A fully distributed asynchronous approach for multi-area coordinated network-constrained unit commitment

Abstract

This paper discusses a consensus-based alternating direction method of multipliers (ADMM) approach to solve the multi-area coordinated network-constrained unit commitment (NCUC) problem in a distributed manner. Due to political and technical difficulties, it is neither practical nor feasible to solve the multi-area coordination problem in a centralized fashion, which requires full access to all the data of individual areas. In comparison, in the proposed fully distributed approach, local NCUC problems of individual areas can be solved independently, and only limited information is exchanged among adjacent areas to facilitate the multi-area coordination. Furthermore, since traditional ADMM can guarantee convergence only for convex problems, this paper discusses several strategies to mitigate oscillations, enhance convergence performance, and derive good-enough feasible solutions, including: (1) a tie-line power-flow-based area coordination strategy is designed to reduce the number of global consensus variables; (2) different penalty parameters ρ are assigned to individual consensus variables and are updated via certain rules during the iterative procedure, which reduces the impact of the initial values of ρ on the convergence performance; (3) heuristic rules are adopted to fix certain unit commitment variables to avoid oscillations during the iterative procedure; and (4) an asynchronous distributed strategy is studied, which solves NCUC subproblems of small areas multiple times and exchanges information with adjacent areas more frequently within one complete run of slower NCUC subproblems of large areas. Numerical cases illustrate the effectiveness of the proposed asynchronous fully distributed NCUC approach, and we investigate key factors that would affect its convergence performance.

Appendix: Illustration of proposed distributed NCUC via a two-area four-bus system

In this “Appendix”, the two-area four-bus system shown in Fig. 10a is used to describe the detailed formulation of the NCUC problem (1) and the tie-line power flow based decomposition strategy proposed in Sect 2.2.

Fig. 10

Two-area four-bus system

The centralized formulation of the NCUC problem for the two-area four-bus system in Fig. 10a is presented below.

$${ \hbox{min} }\mathop \sum \limits_{t = 1}^{NT} \left\{ {\left[ {F_{1} \left( {P_{1,t} } \right) + C_{nl1} \cdot I_{1,t} + SU_{1,t} + SD_{1,t} } \right] + \left[ {F_{2} \left( {P_{2,t} } \right) + C_{nl2} \cdot I_{2,t} + SU_{2,t} + SD_{2,t} } \right]} \right\}$$

(A1)

$$s.t.\quad P_{1}^{min} \cdot I_{1,t} \le P_{1,t} \le P_{1}^{max} \cdot I_{1,t} ,\quad \forall t \in \varvec{T}$$

(A2)

$$\begin{aligned} & \mathop \sum \limits_{t = 1}^{{UT_{1} }} \left( {1 - I_{1,t} } \right) = 0,\quad where\;UT_{1} = max\left\{ {0,min\left[ {NT,\left( {T_{on,1} - ON_{1,0} } \right) \cdot I_{1,0} } \right]} \right\} \\ & \mathop \sum \limits_{\tau = t}^{{t + T_{on,1} - 1}} I_{1,\tau } \ge T_{on,1} \cdot \left( {I_{1,t} - I_{{1,\left( {t - 1} \right)}} } \right),\quad \forall t = UT_{1} + 1, \ldots ,NT - T_{on,1} + 1 \\ & \mathop \sum \limits_{\tau = t}^{NT} \left[ {I_{1,\tau } - \left( {I_{1,t} - I_{{1,\left( {t - 1} \right)}} } \right)} \right] \ge 0,\quad \forall t = NT - T_{on,1} + 2, \ldots ,NT \\ \end{aligned}$$

(A3)

$$\begin{aligned} & \mathop \sum \limits_{t = 1}^{{DT_{1} }} I_{1,t} = 0,\quad where\;DT_{1} = max\left\{ {0,min\left[ {NT,\left( {T_{off,1} - OFF_{1,0} } \right)} \right] \cdot \left( {1 - I_{1,0} } \right)} \right\} \\ & \mathop \sum \limits_{\tau = t}^{{t + T_{off,1} - 1}} \left( {1 - I_{1,\tau } } \right) \ge T_{off,1} \cdot \left( {I_{{1,\left( {t - 1} \right)}} - I_{1,t} } \right),\quad \forall t = DT_{1} + 1, \ldots ,NT - T_{off,1} + 1 \\ & \mathop \sum \limits_{\tau = t}^{NT} \left[ {1 - I_{1,\tau } - \left( {I_{{1,\left( {t - 1} \right)}} - I_{1,t} } \right)} \right] \ge 0,\quad \forall t = NT - T_{off,1} + 2, \ldots ,NT \\ \end{aligned}$$

