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Hydro Thermal Unit Commitment involving Demand Response resources: a MILP formulation

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Abstract

Hydro Thermal Unit Commitment (HTUC) is an issue of concern in current power systems due to various technical and economic concerns. HTUC is a problem with a large number of decision variables and nonlinear complex constraints which may not be readily solved. Therefore, in this paper, HTUC problem is formulated as a Mixed-Integer Linear Programming (MILP) which requires less computational effort and is solvable even for large-scale power systems. In this MILP problem, the models of both hydro and thermal units are considered with a closer look at practical limitations of actual operating conditions along with other common limitations. Besides, the presence of Demand Response (DR) has a significant impact on the market outcomes and commitment of generation units. Hence, this paper has incorporated the DR in a HTUC problem for the energy markets environment based on the concepts of price Responsive Demands (RDs) and Demand Response Service Providers (DRSPs) as one of the contributions. In order to evaluate the correctness and effectiveness of the proposed model, the method has been implemented on IEEE 118 bus system. The simulation studies show satisfactory results.

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Abbreviations

t :

Index of time interval

i :

Index of thermal unit

j :

Index of hydro unit

r :

Index of demand

k :

Index of DRSPs

T:

Number of periods of the scheduling

\(N_{T}\) :

Number of thermal unit

\(N_{H}\) :

Number of hydro unit

\(N_{D}\) :

Number of RDs

\(N_{DP}\) :

Number of DRSPs

\(b_{n} (i)\) :

Marginal cost of the nth block of energy offer by thermal unit i at periodt($/MWh)

\(P_{{}}^{\max } (i)\) :

Maximum power of unit i (MW)

\(P^{\min } (i)\) :

Minimum power of unit i (MW)

\(F(p_{n - 1}^{u} (i))\) :

Cost of generation of (n − 1)th upper limit in fuel cost curve of unit i ($/h)

\(p_{n - 1}^{u} (i)\) :

(n − 1)th upper limit in fuel cost curve of unit i (MW)

\(p_{n}^{d} (i)\) :

nth lower limit in fuel cost curve of unit i (MW)

\(e(i)f(i)\) :

Coefficients of valve loading effect cost curve of unit i

\(T_{{}}^{\min ,off} (i)\) :

Minimum down time of thermal unit i (h)

\(T_{{}}^{\min ,on} (i)\) :

Minimum down time of thermal unit i (h)

\(T_{{}}^{0,on} (i)\) :

Number of hours that the thermal unit at the beginning of the scheduling period has been on (h)

\(T_{{}}^{0,off} (i)\) :

Number of hours that the thermal unit at the beginning of the scheduling period has been off (h)

\(V_{0} (j)\) :

Minimum content of the reservoir of hydro unit j (Hm3)

\(P_{{_{{k_{h} }} }}^{\min } (j)\) :

Minimum power output of the performance curve \(k_{h}\) th of hydro unit j (MW)

\(P_{{}}^{\min } (j)\) :

Minimum power of hydro unit j (MW)

\(P_{{}}^{\max } (j)\) :

Maximum power of hydro unit j (MW)

\(b_{n}^{{k_{h} }} (j)\) :

Slope of power block n of the performance curve \(k_{h}\) th of hydro unit j (MW/m3/s)

\(Rain(j,t)\) :

Forecasted rainfall entry into the water reservoir of hydro unit j (Hm3/h)

\(De_{m,n}\) :

Time delay between reservoirs of hydro units m and n (h)

\(\eta\) :

Conversion factor, 0.0036 (Hm3s/m3h)

\(SU(j)\) :

Start-up cost of hydro unit j ($)

\(\lambda_{n} (r)\) :

Marginal benefit of the nth block of energy bid by demand r at period t ($/MWh)

\(D_{{}}^{\min } (r,t)\) :

Minimum consumption of demand r at period t (MW)

\(B^{\min } (r,t)\) :

Benefit of demand r at the point of \(D^{\min } (r,t)\)

\(\lambda_{{}}^{\max } (r,t)\) :

Maximum bid of demand r at period t ($/MWh)

\(D_{n}^{\max } (r,t)\) :

Maximum consumption at the nth block of benefit function of demand r at period t (MW)

\(am/bm(k)\) :

The customers cost function coefficients

\(a(k)\) :

Supply curve coefficients of kth DRSPs

\(X(b,c)\) :

The transmission line reactance located between the buses b and c

\(Load(t)\) :

Total system load at period t (MW)

\(PL_{{}}^{\min } (b,c)\) :

The minimum transfer power from the line between buses b and c (MW)

\(F(i,t)\) :

Fuel cost of thermal unit i at period t ($)

\(z_{n} (i,t)\) :

Generation at the nth block of fuel cost curve of thermal unit i at periodt(MW)

\(P(i,t)\) :

Generation of thermal unit i at period t (MW)

\(VLC(i,t)\) :

Valve loading effect cost of thermal unit i at period t ($)

\(\psi_{n} (i,t)\) :

Generation at the nth block of valve loading effect cost of thermal unit i at period t (MW)

\(V_{n} (j)\) :

