered} sumlimits_{i}^{{}} {R_{i}^{U} } ,sumlimits_{i = 1}^{M} {left[ {(1 – e_{i} )P_{i}^{{{text{Gmax}}}} – sumlimits_{k}^{{}} {P_{i,k,t}^{{text{G}}} } } right]} + hfill \ sumlimits_{i = M}^{M + N} {(P_{i}^{{{text{Gmax}}}} } , – sumlimits_{k}^{{}} {P_{i,k,t}^{{text{G}}} )} hfill \ end{gathered} right} ge S_{t}^{{text{U}}}$$

(19)

$$min left{ begin{gathered} sumlimits_{i}^{{}} {R_{i}^{D} } ,sumlimits_{i = 1}^{M} {e_{i} P_{i}^{{{text{Gmax}}}} } + hfill \ sumlimits_{i = M}^{M + N} ( sumlimits_{k}^{{}} {P_{i,k,t}^{{text{G}}} – P_{i}^{{{text{Gmin}}}} )} hfill \ end{gathered} right} ge S_{t}^{D}$$

(20)

Here, (S_{t}^{{text{U}}}) and (S_{t}^{{text{D}}}) are the upward and downward spinning reserve requirements, respectively. The spinning reserve constraints ensure that the grid-connected electrical energy can meet demand loads in the event of fluctuations or failures in uncertain renewable energy generation [27].

7. 7)

Minimum start-up and shut-down time constraints

$$T_{i,t}^{U} – (u_{i,t – 1} – u_{i,t} )T_{U} ge 0$$

(21)

$$T_{i,t}^{D} – (u_{i,t} – u_{i,t – 1} )T_{D} ge 0$$

(22)

here (T_{i,t}^{U}) is the continuous start-up time of each unit and (T_{U}) is its minimum value, while (T_{i,t}^{D}) is the continuous shut-down time of each unit and (T_{D}) is its minimum value.

### Mathematical formulation of re-adjustment market clearing

The re-adjustment market clearing mechanism is also formulated as a bi-level market equilibrium problem. Again, the upper level selects the optimal bidding strategy for renewable energy generation enterprises that maximizes their profit according to the winning electrical energy and locational marginal price obtained in the lower level, while the lower level optimizes the social welfare of all market players based on customer benefits, power purchase costs, compensation costs, and the optimal bidding strategy obtained in the upper level. The upper level applies the same constraints in (6) and as those applied in the dual-market clearing mechanism, while applying a new constraint on the bidding electrical energy, adding a further constraint to the bidding price, and omitting constraint (8). The lower level applies the same constraints in (10) that are used in the dual-market clearing process, while omitting the start-up and shut-down constraints.

#### Upper level optimization process

The objective function F applied in the upper level optimization model is given as:

$$begin{gathered} min F = sumlimits_{i = 1}^{M} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( { – lambda_{t,n}^{{text{Re}}} P_{i,t,k}^{{G{text{Re}} }} } right) + sumlimits_{i = M + 1}^{M + N} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} hfill \ left{ {left( { – lambda_{t,n}^{{}} – overline{beta } + lambda_{i,t}^{G} } right)P_{i,t,k}^{{G{text{Re}} }} + (lambda_{t,n}^{{text{Re}}} – lambda_{t,n}^{{}} ) times (P_{i,t,k}^{G} – P_{i,t,k}^{{G{text{Re}} }} )} right} hfill \ end{gathered}$$

(23)

here all terms with the superscript ({text{Re}}) applied in the re-adjustment market clearing formulation refer to the same terms applied in the dual-market clearing formulation discussed above, and ((lambda_{t,n}^{{text{Re}}} – lambda_{t,n}^{{}} ) times (P_{i,t,k}^{G} – P_{i,t,k}^{{G{text{Re}} }} )) is the compensation applied to conventional energy generation providers for their transfer of electrical energy to renewable energy generation providers.

The new constraints applied in the upper level of the model are given as follows:

(1) Bidding electrical energy constraint

$$sumlimits_{k}^{{}} {P_{i,k}^{{{text{Gmax}}{text{Re}} }} } = P_{i}^{{{text{Gmax}}}} – sumlimits_{k}^{{}} {P_{i,k}^{{text{G}}} } ,i in left[ {1,M} right]$$

(24)

Constraint (24) replaces constraint (5) for renewable energy generation units, and ensures that the total maximum declared electrical energy is the unconsumed renewable energy generation after completion of the dual-market clearing process.

(2) Bidding price constraint

$$alpha_{i,k}^{{text{Re}}} le alpha_{i,k}^{{}}$$

(25)

This formulation augments constraint (7) to ensure that the re-adjustment bidding price (alpha_{i,k}^{{text{Re}}}) for renewable energy generation is less than its dual-market bidding price.

#### Lower level optimization process

The objective function F applied in the lower level optimization model is given as:

$$begin{gathered} min F = sumlimits_{i = 1}^{M} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( {alpha_{i,k}^{{text{Re}}} P_{i,t,k}^{{G{text{Re}} }} } right) + hfill \ sumlimits_{i = M + 1}^{M + N} {} sumlimits_{t = 1}^{T} {} sumlimits_{k = 1}^{K} {} left( {alpha_{i,k} + beta_{i} } right)P_{i,t,k}^{{G{text{Re}} }} – sumlimits_{t = 1}^{T} {} lambda_{t}^{D} P_{t}^{D} hfill \ end{gathered}$$

(26)