(A4)

$$SU_{1,t} \ge C_{{su_{1} }} \cdot \left( {I_{1,t} - I_{{1,\left( {t - 1} \right)}} } \right);\quad SD_{1,t} \ge C_{{sd_{1} }} \cdot \left( {I_{{1,\left( {t - 1} \right)}} - I_{1,t} } \right)\quad \forall t \in \varvec{T}$$

(A5)

$$P_{1,t} - P_{{1,\left( {t - 1} \right)}} \le UR_{1} \cdot I_{{1,\left( {t - 1} \right)}} + P_{1}^{min} \cdot \left( {I_{1,t} - I_{{1,\left( {t - 1} \right)}} } \right) + P_{1}^{max} \cdot \left( {1 - I_{1,t} } \right),\quad \forall t \in \varvec{T}$$

(A6)

$$P_{{1,\left( {t - 1} \right)}} - P_{1,t} \le DR_{1} \cdot I_{1,t} + P_{1}^{min} \cdot \left( {I_{{1,\left( {t - 1} \right)}} - I_{1,t} } \right) + P_{1}^{max} \cdot \left( {1 - I_{{1,\left( {t - 1} \right)}} } \right),\quad \forall t \in \varvec{T}$$

(A7)

$$P_{1,t} - PL_{12,t} = 0;\quad \theta_{1,t} = 0,\quad \forall t \in \varvec{T}$$

(A8)

$$PL_{12,t} = \left( {\theta_{1,t} - \theta_{2,t} } \right)/X_{12} ;\quad PL_{12}^{min} \le PL_{12,t} \le PL_{12}^{max} ;\quad \forall t \in \varvec{T}$$

(A9)

$$PL_{12,t} - PL_{23,t} = P_{d1,t} ,\quad \forall t \in \varvec{T}$$

(A10)

$$P_{2}^{min} \cdot I_{2,t} \le P_{2,t} \le P_{2}^{max} \cdot I_{2,t} ;\quad \forall t \in \varvec{T}$$

(A11)

$$\begin{aligned} & \mathop \sum \limits_{t = 1}^{{UT_{2} }} \left( {1 - I_{2,t} } \right) = 0,\quad where\;UT_{2} = max\left\{ {0,min\left[ {NT,\left( {T_{on,2} - ON_{2,0} } \right) \cdot I_{2,0} } \right]} \right\} \\ & \mathop \sum \limits_{\tau = t}^{{t + T_{on,2} - 1}} I_{2,\tau } \ge T_{on,2} \cdot \left( {I_{2,t} - I_{{2,\left( {t - 1} \right)}} } \right),\quad \forall t = UT_{2} + 1, \ldots ,NT - T_{on,2} + 1 \\ & \mathop \sum \limits_{\tau = t}^{NT} \left[ {I_{2,\tau } - \left( {I_{2,t} - I_{{2,\left( {t - 1} \right)}} } \right)} \right] \ge 0,\quad \forall t = NT - T_{on,2} + 2, \ldots ,NT \\ \end{aligned}$$

(A12)

$$\begin{aligned} & \mathop \sum \limits_{t = 1}^{{DT_{2} }} I_{2,t} = 0,\quad where\;DT_{2} = max\left\{ {0,min\left[ {NT,\left( {T_{off,2} - OFF_{2,0} } \right)} \right] \cdot \left( {1 - I_{2,0} } \right)} \right\} \\ & \mathop \sum \limits_{\tau = t}^{{t + T_{off,2} - 1}} \left( {1 - I_{2,\tau } } \right) \ge T_{off,2} \cdot \left( {I_{{2,\left( {t - 1} \right)}} - I_{2,t} } \right),\quad \forall t = DT_{2} + 1, \ldots ,NT - T_{off,2} + 1 \\ & \mathop \sum \limits_{\tau = t}^{NT} \left[ {1 - I_{2,\tau } - \left( {I_{{2,\left( {t - 1} \right)}} - I_{2,t} } \right)} \right] \ge 0,\quad \forall t = NT - T_{off,2} + 2, \ldots ,NT \\ \end{aligned}$$

(A13)

$$SU_{2,t} \ge C_{{su_{2} }} \cdot \left( {I_{2,t} - I_{{2,\left( {t - 1} \right)}} } \right);\quad SD_{2,t} \ge C_{{sd_{2} }} \cdot \left( {I_{{2,\left( {t - 1} \right)}} - I_{2,t} } \right),\quad \forall t \in \varvec{T}$$

(A14)

$$P_{2,t} - P_{{2,\left( {t - 1} \right)}} \le UR_{2} \cdot I_{{2,\left( {t - 1} \right)}} + P_{2}^{min} \cdot \left( {I_{2,t} - I_{{2,\left( {t - 1} \right)}} } \right) + P_{2}^{max} \cdot \left( {1 - I_{2,t} } \right),\quad \forall t \in \varvec{T}$$