Maximum volume of the reservoir of hydro unit j associated with nth variable head (Hm3)

\(V(j,t)\) :

Volume of the reservoir of hydro unit j at periodt(Hm3)

\(P(j,t)\) :

Generation of hydro unit j at periodt(MW)

\(Q_{n} (j,t)\) :

Water discharge of block n of hydro unit j at period t (m3/s)

\(Q(j,t)\) :

Water discharge of hydro unit j at period t (m3/s)

\(S(j,t)\) :

Spillage of the reservoir of hydro unit j at period t (m3/s)

\(SUC(j,t)\) :

Start-up cost of thermal unit i at period t ($)

\(B(r,t)\) :

Amount of benefit function for demand r at period t ($)

\(D_{n} (r,t)\) :

Consumption at the nth block of benefit function of demand r at period t (MW)

\(D(r,t)\) :

Consumption of demand r at period t (MW)

\(DR(k,t)\) :

Amount of sold DR by DRSP k at period t (MW)

\(DP(t)\) :

Demand response clearing price ($/MWh)

\(b_{dr} (k,t)\) :

Dynamic coefficients in supply curve of DRSP k at period t

\(PF_{dr} (k,t)\) :

Benefit of DRSP k at period t ($)

\(F_{dr} (k,t)\) :

Income of DRSP k at period t ($)

\(PL(b,c,t)\) :

Transmission power of the line between buses b and c at period t (MW)

\(\delta (b,t)\) :

Angle of bus b at period t (Rad)

\(F_{dr} (k,t)\) :

Income of DRSP k at period t ($)

\(\delta_{n} (i,t)\) :

If block n of fuel cost curve of thermal unit i at period t has been selected, this equals to 1

\(I(i,t)\) :

If thermal unit i at period t is online, this equals to 1

\(\chi_{n} (i,t)\) :

If power output of thermal unit i at period t has exceeded block n, this equals to 1

\(\beta_{n} (j,t)\) :

If variable head n + 1 of hydro unit j at period t has been selected, this equals to 1

\(I(j,t)\) :

If hydro unit j at period t is online, this equals to 1

\(\alpha_{n} (j,t)\) :

If volume of reservoir of hydro unit j at period t is greater than the nth block of performance curve, this equals to 1

\(I(r,t)\) :

If demand r at period t is scheduled, this equals to 1

\(I(k,t)\) :

If DRSP k at period t is scheduled, this equals to 1

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Authors and Affiliations

  1. Faculty of Electrical and Computer Engineering, Tarbiat Modares University, P.O Box 14115-194, Tehran, Iran

    Mahdi Nozarian, Hossein Seifi, Mohamad Kazem Sheikh-El-Eslami & Hamed Delkhosh

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  1. Mahdi Nozarian
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Appendix: Linearization of multiplication of two variables

Appendix: Linearization of multiplication of two variables

The multiplication of the clearing price and the amount of supply of DRSPs are shown by (40). This relationship is linearized using (41)–(47) [40]. Also, the multiplication of the amount of supply variable in the binary variable indicating the presence or absence of DRSPs is presented in (48). By using (48)–(51) instead of (48), the model would be in MILP format. In the following relationships \(x(s,k,t)\) is a binary variable and S and M are large positive values.

$$ F_{dr} (k,t) = DP(t)DR(k,t),\forall k,\forall t $$
(40)
$$ {(\underline{DR}} (k,t))^{2} + \sum\limits_{s = 1}^{S} x (s,k,t)v(k,t) \le DR(k,t) $$
(41)
$$ DR(k,t) \le {(\underline{DR}} (k,t))^{2} + v(k,t) + \sum\limits_{s = 1}^{S} x (s,k,t)v(k,t) $$
(42)
$$ x(s,k,t) \le x(s - 1,k,t),x(s,k,t) \in \left\{ {0,1} \right\} $$
(43)
$$ F_{dr} (k,t) = ((\underline{DR} (k,t))^{2} + \frac{v(k,t)}{2})DP(t) + \sum\limits_{s = 1}^{S} P (s,k,t) $$
(44)
$$ 0 \le v(k,t)DP(t) - P(s,k,t) \le v(k,t) {(\overline{DP}} (t))^{2} (1 - x(s,k,t)) $$
(45)
$$ 0 \le P(s,k,t) \le v(k,t) {(\overline{DP}} (t))^{2} x(s,k,t) $$
(46)
$$ v(k,t) = \frac{{\overline{DR} (k,t) - \underline{DR} (k,t)}}{S + 1} $$
(47)
$$ DRI(k,t) = DR(k,t)I(k,t),\forall k,\forall t $$
(48)
$$ DRI(k,t) \le I(k,t)*M $$
(49)
$$ DRI(k,t) \le DR(k,t) $$
(50)
$$ DRI(k,t) \ge DR(k,t) - (1 - I(k,t))*M $$
(51)

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Nozarian, M., Seifi, H., Sheikh-El-Eslami, M.K. et al. Hydro Thermal Unit Commitment involving Demand Response resources: a MILP formulation. Electr Eng 105, 175–192 (2023). https://doi.org/10.1007/s00202-022-01651-z

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