(A15)

$$P_{{2,\left( {t - 1} \right)}} - P_{2,t} \le DR_{2} \cdot I_{2,t} + P_{2}^{min} \cdot \left( {I_{{2,\left( {t - 1} \right)}} - I_{2,t} } \right) + P_{2}^{max} \cdot \left( {1 - I_{{2,\left( {t - 1} \right)}} } \right),\quad \forall t \in \varvec{T}$$

(A16)

$$P_{2,t} + PL_{34,t} = 0;\quad PL_{34,t} = \left( {\theta_{3,t} - \theta_{4,t} } \right)/X_{34} ;\quad PL_{34}^{min} \le PL_{34,t} \le PL_{34}^{max} ,\quad \forall t \in \varvec{T}$$

(A17)

$$PL_{23,t} - PL_{34,t} = P_{d2,t} ;\quad PL_{23,t} = \left( {\theta_{2,t} - \theta_{3,t} } \right)/X_{23} ;\quad PL_{23}^{min} \le PL_{23,t} \le PL_{23}^{max} ,\quad \forall t \in \varvec{T}$$

(A18)

Objective function (A1) minimizes the total operation cost of the entire system, which is composed of two areas S₁ and S₂. Constraints (A2)–(A10) correspond to the operation and security requirements of area S₁. Specifically, constraints (A2)–(A7) are local operation constraints for the generator in S₁ and constraints (A8)–(A9) are local network security constraints, both of which include only local variables in area S₁. On the other hand, constraint (A10) is a coupling constraint which includes the tie-line power flow variable PL_23,t. Similarly, for area S₂, constraints (A11)–(A16) are local generator operation constraints, (A17) are local network security constraints, and (A18) are coupling constraints which include the tie-line power flow variable PL_23,t.

As shown in Fig. 10b, the original system can be decomposed into two independent subsystems after duplicating bus 2/3 as bus 2′/3′ in the adjacent subsystem S₂/S₁. The NCUC problems for the two subsystems are as follows, where $\widetilde{PL}_{{23^{\prime},t}}$ and $\widetilde{PL}_{{2^{\prime}3,t}}$ are respectively the power flow variables of the same tie-line at time t in S₁ and S₂ (i.e., $\widetilde{PL}_{{23^{\prime},t}} = \widetilde{PL}_{{2^{\prime}3,t}}$).

NCUC problem for S1	NCUC problem for S2
$\hbox{min} \mathop \sum \limits_{t = 1}^{T} \left( {F_{1} \left( {P_{1,t} } \right) + C_{nl1} \cdot I_{1,t} + SU_{1,t} + SD_{1,t} } \right)$	${ \hbox{min} }\mathop \sum \limits_{t = 1}^{T} \left( {F_{2} \left( {P_{2,t} } \right) + C_{nl2} \cdot I_{2,t} + SU_{2,t} + SD_{2,t} } \right)$
s.t. Constraints (A1)–(A12)	s.t. Constraints (A14)–(A23)
$PL_{12,t} - \widetilde{PL}_{{23^{\prime},t}} = P_{d1,t},\quad \forall t \in \varvec{T}$	$\widetilde{PL}_{{2^{\prime}3,t}} - PL_{34,t} = P_{d2,t},\quad \forall t \in \varvec{T}$
$\widetilde{PL}_{{23^{\prime},t}} = \frac{{\left({\theta_{2,t} - \theta_{{3^{\prime},t}}} \right)}}{{X_{23}}},\quad \forall t \in \varvec{T}$	$\widetilde{PL}_{{2^{\prime}3,t}} = \frac{{\left({\theta_{{2^{\prime},t}} - \theta_{3,t}} \right)}}{{X_{23}}},\quad \forall t \in \varvec{T}$
$PL_{23}^{min} \le \widetilde{PL}_{{23^{\prime},t}} \le PL_{23}^{max},\quad \forall t \in \varvec{T}$	$PL_{23}^{min} \le \widetilde{PL}_{{2^{\prime}3,t}} \le PL_{23}^{max},\quad \forall t \in \varvec{T}$
$\widetilde{PL}_{{23^{\prime},t}} = Z_{1,t},\quad \forall t \in \varvec{T}$	$\widetilde{PL}_{{2^{\prime}3,t}} = Z_{1,t},\quad \forall t \in \varvec{T}$

Indexing $\widetilde{PL}_{{23^{\prime},t}}$ as the second power flow variable in S₁ at time t (i.e., $\widetilde{\varvec{PL}}_{{S_{1},t}} \left(2 \right) = \widetilde{PL}_{{23^{\prime},t}}$) and $\widetilde{PL}_{{2^{\prime}3,t}}$ as the second power flow variable in S₂ at time t (i.e., $\widetilde{\varvec{PL}}_{{S_{2},t}} \left(2 \right) = \widetilde{PL}_{{2^{\prime}3,t}}$), the mapping G _t (n,w) = g between the global variables Z_g,t and the duplicated local variables $PL_{n,t} \left( w \right)$ defined in Section II.B can be represented as G _t (S₁,2) = 1 and G _t (S₂,2) = 1 for each time $t \in \varvec{T}$. To match the consensus-based ADMM from (7), the coupling constraints $\widetilde{PL}_{{23^{\prime},t}} = Z_{1,t}$ in NCUC problem 1 and $\widetilde{PL}_{{2^{\prime}3,t}} = Z_{1,t}$ in NCUC problem 2 are relaxed into the corresponding objective functions, as shown in the following subproblems 1 and 2.

Subproblem 1	Subproblem 2
$\hbox{min} \mathop \sum \limits_{t = 1}^{T} \left[ {F_{1} \left( {P_{1,t} } \right) + C_{nl1} \cdot I_{1,t} + SU_{1,t} + SD_{1,t} + \lambda_{1t} \left( {PL_{{23^{\prime},t}} - Z_{1t} } \right) + \rho \left( {PL_{{23^{\prime},t}} - Z_{1t} } \right)^{2} } \right]$	$\hbox{min} \mathop \sum \limits_{t = 1}^{T} \left[ {F_{2} \left( {P_{2,t} } \right) + C_{nl2} \cdot I_{2,t} + SU_{2,t} + SD_{2,t} + \lambda_{2t} \left( {PL_{{2^{\prime}3,t}} - Z_{1t} } \right) + \rho \left( {PL_{{2^{\prime}3,t}} - Z_{1t} } \right)^{2} } \right]$
s.t. Constraints (A1)–(A12)	s.t. Constraints (A14)–(A23)
$PL_{12,t} - \widetilde{PL}_{{23^{\prime},t}} = P_{d1,t},\quad \forall t \in \varvec{T}$	$\widetilde{PL}_{{2^{\prime}3,t}} - PL_{34,t} = P_{d2,t},\quad \forall t \in \varvec{T}$
$\widetilde{PL}_{{23^{\prime},t}} = \frac{{\left({\theta_{2,t} - \theta_{{3^{\prime},t}}} \right)}}{{X_{23}}},\quad \forall t \in \varvec{T}$	$\widetilde{PL}_{{2^{\prime}3,t}} = \frac{{\left({\theta_{{2^{\prime},t}} - \theta_{3,t}} \right)}}{{X_{23}}},\quad \forall t \in \varvec{T}$
$PL_{23}^{min} \le \widetilde{PL}_{{23^{\prime},t}} \le PL_{23}^{max},\quad \forall t \in \varvec{T}$	$PL_{23}^{min} \le \widetilde{PL}_{{2^{\prime}3,t}} \le PL_{23}^{max},\quad \forall t \in \varvec{T}$

Finally, the consensus-based ADMM procedure (i.e., Algorithm 1 in Sect. 2.2) for this two-area NCUC problem can be expressed as follows:

• 1. We initialize λ_1,t, λ_2,t and Z_1,t.

• 2. S₁ solves subproblem 1 and S₂ solves subproblem 2 in parallel.

• 3. S₁ sends its $\widetilde{PL}_{{23^{\prime},t}}$ solution to S₂, and S₂ sends its $\widetilde{PL}_{{2^{\prime}3,t}}$ solution to S₁.

• 4. Both areas S₁ and S₂ update Z_1,t via equation $Z_{1,t} = \frac{{\left({\widetilde{PL}_{{23^{\prime},t}} + \widetilde{PL}_{{2^{\prime}3,t}}} \right)}}{2},\quad \forall t \in \varvec{T}$.

• 5. S₁ updates λ_1,t and S₂ updates λ_2,t via:

  $$\begin{aligned} \lambda_{1,t} =& \lambda_{1,t} + \rho \cdot \left({\widetilde{PL}_{{23^{\prime},t}} - Z_{1,t}} \right),\quad \forall t \in \varvec{T} \hfill \\ \lambda_{2,t} =& \lambda_{2,t} + \rho \cdot \left({\widetilde{PL}_{{2^{\prime}3,t}} - Z_{1,t}} \right),\quad \forall t \in \varvec{T} \hfill \\ \end{aligned}$$

• 6. If the stopping criterion (7e) is satisfied, we terminate; otherwise, we return to Step (2).